\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 64, pp. 1--22.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/64\hfil Weighted eigenvalue problems] {Weighted eigenvalue problems for the $p$-Laplacian with weights in weak Lebesgue spaces} \author[T. V. Anoop\hfil EJDE-2011/64\hfilneg] {T. V. Anoop} % in alphabetical order \address{T. V. Anoop \newline The Institute of Mathematical Sciences \\ Chennai 600113, India} \email{tvanoop@imsc.res.in} \thanks{Submitted November 11, 2011. Published May 17, 2011.} \subjclass[2000]{35J92, 35P30, 35A15} \keywords{Lorentz spaces; principal eigenvalue; radial symmetry; \hfill\break\indent Ljusternik-Schnirelmann theory} \begin{abstract} We consider the nonlinear eigenvalue problem $-\Delta_p u= \lambda g |u|^{p-2}u,\quad u\in \mathcal{D}^{1,p}_0(\Omega)$ where $\Delta_p$ is the p-Laplacian operator, $\Omega$ is a connected domain in $\mathbb{R}^N$ with $N>p$ and the weight function $g$ is locally integrable. We obtain the existence of a unique positive principal eigenvalue for $g$ such that $g^+$ lies in certain subspace of weak-$L^{N/p}(\Omega)$. The radial symmetry of the first eigenfunctions are obtained for radial $g$, when $\Omega$ is a ball centered at the origin or $\mathbb{R}^N$. The existence of an infinite set of eigenvalues is proved using the Ljusternik-Schnirelmann theory on $\mathcal{C}^1$ manifolds. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Porposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} For given $N\geq 2$, $11$, when $g\equiv 1$ and the domain $\Omega$ bounded. Later, Azorero and Alonso \cite{Azorero} identified infinitely many eigenvalues of \eqref{eq1}, for $p\not =2$, using the Ljusternik-Schnirelmann type minmax theorem. Many authors have given sufficient conditions on $g$ for the existence of a positive principal eigenvalue for \eqref{eq1}, when $\Omega=\mathbb{R}^N$, for example Brown et. al. \cite{BroCosFle} and Allegretto \cite{All1} for $p=2$, Huang \cite{Huang}, Allegretto and Huang \cite{AllHua1} for the respective generalization to $p\neq 2$. Fleckinger et al. \cite{FleGosThe}, studied the problem \eqref{eq1} for general $p$. All these earlier results assume that either $g$ or $g^+$ should be in $L^{N/p}(\mathbb{R}^N)$. In \cite{SWi}, Willem and Szulkin enlarged the class of weight functions beyond the Lebesgue space $L^{N/p}(\mathbb{R}^N)$. They obtained the existence of positive principal eigenvalue, even for the weights whose positive part has a faster decay than $1/|x|^p$ at infinity and at all the points in the domain (see \eqref{eq:SzulinWillem}). For $p=2$, there are some results available for the weights in Lorentz spaces, for example, Visciglia in \cite{Visciglia} looked at \eqref{eq1} in the context of generalized Hardy-Sobolev inequality for the positive weights in certain Lorentz spaces. Following this direction, Mythily and Marcello in \cite{ML-MR} showed the existence of a unique positive principal eigenvalue for \eqref{eq1}, when $g$ is in certain Lorentz spaces. Anoop, Lucia and Ramaswamy \cite{ALM} unified the sufficient conditions given in \cite{All1,BroCosFle,ML-MR,SWi} by showing the existence of a positive principal eigenvalue for \eqref{eq1}, when $g^+$ lies in a suitable subspace of weak-$L^{\frac{N}{2}}(\Omega)$. In this paper we obtain an analogous result that unify the sufficient conditions given in \cite{AllHua1,Huang,FleGosThe,SWi} for the existence of a positive eigenvalue for \eqref{eq1} by considering weights in a suitable subspace of the weak- $L^{N/p}(\Omega)$. For $p=2$, the existence of a positive principal eigenvalue for more general positive weights is obtained in \cite{Rozen} using certain capacity conditions of Maz\textquoteright ja \cite{Mazja} and in \cite{Te} using the concentration compactness lemma. However, their eigenfunctions are only a distributional solutions of \eqref{sol1} and the first eigenvalue lacks certain qualitative properties. Indeed, here we obtain a unique positive principal eigenvalue and an infinite set of eigenvalues for \eqref{eq1} for the weights in a suitable subspace of the Lorentz space $L({\frac{N}{p}},\infty)$. Here we fix the solution space as $\mathcal{D}^{1,p}_0(\Omega)$, which fits very well with the weak formulation of boundary value problems in the unbounded domains. Furthermore, when $1 0 \}. Let \begin{gather} M:=\{ u \, \in \mathcal{D}^{1,p}_0(\Omega): \int_\Omega g |u|^p =1 \}, \\ J(u) := \frac{1}{p} \int_{\Omega}|\nabla u|^p \end{gather} If$R$is$\mathcal{C}^1$, then we arrive at \eqref{eq1} as the Euler-Lagrange equation corresponding to the critical points of$R$on$\mathcal{D}^+(g)$, with the critical values as the eigenvalues of \eqref{eq1}. Moreover, there is a one to one correspondence between the critical points of$R$over$\mathcal{D}^+(g)$and the critical points of$J$over$M$. Thus we look for the sufficient conditions on$g^+$for the existence of a critical points of$J$on$M$. As in \cite{ALM}, here we consider the space $\mathcal{F}_{N/p}:= \text{closure of \mathcal{C}_c^{\infty}(\Omega) in L(N/p,\infty)}$ Now we state one of our main results. \begin{theorem}\label{Exist} Let$\Omega$be an open connected subset of$\mathbb{R}^N$with$p\in(1,N)$. Let$g\in L^1_{\rm loc}(\Omega)$be such that$g^+\in \mathcal{F}_{N/p}\setminus\{0\}$. Then $$\lambda_1=\inf\{J(u):u\in M \}$$ is the unique positive principal eigenvalue of \eqref{eq1}. Furthermore, all the eigenfunctions corresponding to$\lambda_1$are of the constant sign and$\lambda_1$is simple. \end{theorem} Note that$g^-$is only locally integrable and hence the map$G$defined as $$G(u)=\int_{\Omega}g |u|^p$$ may not even be continuous and hence$M$may not even be closed in$\mathcal{D}^{1,p}_0(\Omega)$. Nevertheless, we show that the weak limit of a minimizing sequence of$J$on$M$lies in$M$. In general the eigenfunctions are only in$W^{1,p}_{\rm loc}(\Omega)$and hence the classical tools for proving the qualitative properties of$\lambda_1$are not applicable, as they require more regularity for the eigenfunctions. However, Kawohl, Lucia and Prashanth \cite{KPL} developed a weaker version of strong maximum principle for quasilinear operator analogous to the result in \cite{BrezisPonce}. Further, we discuss the sufficient conditions on$g$for the radial symmetry of the eigenfunctions corresponding$\lambda_1$, when$\Omega$is a ball centered at origin or$\mathbb{R}^N$. This generalizes the result of Bhattacharya \cite{Bhattacharya}, who proved the radial symmetry of the first eigenfunctions of \eqref{eq1}, when$\Omega$is a ball centered at origin and$g\equiv 1$. \begin{theorem}\label{radial} Let$\Omega$be a ball centered at origin or$\mathbb{R}^N$. Let$g$be nonnegative, radial and radially decreasing measurable function. If$\lambda_1$is an eigenvalue of \eqref{eq1}, then any positive eigenfunction corresponding to$\lambda_1$is radial and radially decreasing. \end{theorem} A sufficient condition on$g$, for the existence of infinitely many eigenvalues of \eqref{eq1} is also discussed here. Let us point out that a complete description of the set of all eigenvalues of$p$-Laplacian is widely open for$p\neq 2$. The question of discreteness, countability of the set of all eigenvalues of$p$-Laplacian is not known, even in the simplest case:$g\equiv 1$and$\Omega$is a ball. However there are several methods that exhibit infinite number of eigenvalues goes to infinity. For$p\neq 2$, the existence of infinitely many eigenvalues is obtained in \cite{AllHua1,Huang, SWi}, using the Ljusternik-Schnirelmann minimax theorem. In this direction we have the following result under certain weaker assumptions on$g^+$. \begin{theorem}\label{infinite} Let$\Omega$be an open connected subset of$\mathbb{R}^N$with$p\in(1,N)$. Let$g\in L^1_{\rm loc}(\Omega)$be such that$g^+\in \mathcal{F}_{N/p}\setminus \{0 \}$. Then \eqref{eq1} admits a sequence of positive eigenvalues going to$\infty$. \end{theorem} The classical Ljusternik-Schnirelmann minimax theorem requires a deformation homotopy that is available when$M$is at least a$\mathcal{C}^{1,1}$manifold(i.e, transition maps are$\mathcal{C}^1$and its derivative is locally Lipschitz). The set$M$that we are considering here is$\mathcal{C}^1$but generally not$\mathcal{C}^{1,1}$. Szulkin \cite{Szulkin} developed the Ljusternik-Schnirelmann theorem on$\mathcal{C}^1$manifold using the Ekeland variational principle. We use Szulkin's result to obtain an increasing sequence of positive eigenvalues of \eqref{eq1} that going to infinity. This paper is organized as follows. In Section 2, we recall certain basic properties of the symmetric rearrangement of a function and the Lorentz spaces. Section 3 deals with several characterizations of the spaces$\mathcal{F}_d, d>1$. The examples of functions belonging to$\mathcal{F}_{N/p}$are also given in Section 3. In Section 4, we present a proof of the existence and other qualitative properties of the first eigenvalue like, simplicity, uniqueness. The radial symmetry of the eigenfunctions corresponding to$\lambda_1$is discussed in Section 4. In section 5, we discuss the Ljusternik-Schirelmann theory on$\mathcal{C}^1$Banach manifold and give a proof for the existence of infinitely many eigenvalues of \eqref{eq1}. Further extensions and the applications of weighted eigenvalue problems for the$p$-Laplacian are indicated in Section 6. \section{Prerequisites} \subsection{Symmetrization} First, we recall the definition of the symmetrization of a function and its properties. Then we state certain rearrangement inequalities needed for the subsequent sections, for more details on symmetrization we refer to \cite{Lieb,kesh1,EdEv}. Let$\Omega$be a domain in$\mathbb{R}^N$. Given a measurable function$f$on$\Omega$, we define distribution function$\alpha_f$and decreasing rearrangement$f^*$of$f$as below $$\alpha_f(s) : = \big\vert \{x \in \Omega : |f(x)| >s\} \big\vert, \quad f^*(t) : =\inf \{s>0 : \alpha_f(s)\leq t \}. \label{2.1}$$ In the following proposition we summarize some useful properties of distribution and rearrangements. \begin{proposition}\label{properties-f*} Let$\Omega$be a domain and$f$be a measurable function on$\Omega$. Then \begin{itemize} \item[(i)]$\alpha_f, f^*$are nonnegative, decreasing and right continuous. \item[(ii)]$f^* (\alpha_f(s_0))\leq s_0$,$\alpha_f (f^*(t_0))\leq t_0$; \item[(iii)]$f^*(t)\leq s$if and only if$\alpha_f(s)\leq t$, \item[(iv)]$f$and$f^*$are equimeasurable; i.e,$\alpha_f(s)=\alpha_{f^*}(s)$for all$s>0$. \item[(v)] Let$c,s,t>0$such that$c=s t^{1/p}$. Then $$\label{relation1} t^{1/p}f^*(t) \leq c \quad \text{if and only if} \quad s(\alpha_f(s))^{1/p} \leq c.$$ \end{itemize} \end{proposition} \begin{proof} For a proof of (i), (ii) and (iii), see \cite[Propositions 3.2.2 and 3.2.3]{EdEv}. Item (iv) follows from (iii) as follows $\alpha_{f^*}(s)= |\{t:f^*(t)> s\}|= |\{t:t<\alpha_f(s)\}|=\alpha_f(s).$ (v) Taking$s=c t^{\frac{-1}{p}}$in (iii) one deduces that $t^{1/p}f^*(t) \leq c \quad \text{if and only if} \quad \alpha_f(s) \leq t .$ Now as$t=(c/s)^p$, we obtain $\alpha_f(s) \leq t \quad \text{if and only if} \quad s (\alpha_f(s))^{1/p}\leq c.$ \end{proof} Next we define Schwarz symmetrization of measurable sets and functions, see \cite{Lieb} for more details. \begin{definition} \label{def2.2} \rm Let$A\subset \mathbb{R}^N$be a Borel measurable set of finite measure. We define$A_*$, the symmetric rearrangement of the set$A$, to be the open ball centered at origin having the same measure that of$A$. Thus $A_*=\{x:|x|0. Then we define the \emph{symmetric decreasing rearrangement} f_* of f on \Omega_* as \[ f_*(x)= \int_0^\infty \chi_{\{{|f|>s\}}_*}(x)ds$ Next we list a few inequalities concerning$f_*$that we use for proving the radial symmetry of the eigenfunctions corresponding to the first eigenvalue. For a proof see \cite[Section 3.3]{Lieb}. \begin{proposition}\label{symmetric} Let$\Omega$be a ball centered at origin or$\mathbb{R}^N$. Let$f$be a nonnegative measurable function on$\Omega$such that$\alpha_f(s) <\infty$for each$s>0$. \begin{itemize} \item[(a)] If$f$is radial and radially decreasing then$f=f_*$a.e. \item[(a)] Let$F: \mathbb{R}^+ \to \mathbb{R}$be a nonnegative Borel measurable function. Then $$\int_{\mathbb{R}^N} F(f_*(x))dx=\int_{\mathbb{R}^N} F(f(x))dx.$$ \item[(b)] If$\Phi:\mathbb{R}^+ \to \mathbb{R}$is nonnegative and nondecreasing then $$(\Phi\circ f)_*=\Phi \circ f_* \quad \text{a.e.}$$ \end{itemize} \end{proposition} \subsection{Lorentz Spaces} In this section, we recall the definition and the main properties of the Lorentz spaces. For more details on Lorentz spaces see \cite{Adams,EdEv,RH}. Given a measurable function$f$and$p,q \in [1,\infty]$, we set $$\|f\|_{(p,q)} := \|t^{\frac{1}{p}-\frac{1}{q}} f^* (t) \|_{q;(0,\infty)}$$ and the Lorentz spaces are defined by$ L(p,q) := \{ f : \|f\|_{(p,q)}<\infty \}$. In particular for$q=\infty$, we obtain $$\|f\|_{(p,\infty)}=\sup_{t>0}t^{1/p}f^*(t).$$ For$p> 1$, the weak-$L^p$space is defined as $\text{weak-}L^p:=\{ f:\sup_{s>0}s(\alpha_f(s))^{1/p}<\infty \}.$ The following lemma identifies the Lorentz space$L(p,\infty)$with the weak-$L^p$space. \begin{lemma} \label{equivsup} Let$\Omega$be a domain in$\mathbb{R}^N$and$f$be a measurable function on$\Omega$. For each$ p>1$, we have $$\sup_{t>0}t^{1/p}f^*(t)=\sup_{s>0}s(\alpha_f(s))^{1/p}.$$ \end{lemma} \begin{proof} Let $$\label{relation0} c_1=\sup_{t>0}t^{1/p}f^*(t),\quad c_2=\sup_{s>0}s(\alpha_f(s))^{1/p}.$$ Without loss of generality we may assume that$c_1$is finite. Now for$s>0$, take${t}=(\frac{c_1}{{s}})^p$. Thus${t}^{1/p}f^*({t})\leq c_1$. Now by taking$c=c_1$in \eqref{relation1}, with$c_1= s t^\frac{1}{p}$, one can deduce that${s} (\alpha_f({s}))^{1/p}\leq c_1$, for all$s>0$. Hence$c_2\leq c_1$. The other way inequality follows in a similar way. \end{proof} The functional$\| \cdot \|_{(p,q)}$is not a norm on$L(p,q)$. To obtain a norm, we set$f^{**}(t):=\frac{1}{t}\int^t_0 f^* (r) dr$and define $$\|f\|^*_{(p,q)} := \|t^{\frac{1}{p}-\frac{1}{q}} f^{**} (t) \|_{q\,;\,(0,\infty)},\quad \text{for } 1\leq p, q \leq \infty.$$ For$p>1$, the functional$\| \cdot \|^*_{(p,q)}$defines a norm in$L(p,q)$equivalent to$\|.\|_{(p,q)}$(see \cite[Lemma 3.4.6]{EdEv}). Endowed with this norm$L(p,q)$is a Banach space, for$p,q\geq 1$. In the following proposition we summarize some of the properties of$L(p,q)$spaces, see \cite{EdEv,RH} for the proofs. \begin{proposition}\label{propertiesLorentz} \begin{itemize} \item[(i)] If$p\,>\,0$and$q_2 \,\geq \, q_1 \, \geq 1$, then$L(p,q_1) \hookrightarrow L(p, q_2)$\item[(ii)] If$ p_2\,>p_1\,\geq 1$and$q_1,\,q_2\geq 1$, then$L(p_2,q_2)\hookrightarrow L_{\rm loc}(p_1,q_1)$. \item[(iii)] H\"{o}lder inequality: Given$(f,g) \in L(p_1, q_1)\times L(p_2, q_2)$and$(p,q) \in (1,\infty)\allowbreak\times [1,\infty]$such that$1/p = 1/p_1 + 1/p_2$,$\, 1/q \leq 1/q_1+ 1/q_2$, then $$\label{Holder} \|f g\|_{(p,q)} \leq C \|f\|_{(p_1, q_1)} \;\;\|g\|_{(p_2,q_2)},$$ where$C$depends only on$p$. \item[(iv)] Let$(p,q) \in (1, \infty) \times (1, \infty)$. Then the dual space of$L(p,q)$is isomorphic to$L(p' , q')$where$ 1/p + 1/p' =1$and$1/q + 1/q' =1$. \item[(v)] Let$\gamma>0$. Then $$\label{eq4} \big\| |f|^\gamma \big\|_{(p,q)} =\| f \|^{\gamma}_{(\frac{p}{\gamma},\frac{q}{\gamma})}$$ \end{itemize} \end{proposition} As mentioned before the main interest of considering the Lorentz spaces is that the usual Sobolev embedding, the embedding of$\mathcal{D}^{1,p}_0(\Omega)$in to$L^{p^{*}} (\Omega)$, can be improved as below (see for example, appendix in \cite{ALT}): \begin{proposition}[Lorentz-Sobolev embedding]\label{prop:Sobolev} We have$\mathcal{D}^{1,p}_0(\Omega) \hookrightarrow L(p^*, p)$;\\ i.e., there exists$C>0$such that $$\| u\|_{(p^*,\,p)} \leq C \| \nabla u\|_{p},\quad \forall \;u \in \mathcal{D}^{1,p}_0(\Omega).$$ \end{proposition} \section{The function space$\mathcal{F}_d$} For$(d,q) \in[1,\infty)\times [1, \infty)$,$C_c^{\infty} (\Omega)$is dense in the Banach space$L(d, q)$. However, the closure of$C_c^{\infty} (\Omega)$in$L(d, \infty)$is a closed proper sub space of$L(d, \infty)$that will henceforth be denoted by $$\mathcal{F}_d := \overline{C_c^{\infty} (\Omega)}^{ \|\cdot \|_{(d,\infty)} } \subset L(d,\infty).$$ Next we list some of the properties of the space$\mathcal{F}_d$, see \cite[Proposition 3.1]{ALM} for a proof. \begin{proposition}\label{prop1} \begin{itemize} \item[(i)] For each$d>1$,$L(d,q) \subset \mathcal{F}_d$when$ 1\leq q < \infty$. \item[(ii)] For each$a \in \Omega$, the Hardy potential$x \mapsto |x-a|^{\frac{-N}{d}}$does not belong to$\mathcal{F}_{d}$. \end{itemize} \end{proposition} Recall that$L(d,d)=L^d(\Omega)$, hence from (i) it follows that$L^{N/p}(\Omega)$is contained in$F_{N/p}$. Thus Theorem \ref{Exist} readily extends the results in \cite{AllHua1,FleGosThe}, since$g\in L^{N/p}(\Omega)$is a part of their assumptions. Similarly the result in \cite{Huang} follows as the positive part of weights he considered is bounded and compactly supported. Note that (ii) shows that$\mathcal{F}_d$is a proper subspace of the Lorentz space$L(d,\infty)$. Now we state a few useful characterizations of the space$\mathcal{F}_d$. \begin{proposition}\label{equivalent} The following statements are equivalent \begin{itemize} \item[(i)]$f \in\mathcal{F}_d$, \item[(ii)]$f^*(t)=o(t^{-1/d})$at 0 and$\infty$; i.e., $$\label{eq:Tertikas} \lim_{t \to 0_+} t^{1/d}f^\ast(t) =0 = \lim_{t \to \infty} t^{1/d}f^\ast(t).$$ \item[(iii)]$\alpha_f(s)=o(s^{-d})$at 0 and$\infty$; i.e., $$\label{newchar} \lim_{s \to 0_+}s (\alpha_f(s))^{1/d} = 0 = \lim_{s \to \infty}s (\alpha_f(s))^{1/d}.$$ \end{itemize} \end{proposition} \begin{proof} (i)$\Rightarrow$(ii): See the first part of \cite[Theorem 3.3]{ALM}. (ii)$\Rightarrow$(iii): Let (ii) hold. Thus for given$\varepsilon>0$, there exist$t_1,t_2>0$such that $$\label{relation2} t^{1/d} f^\ast(t)<\varepsilon,\quad \forall \,t \in (0, t_1) \cup (t_2,\infty).$$ Let$s_1=\varepsilon\,(t_1)^{-1/d}$and$s_2=\varepsilon\,(t_2)^{-1/d}$. Note that $$\text{If s \in (0,s_2)\cup (s_1,\infty), then t=(\frac{\varepsilon}{s})^d\in (0, t_1) \cup (t_2,\infty).}$$ Now using \eqref{relation2} and \eqref{relation1} with$c=\varepsilon$, we obtain $s (\alpha_f(s))^{1/d}<\varepsilon, \quad \forall s \in (0,s_2) \cup (s_1,\infty).$ This shows that$\alpha_f(s)=o(s^{-d})$at 0 and$\infty$. (iii)$\Rightarrow$(i): Assume (iii). Then for a given$\varepsilon>0$, there exist$s_1,s_2$such that $$\label{eq12} s(\alpha_f(s))^{1/d}<\varepsilon,\quad \forall s \in(0,s_1] \cup [s_2,\infty).$$ We use \cite[Proposition 3.2]{ALM} to show that$f$is in$\mathcal{F}_d$. Let $$A_\varepsilon:= \{x: s_1\leq f(x)< s_2\},\quad f_\varepsilon:=f \chi_{A_\varepsilon}.$$ Note that$|A_\varepsilon|\leq \alpha_f(s_1)<\infty$and$f_\varepsilon\in L^\infty(\Omega)$. Let$g=f\chi_{A_\varepsilon^c}$. Thus it is enough to prove $$\|f-f_\varepsilon\|_{(d,\infty)}=\|g\|_{(d,\infty)}<\varepsilon.$$ Observe that, for$s\in (s_1,s_2)$,$ \alpha_g(s)=\alpha_f(s_2)$and hence $$\label{eq11} s(\alpha_g(s))^{1/d} < s_2(\alpha_f(s_2))^{1/d}<\varepsilon, \,\, \forall s\in (s_1,s_2).$$ Since$|g|\leq|f|$, we have$\alpha_g(s)\leq \alpha_f(s)$, for all$s>0$. Now by combining \eqref{eq12} and \eqref{eq11} we obtain $s(\alpha_g(s))^{1/d} < \varepsilon,\quad \forall s>0.$ Hence by lemma \ref{equivsup} we obtain$\|g\|_{(d,\infty)}<\varepsilon$. \end{proof} Next we give another sufficient condition similar to a condition of Rozenblum, see \cite[(2.19)]{Rozen}, for a function to be in$\mathcal{F}_d$. \begin{lemma}\label{IntegrabilityCondn} Let$h\in L({d}, \infty)$and$h>0$. If$f$is such that$\int_\Omega h^{d-q}|f|^q< \infty$for some$q\geq d$. Then$f\in L(d,q)$and hence in$\mathcal{F}_d$. \end{lemma} \begin{proof} The result is obvious when$q=d$. For$q>d$, let$g=h^{\frac{d}{q}-1}f$. Then the above integrability condition yields$g\in L^q(\Omega)$. Using property \eqref{eq4} we obtain$h^{1-\frac{d}{q}}\in L(\frac{dq}{q-d},\infty)$. Now by H\"{o}lder inequality \eqref{Holder} we obtain$f\in L(d,q)$and hence in$\mathcal{F}_d$as$L(d,q)\subset \mathcal{F}_d$. \end{proof} \begin{remark} \label{rmk3.4} \rm Let$g\in L^q(\mathbb{R}^N)$with$q\geq d$and let $f(x)= |x|^{(\frac{1}{q}-\frac{1}{d} )N}g.$ Then using the above lemma one can easily verify that$f\in L(d,q)$. In general for any$h\in L(d,\infty)$with$h>0$,$f=g h^{1-\frac{d}{q}}\in L(d,q)$. Thus we can obtain Lorentz spaces by interpolating Lebesgue and weak-Lebesgue spaces suitably. \end{remark} Another class of functions contained in$\mathcal{F}_{N/p}$is provided by the work of Szulkin and Willem \cite{SWi}. More specifically they consider the weights$g$defined by the conditions: $$\label{eq:SzulinWillem} \begin{gathered} g \in L^1_{\rm loc} (\Omega), \quad g^+=g_1+g_2\not\equiv 0, \quad g_1 \in L^{N/p} (\Omega), \\ \lim_{|x|\to \infty,\, x \in \Omega} |x|^p g_2(x)=0, \quad \lim_{x\to a ,\, x\in \overline{\Omega}} | x- a |^p g_2(x)=0 \quad \forall a \in {\overline \Omega}. \end{gathered}$$ The following lemma can be proved using similar arguments as in \cite[Lemma 4.1]{ALM}. \begin{lemma}\label{Willem} Let$g:\Omega\to \mathbb{R}$be a measurable function such that $$(i)\lim_{|x|\to \infty,\, x \in \Omega} |x|^p g(x)=0, \quad (ii) \lim_{x\to a ,\, x\in \overline{\Omega}} | x- a |^p g(x)=0, \quad \forall a \in {\overline \Omega}.$$ Then there exist finite number of points$a_1 , \dots, a_m \in \overline{\Omega} $with the following property: For every$\varepsilon>0$there exists$R:= R(\varepsilon) >0$such that \begin{gather} |g(x)| < \frac{\varepsilon}{|x|^p} \quad \text{a.e. } x \in \Omega \setminus B(0,R) \label{eqn2.2}\\ |g(x)| < \frac{\varepsilon}{|x- a_i|^p} \quad \text{a.e. } x \in \Omega \cap B(a_i, R^{-1}) \,, \; i=1, \dots, m, \label{eqn2.3}\\ g \in L^{\infty} ( \Omega \setminus A_{\varepsilon} ), \label{eqn2.4} \end{gather} where$A_{\varepsilon} := \bigcup_{i=1}^m B (a_i, R^{-1})\cap \Omega$. \end{lemma} \begin{theorem} \label{WillemSzulkin} Let$g:\Omega\to \mathbb{R}$be as in the previous lemma. Then$g\in \mathcal{F}_{N/p}$. \end{theorem} \begin{proof} We use Proposition \ref{equivalent}(iii) to show that$g \in \mathcal{F}_{N/p}$. For$\varepsilon >0$, let$R$be given as in the previous lemma. Let$s_1 :=\varepsilon R^{-p}$. We first show that $$s(\alpha_g(s))^{p/N}<\varepsilon,\quad \forall s s \} \big| \leq \omega_N (\frac{\varepsilon}{s})^{N/p} \,,$$ where$\omega_N$is the volume of unit ball in$\mathbb{R}^N$. Thus \label{Willemeq1} s(\alpha_g(s))^{p/N}< C_1\varepsilon,\quad \forall\, s s_2$, using \eqref{eqn2.3} the distribution function can be estimated as follows: \begin{align*} \alpha_{g} (s) &= \big| \{ x \in \Omega : |g (x)| > s \} \big| = \big| \{ x \in A_{\varepsilon} : |g (x)| > s \} \big|\\ &\leq \sum_{i=1}^m \big| \{ x \in B (a_i, R^{-1} ) \cap \Omega : |g (x)| > s \} \big| \\ &\leq \sum_{i=1}^m \big| \{ x \in B (a_i, R^{-1} ) : \varepsilon |x-a_i|^{-p} > s \}\big|\\ &= \sum_{i=1}^m \omega_N ( \frac{\varepsilon}{s} )^{N/p}. \end{align*} Therefore, $$\label{Willemeq2} s (\alpha_g(s))^\frac{p}{N} \leq C_2 \varepsilon \quad \forall s > s_2,$$ where $C_2$ is independent of $\varepsilon$. Now proof follows using condition (iii) of proposition \ref{equivalent} together with \eqref{Willemeq1} and \eqref{Willemeq2}. \end{proof} As an immediate consequence we have the following remark. \begin{remark} \label{rmk37} \rm The positive part of any function satisfying \eqref{eq:SzulinWillem} belongs to the space $\mathcal{F}_{N/p}$. In particular Theorem \ref{Exist} summarizes the result by Willem and Szulkin \cite{SWi}. \end{remark} \subsection{Examples} Now we consider examples of weights that admit a positive principal eigenvalue for \eqref{eq1} to understand how the conditions \eqref{eq:SzulinWillem} and the properties that define the space $\mathcal{F}_{N/p}$ are related to one another. First, we consider the following functions: \begin{gather}\label{eq:ModifiedW} {g}_1(x) = \frac{1}{\big( \log (2+|x|^2) \big)^{p/N}{(1+|x|^2)^{p/2}}},\\ { g}_2 (x) = \frac{1}{|x|^p(1+|x|^2)^{p/2} \big( \log (2+\frac{1}{|x|^2}) \big)^{p/N}}. \end{gather} One can verify that ${g}_1,\, { g}_2$ satisfy \eqref{eq:SzulinWillem} and hence belong to $\mathcal{F}_{N/p}$ and none of them lies in $L^{N/p}(\mathbb{R}^N)$. Next we give an example of a weight which is in $\mathcal{F}_{N/p}$ but does not satisfy the condition \eqref{eq:SzulinWillem}. \begin{example} \label{exa4} \rm In the cube $\Omega = \{ (x_1,\dots, x_N) \in \mathbb{R}^N : |x_i| < R \}$ with $0< R <1$ consider the function defined by $$\label{eq:W3} g_3(x)= \big| x_1 \log (|x_1|) \big|^{-p/N}, \quad x_1 \ne 0.$$ Using the condition \eqref{IntegrabilityCondn}, one can verify that $g_3 \in L (\frac{N}{p}, q)$, for $q>\frac{N}{p}$. But $g_3$ does not satisfy~\eqref{eq:SzulinWillem}. Indeed along the curve $x_2 = (x_1)^{\frac{1}{2N}}$, the limit of $|x|^p g_3(x)$ is infinity as $x$ tends to $0$ and this limit is zero as $x$ tends to $0$ along the $x_1$ axis. Thus $g_3$ does not satisfy the condition~\eqref{eq:SzulinWillem}. \end{example} \section{Existence of an eigenvalue and its properties} \label{Sec:existence} In this section we prove the existence and the uniqueness of the positive principal eigenvalue for \eqref{eq1} for $g$ for which $g^+\in \mathcal{F}_{N/p}\setminus \{0\}$. Moreover we prove a few qualitative properties of that positive principal eigenvalue. \subsection{The existence of a minimizer} We prove the existence using a direct variational principle. First, we recall the following sets and functional: \begin{gather*} \mathcal{D}^+(g)= \{ u\in \mathcal{D}^{1,p}_0(\Omega):\int_{\Omega}g|u|^p>0 \},\quad M= \{ u\in \mathcal{D}^{1,p}_0(\Omega):\int_{\Omega}g|u|^p=1 \},\\ J(u)=\frac{1}{p}\int_{\Omega}|{\nabla }u|^p,\quad G(u) =\frac{1}{p}\int_{\Omega}g|u|^p. \end{gather*} From the definition of the space $\mathcal{D}^{1,p}_0(\Omega)$, it is obvious that $J$ is coercive and weakly lower semi-continuous. Due to the weak assumption on $g^-$, the map $G$ may not be even continuous. However the map $$G^+(u):= \frac{1}{p}\int_{\Omega}g^+|u|^p$$ is continuous and compact on $\mathcal{D}^{1,p}_0(\Omega)$. \begin{lemma}\label{compactness} Let $g^+\in F_{N/p}\setminus\{0\}$. Then $G^+$ is compact. \end{lemma} \begin{proof} Let $\{u_n\}$ converge weakly to $u$ in $X$. We show that $G^+(u_n)\to G^+(u)$, up to a subsequence. % pact setapproximate $g^+$ with a $\mathcal{C}_c^\infty(\Omega)$ function in $L(\frac{N}{p}, \infty)$ For $\phi\in\mathcal{C}_c^{\infty}(\Omega)$, we have $$\label{eq5} p(G^+(u_n)-G^+(u)) = \int_{\Omega}\phi\,(|u_n|^p-|u|^p) + \int_{\Omega}(g^+-\phi)\,(|u_n|^p-|u|^p).$$ We estimate the second integral using the Lorentz-Sobolev embedding and the H\"{o}lder inequality as below $$\label{eq7} \int_{\Omega}|(g^+-\phi)|\,\big|(|u_n|^p-|u|^p)\big | \leq C\|g^+-\phi\|_{(N/p,\infty)} \big (\|u_n\|_{(p^*,p)}^p+\|u\|_{(p^*,p)}^p \big )$$ where C is a constant which depends only on $N,p$. Clearly $\{u_n\}$ is a bounded sequence in $L(p^*,p)$. Let $$m:=\sup_{n}\{\|u_n\|_{(p^*,p)}^p+\|u\|_{(p^*,p)}^p\}.$$ Now using the definition of the space $F_{N/p}$, for a given $\varepsilon>0$, we choose $g_\varepsilon \in \mathcal{C}_c^{\infty}(\Omega)$ so that $$\|g^+-g_{\varepsilon}\|_{(N/p,\infty)}<\frac{p \;\varepsilon}{2 m C}.$$ Thus by taking $\phi=g_\varepsilon$ in \eqref{eq7} we obtain $\int_{\Omega}|(g^+-g_\varepsilon)|\,\big|(|u_n|^p-|u|^p)\big | <\frac{p\;\varepsilon}{2}$ Since $X\hookrightarrow L^p_{\rm loc}(\Omega)$ compactly, the first integral in \eqref{eq5} can be made arbitrary small for large $n$. Thus we choose $n_0\in \mathbb{N}$ so that $\int_{\Omega}g_\varepsilon(|u_n|^p-|u|^p)<\frac{p\varepsilon}{2},\quad \forall n>n_0.$ Hence $|G^+(u_n)-G^+(u)|<\varepsilon$, for $n>n_0$. \end{proof} Now we are in a position to prove the existence of a minimizer for $J$ on $M$. \begin{theorem}\label{Existence} Let $\Omega$ be a domain in $\mathbb{R}^N$ with $N>p$. Let $g\in L^1_{\rm loc}(\Omega)$ and $g^+\in \mathcal{F}_{N/p} \setminus \{0\}$. Then $J$ admits a minimizer on $M$. \end{theorem} \begin{proof} Since $g \in L^1_{\rm loc} (\Omega)$ and $g^{+} \neq 0$, there exists $\varphi \in \mathcal{C}_c^{\infty} (\Omega)$ such that $\int_{\Omega} g|\varphi|^p >0$ (see for example, \cite[Proposition 4.2]{KPL}) and hence $M\neq \emptyset$. Let $\{u_n\}$ be a minimizing sequence of $J$ on $M$; i.e., $$\lim_{n \to \infty} J(u_n)=\lambda_1:=\inf_{u\in M}J(u).$$ By the coercivity of $J,\{u_n\}$ is bounded in $\mathcal{D}^{1,p}_0(\Omega)$ and hence using the reflexivity of $\mathcal{D}^{1,p}_0(\Omega)$ we obtain a subsequence of $\{u_n\}$ that converges weakly. We denote the weak limit by $u$ and the subsequence by $\{u_n\}$ itself. Now using the compactness of $G^+$, we obtain $$\lim_{n\to\infty}\int_{\Omega}g^+ |u_n|^p = \int_{\Omega}g^+ |u|^p.$$ Now as $u_n\in M$ we write, $$\int_{\Omega}g^- |u_n|^p=\int_{\Omega}g^+ |u_n|^p-1$$ Since the embedding $\mathcal{D}^{1,p}_0(\Omega) \hookrightarrow L^p_{\rm loc}(\Omega)$ is compact, up to a subsequence $u_n\to u$ a.e. in $\Omega$. Hence by applying Fatou's lemma, $$\int_{\Omega}g^- |u|^p\leq\int_{\Omega}g^+ |u|^p-1,$$ which shows that $\int_{\Omega}g |u|^p\geq 1$. Setting ${\widetilde u} := u/(\int_{\Omega}g |u|^p)^{1/p}$, the weak lower semi continuity of $J$ yields \begin{align*} \lambda_1\leq J(\widetilde u) =\frac{J(u)}{\int_{\Omega}g |u|^p}\leq J(u)\leq \liminf_{n}J(u_n) = \lambda_1 \end{align*} Thus the equality must hold at each step and hence $\int_{\Omega}g |u|^p = 1$, which shows that $u\in M$ and $J (u)= \lambda_1$. \end{proof} Note that $R$ is not sufficiently regular to conclude that $u$ is an eigenfunction of \eqref{sol1} corresponding to $\lambda_1$, using critical point theory. \begin{proposition}\label{Mineigen} Let $u$ be a minimizer of $R$ on $\mathcal{D}^+(g)$. Then $u$ is an eigenfunction of \eqref{eq1} \end{proposition} \begin{proof} For each $\phi \in \mathcal{C}_c^\infty(\Omega)$, using dominated convergence theorem one can verify that $R$ admits directional derivative along $\phi$. Now since $u$ is a minimizer of $J$ on $\mathcal{D}^+(g)$ we obtain $$\frac{d}{dt}R(u+t\phi)\vert_{t=0}=0.$$ Therefore, $$\int_\Omega |{\nabla }u| ^{p-2} {\nabla }u\cdot{\nabla }\phi = \lambda_1 \int_\Omega g\, |u|^{p-2} u\, \phi, \quad \forall \,\phi \in \mathcal{C}_c^\infty(\Omega).$$ Now we use the density of $\mathcal{C}_c^\infty(\Omega)$ in $\mathcal{D}^{1,p}_0(\Omega)$ to conclude that $$\int_\Omega |{\nabla }u| ^{p-2} {\nabla }u\cdot{\nabla }v = \lambda_1 \int_\Omega g\, |u|^{p-2} u\, v, \quad \forall \,v \in \mathcal{D}^{1,p}_0(\Omega).$$ \end{proof} \subsection{Qualitative properties of $\lambda_1$} First we prove that the eigenfunctions corresponding to $\lambda_1$ are of constant sign. Since the eigenfunctions are not regular enough, the classical strong maximum principle is not applicable here. In \cite{ALM}, for $p=2$, we use a strong maximum principle due to Brezis and Ponce \cite{BrezisPonce} to show that first eigenfunctions are of constant sign. A similar strong maximum principle is obtained in \cite{KPL}, for quasilinear operators. From \cite[Proposition 3.2]{KPL} we have the following lemma. \begin{lemma}[Strong Maximum principle for $\Delta_p$]\label{StMax} Let $u\in\mathcal{D}^{1,p}_0(\Omega)$, $V\in L^1_{\rm loc}(\Omega)$ be such that $u, V\geq 0$ a.e in $\Omega$. If $V|u|^{p-1}\in L^1_{\rm loc}(\Omega)$ and $u$ satisfies the following differential inequality( in the sense of the distributions) $-\Delta_p(u)+V(x)u^{p-1}\geq 0 \quad \text{in } \Omega,$ then either $u\equiv 0$ or $u>0$ a.e. \end{lemma} Now using the above lemma we prove the following result. \begin{lemma}\label{Constantsign} The eigenfunctions of \eqref{eq1} corresponding to $\lambda_1$ are of constant sign. \end{lemma} \begin{proof} It is clear that the eigenfunctions corresponding to $\lambda_1$ are the minimizers of $R_p$ on $\mathcal{D}_p^+(g)$. Let $u$ be a minimizer of $R_p$ on $\mathcal{D}_p^+(g)$. Since $u\neq 0$ either $u^+$ or $u^-$ is non zero. Without loss of generality we may assume that $u^+\neq 0$. Now by taking $u^+$ as a test function in \eqref{sol1}, we see that $u^+$ also minimizes $R_p$ on $\mathcal{D}_p^+(g)$. Thus by Proposition \ref{Mineigen}, $u^+$ also solves \eqref{eq1} in the weak sense, $-\Delta_p u^{+}-\lambda_1 g {(u^{+})}^{p-1}=0, \quad \text{in } \Omega.$ In particular, we have the following differential inequality in the sense of distributions: $$-\Delta_p u^{+}+\lambda_1 g^-{(u^{+})}^{p-1} = \lambda_1 g^+{(u^{+})}^{p-1}\geq0, \quad \text{in } \Omega.$$ It is clear that $g^-$ and $u^+$ satisfy all the assumptions of Lemma \ref{StMax}, provided $g^-(u^+)^{p} \in L^1_{\rm loc}(\Omega)$. Since $g|u|^p\in L^1(\Omega)$, we have $(g^-)^{1/q}(u^+)^{p-1} \in L^q(\Omega)$, where $q$ is the conjugate exponent of $p$. Further, $(g^-)^{1/p}\in L^p_{\rm loc}(\Omega)$, since $g\in L^1_{\rm loc}(\Omega)$. Let us write $$g^-{(u^{+})}^{p-1}=(g^-)^{1/p}(g^-)^{1/q}(u^+)^{p-1}.$$ Now we use H\"{o}lder inequality to conclude that $g^-{(u^{+})}^{p-1}\in L^1_{\rm loc}(\Omega)$. Now in view of Lemma \ref{StMax} we obtain $u^+>0$ a.e. and hence $u=u^+$. Moreover, the zero set of $u$ is of measure zero. \end{proof} Indeed, the above lemma shows that $\lambda_1$ is a principal eigenvalue of \eqref{eq1}. Next we prove the uniqueness of the positive principal eigenvalue, using the Picone's identity for the p-Laplacian. In \cite{AllHua2}, Picone's identity is proved for $\mathcal{C}^1$ functions. However it is not hard to obtain a similar identity for less regular functions. \begin{lemma}[Picone's identity] \label{picone} Let $u\geq 0, v>0$ a.e. and let $|\nabla v|,|\nabla u|$ exist as measurable functions. Then the following identity holds a.e. \begin{align*} &|\nabla u|^p +(p-1)\frac{u^p}{v^p}|\nabla v|^p -p\frac{u^{p-1}}{v^{p-1}}|\nabla v|^{p-2}\nabla v \\ &= |\nabla u|^p - \nabla (\frac{u^p}{v^{p-1}})\cdot|\nabla v|^{p-2} \nabla v. \end{align*} Further, the left hand side of the above identity is nonnegative. \end{lemma} Now we prove the uniqueness of the positive principal eigenvalue. \begin{lemma}\label{principal} Let $g\in L(N/p,\infty)$ and let $\lambda>0$ be a positive principal eigenvalue of \eqref{eq1}. Then $\lambda=\lambda_1=\inf\{\int_\Omega|\nabla u |^p: u \in M \}.$ \end{lemma} \begin{proof} Let $v\in \mathcal{D}^{1,p}_0(\Omega)$ be a positive eigenfunction of \eqref{eq1} corresponding to $\lambda$. Let $u\in M$ and let $\{\phi_n\}$ in $\mathcal{C}_c^\infty(\Omega)$ be such that $\|u-\phi_n\|_{\mathcal{D}^{1,p}_0(\Omega)}\to 0$ and $\int_{\Omega}g |u|^p=1$. Note that $\frac{|\phi_n|^p}{v+\varepsilon}\in \mathcal{D}^{1,p}_0(\Omega)$. Thus by the Picone's identity (see Lemma \ref{picone}), we have $$\label{principal:eqn1} 0\leq\int_\Omega|\nabla\phi_n|^p-\int_\Omega|\nabla v|^{p-2} \nabla v\cdot \nabla \big(\frac{|\phi_n|^p}{(v+\varepsilon)^{p-1}}\big).$$ Since $v$ is an eigenfunction of \eqref{eq1} corresponding to $\lambda$, we have $$\label{principal:eqn2} \int_\Omega|\nabla v|^{p-2} \nabla v\cdot \nabla \Big(\frac{\phi_n^p}{(v+\varepsilon)^{p-1}}\Big) =\lambda\int_\Omega gv^{p-1}\frac{|\phi_n|^p}{(v+\varepsilon)^{p-1}}.$$ Now from \eqref{principal:eqn1} and \eqref{principal:eqn2} we $$0\leq \int_\Omega|\nabla\phi_n|^p -\lambda\int_\Omega gv^{p-1}\frac{|\phi_n|^p}{(v+\varepsilon)^{p-1}}.$$ By letting $\varepsilon \to 0$, the dominated convergence theorem yields $0\leq \int_\Omega |\nabla\phi_n|^p-\lambda\int_\Omega g |\phi_n|^p.$ Now we let $n \to \infty$ to obtain the inequality $0\leq \int_\Omega|\nabla u|^p-\lambda\int_\Omega {g}u^p.$ Therefore, $$\lambda \leq \int_\Omega|\nabla u|^p, \quad \; \forall\, u\in M.$$ This completes the proof. \end{proof} \begin{remark} \label{remark:principal} \rm Using Lemma \ref{Constantsign}, we see that $\lambda_1$ is a positive principal eigenvalue and Lemma \ref{principal} shows that $\lambda_1$ is the unique positive principal eigenvalue of \eqref{eq1}. In particular, the eigenfunctions corresponding to other eigenvalues of \eqref{eq1} must change sign. \end{remark} When $\Omega$ is connected, for the simplicity of $\lambda_1$, we refer to \cite[Theorem 1.3]{KPL}. There, the authors obtained the simplicity of the first eigenvalue of \eqref{eq1}, if it exists, even for $g$ in $L^1_{\rm loc}(\Omega)$. \subsection{Radial symmetry of the eigenfunctions} Now we give sufficient conditions for the radial symmetry of the eigenfunctions corresponding to the eigenvalue $\lambda_1$ of \eqref{eq1}. Here we assume that the domain $\Omega$ is a ball centered at origin or $\mathbb{R}^N$. Bhattacharya \cite{Bhattacharya} proved the radial symmetry of the first eigenfunctions of \eqref{eq1}, when $g\equiv 1$ and $\Omega$ is ball. Here we prove that all the positive eigenfunctions corresponding to $\lambda_1$ are radial and radially decreasing, provided $g$ is nonnegative, radial and radially decreasing. Thus our result is a two fold generalization of results of Bhattacharya, as we allow more general weight functions and the domain can be $\mathbb{R}^N$. Our result uses certain rearrangement inequalities. We emphasize that here we are not assuming any conditions on $g$ that ensures $\lambda_1$ is an eigenvalue. \begin{theorem} Let $\Omega$ be a ball centered at origin or $\mathbb{R}^N$. Let $g$ be nonnegative, radial and radially decreasing measurable function. If $\lambda_1$ is an eigenvalue of \eqref{eq1}, then any positive eigenfunction corresponding to $\lambda_1$ is radial and radially decreasing. \end{theorem} \begin{proof} Let $u$ be a positive eigenfunction of \eqref{eq1} corresponding to $\lambda_1$. Let $u_*$ and $g_*$ be the symmetric decreasing rearrangement of $u$ and $g$ respectively. Since $g$ is nonnegative, radial and radially decreasing, we use property (a) of Proposition \ref{symmetric} to conclude that $g=g_*$ a.e. Further, as $u$ is positive by property (c) of Proposition~\ref{symmetric} we obtain $(u^p)_*=(u_*)^p$ a.e. Now by the Hardy-Littlewood inequality, $$\int_{\Omega}g\, u^p \leq \int_{\Omega}g_* (u^p)_* = \int_{\Omega}g (u_*)^p .$$ Also due to Polya-Szego, we have the following inequality: $$\int_{\Omega} |{\nabla }u_*|^p\leq \int_{\Omega} |{\nabla }u|^p.$$ Thus $$\label{eq6} \frac{1}{\int_{\Omega}g (u_*)^p} \int_{\Omega} |{\nabla }u_*|^p \leq \frac{1}{\int_{\Omega}g (u)^p}\int_{\Omega} |{\nabla }u|^p.$$ Since $u$ is a minimizer of $R_p$ on $\mathcal{D}_p^+(g)$, equality holds in \eqref{eq6} and hence $u_*$ also minimizes $R_p$ on $\mathcal{D}_p^+(g)$. Now as $\lambda_1$ is simple, we obtain $u_*=\alpha u$ a.e. for some $\alpha>0$. This shows that $u$ is radial, radially decreasing. \end{proof} Using the above lemma we see that for $g(x))=\frac{1}{|x|^p},\, x\in \mathbb{R}^N$ \eqref{eq1} does not admit a positive principal eigenvalue. A proof for the case $p=2$ is given in \cite{Solimini}. \begin{proposition} Let $g(x)=1/|x|^p$, $x\in \mathbb{R}^N$. Then \eqref{eq1} does not admit a positive principal eigenvalue. \end{proposition} \begin{proof} From Lemma \ref{principal}, we know that, if $\lambda>\lambda_1$ then $\lambda$ is not a principal eigenvalue of \eqref{eq1}. Thus, it is enough to show that $\lambda_1$ is not an eigenvalue of \eqref{eq1}, when $g(x)=\frac{1}{|x|^p}$. By \cite[Theorem 1.3]{KPL}, if $\lambda_1$ is an eigenvalue of \eqref{eq1}, then $\lambda_1$ is simple. Further, if $u$ is an eigenfunction of \eqref{eq1} corresponding $\lambda_1$, then using the scale invariance of \eqref{eq1}, for each $\alpha\in \mathbb{R}$, one can verify that $$v_\alpha(x)=u(\alpha x)$$ is also an eigenfunction of \eqref{eq1} corresponding to $\lambda_1$. Now using the simplicity of $\lambda_1$, we obtain $u(x)=|x|^{1-\frac{N}{p}}u(1).$ A contradiction as $|x|^{1-\frac{N}{p}}\not\in \mathcal{D}^{1,p}_0(\mathbb{R}^N)$. \end{proof} \begin{remark} \label{rmk4.11} \rm In particular, the above Lemma shows that the best constant in the Hardy's inequality $\int_{\mathbb{R}^N}|\nabla u|^p \leq C \int_{\mathbb{R}^N} \frac{1}{|x|^p}|u|^p$ is not attained for any $u\in \mathcal{D}^{1,p}_0(\mathbb{R}^N)$. \end{remark} \section{An infinite set of eigenvalues} In this section we discuss the existence of infinitely many eigenvalues of \eqref{eq1}, using the Ljusternik-Schnirelmann theory on $\mathcal{C}^1$ manifold due to Szulkin \cite{Szulkin}. Before stating his result we briefly describe the notion of P.S. condition and genus. Let $\mathcal{M}$ be a $\mathcal{C}^1$ manifold and $f\in \mathcal{C}^1(\mathcal{M};\mathbb{R})$. Denote the differential of $f$ at $u$ by $df(u)$. Then $df(u)$ is an element of $(T_u \mathcal{M})^*$, the cotangent space of $\mathcal{M}$ at $u$ (see \cite[section 27.4]{Deimling} for definition and properties). We say that a map $f\in \mathcal{C}^1(\mathcal{M};\mathbb{R})$ satisfies Palais-Smale ( P.S. for short) condition on $\mathcal{M}$, if a sequence $\{u_n\}\subset \mathcal{M}$ is such that $f(u_n)\to \lambda$ and $df(u_n)\to 0$ then $\{u_n\}$ possesses a convergent subsequence. Let $A$ be a closed symmetric (i.e, $-A=A$) subset of $\mathcal{M}$, the \emph{krasnoselski} genus $\gamma(A)$ is defined to be the smallest integer $k$ for which there exists a non-vanishing odd continuous mapping from $A$ to $\mathbb{R}^k$. If there exists no such map for any $k$, then we define $\gamma(A)=\infty$ and we set $\gamma(\emptyset)=0$. For more details and properties of genus we refer to \cite{Rabinowitz}. From \cite[Corollary 4.1]{Szulkin} one can deduce the following theorem. \begin{theorem}\label{Cuesta} Let $\mathcal{M}$ be a closed symmetric $\mathcal{C}^1$ submanifold of a real Banach space X and $0\notin \mathcal{M}$. Let $f\in \mathcal{C}^1(\mathcal{M};R)$ be an even function which satisfies P.S. condition on $\mathcal{M}$ and bounded below. Define $$c_j:=\inf_{A\in \varGamma_j}\sup_{x\in A}f(x),$$ where $\varGamma_j=\{A\subset \mathcal{M}: A$ is compact and symmetric about origin, $\gamma(A)\geq j\}$. If for a given $j$, $c_j=c_{j+1}\dots =c_{j+p}\equiv c$, then $\gamma(K_c)\geq p+1$, where $K_c= \{ x\in M: f(x)=c\,, df(x)=0\}$. \end{theorem} Note that the set $M=\{ u \in \mathcal{D}^{1,p}_0(\Omega): \int_\Omega g |u|^p =1 \}$ may not even possess a manifold structure from the topology of $\mathcal{D}^{1,p}_0(\Omega)$, due to the weak assumptions on $g^-$. However, we show that $M$ admits a $\mathcal{C}^1$ Banach manifold structure from a subspace contained in $\mathcal{D}^{1,p}_0(\Omega)$. For $g^-\in L^1_{\rm loc}(\Omega)$, we define \begin{gather*} \|u\|_X^p:=\int_\Omega |{\nabla }u|^p + \int_\Omega g^-|u|^p.\\ X:=\{u\in \mathcal{D}^{1,p}_0(\Omega): \|u\|_X<\infty\}. \end{gather*} Then one can easily verify the following: \begin{itemize} \item $X$ is a Banach space with the norm $\|\cdot\|_X$ and $X$ is reflexive. \item Since $g^-$ is locally integrable, $\mathcal{C}_c^\infty(\Omega)$ is contained in $X$. \item Let $g\in L^1_{\rm loc}(\Omega)$ and $g^+\in \mathcal{F}_{N/p}$. Then $\mathcal{D}_p^+(g)$ is contained in $X$. This can be seen as $$\label{bound} \int_{\Omega}g^-|u|^p< \int_{\Omega}g^+|u|^p \leq C \|g^+\|_{(\frac{N}{p}, \infty)}\|u\|_{\mathcal{D}^{1,p}_0(\Omega)}^p<\infty,$$ where $C$ is the constant involving the constants that are appearing in the Lorentz-Sobolev embedding and the H\"{o}lder inequality. Note that the first inequality follows as $\int_{\Omega}g|u|^p>0$, for $u\in \mathcal{D}^+(g)$. \item $X$ is continuously embedded into $\mathcal{D}^{1,p}_0(\Omega)$. Thus $X$ embedded continuously into the Lorentz space $L(p^*,p)$ and embedded compactly into $L^p_{\rm loc}(\Omega)$. \end{itemize} We denote the dual space of $X$ by $X'$ and the duality action by $\langle\cdot,\cdot\rangle$. Using the definition of the norm one can easily see that, the map $G_p^-$, defined by $G_p^-(u):= \frac{1}{p}\int_{\Omega}g^-|u|^p,$ is continuous on $X$. Further, using the dominated convergence theorem one can verify that $G_p^-$ is continuously differentiable on $X$ and its derivative is given by $\langle{G_p^-}'(u),v\rangle=\int_\Omega g^-|u|^{p-2}u\,v.$ Similarly using the Sobolev embedding and the H\"{o}lder inequality one can easily verify that $G_p^+$ is $\mathcal{C}^1$ in $\mathcal{D}^{1,p}_0(\Omega)$ and in particular on $X$. The derivative of $G_p^+$ is given by $\langle{G_p^+}'(u),v\rangle=\int_\Omega g^+|u|^{p-2}u\,v.$ Note that for $u\in M$, $\langle G_p'(u),u\rangle=p$ and hence the map $G_p'(u)\neq0$. Recall that, $c\in \mathbb{R}$ is called a regular value of $G_p$, if $G_p'(u)\neq0$ for all $u$ such that $G_p(u)=c$. Thus we have the following lemma. \begin{lemma} Let $\Omega$ be a domain in $\mathbb{R}^N$ with $N>p$. Let $g\in L_{\rm loc}^1(\Omega)$ be such that $g^+\in \mathcal{F}_{N/p} \setminus \{0\}$. Then the map $G_p$ is in $\mathcal{C}^1(X;\mathbb{R})$ and $G_p':X\to X'$ is given by $\langle G_p'(u),v\rangle=\int_\Omega g|u|^{p-2}u\,v.$ Further, 1 is a regular value of $G_p$. \end{lemma} \begin{remark} \label{rmk5.3} \rm In view of \cite[Example 27.2]{Deimling}, the above lemma shows that $M$ is a $\mathcal{C}^1$ Banach submanifold of $X$. Note that $M$ is symmetric about the origin as the map $G_p$ is even. \end{remark} Next we show that $J_p$ satisfies all the conditions to apply Theorem \ref{Cuesta}. \begin{lemma} $J_p$ is a $\mathcal{C}^1$ functional on $M$ and the derivative of $J_p$ is given by $\langle J_p'(u),v\rangle=\int_\Omega|{\nabla }u|^{p-2} {\nabla }u\cdot{\nabla }v$ \end{lemma} The proof is straight forward and is omitted. \begin{remark} \label{rmk5.5} \rm Using \cite[Proposition 6.4.35]{Drabek}, one can deduce that $$\label{representation} \|dJ_p(u)\|=\min_{\lambda\in\mathbb{R}}\|J_p'(u)- \lambda G_p'(u)\|.$$ Thus $dJ_p(u_n)\to 0$ if and only if there exists a sequence $\{\lambda_n\}$ of real numbers such that $J_p'(u_n)- \lambda_n G_p'(u_n)\to 0$. \end{remark} In the next lemma we prove the compactness of the map $G_p^{+}$, that we use for showing that the map $J_p$ satisfies P.S. condition on $M$. \begin{lemma}\label{weakcontinuity} The map ${G_p^+}':X\to X'$ is compact. \end{lemma} \begin{proof} Let $u_n\rightharpoonup u$ in $X$ and $v\in X$. Let $q$ be the conjugate exponent of $p$. Now using the Lorentz-Sobolev embedding and the H\"{o}lder inequality available for the Lorentz spaces, one can verify the following: \begin{gather*} (|u_n|^{p-2}u_n-|u|^{p-2}u)\in L\Big(\frac{p^*}{p-1},\,\frac{p}{p-1}\Big),\\ (g^+)^{1/q}(|u_n|^{p-2}u_n-|u|^{p-2}u) \in L(\frac{p}{p-1},\,\frac{p}{p-1})\\ (g^+)^{1/p}|v| \in L(p\,,p)\\ \big \|(g^+)^{1/p}v \big \|_p \leq C\|g^+\|_{(N/p,\infty)}^{1/p} \|v\|_{(p^*,p)} \end{gather*} where $C$ is a constant that depends only on $p,N$. Now by using the usual H\"{o}lder inequality we obtain \begin{align*} & |\langle G_p'(u_n)-G_p'(u), v \rangle|\\ &\leq \int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u |\, |v|\\ &\leq \Big(\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u ) |^{p/(p-1)}\Big)^{(p-1)/p} \Big(\int_\Omega g^+|v|^p\Big)^{1/p} \\ &\leq \|g^+\|_{(N/p,\infty)}^{1/p} \|v\|_{(p^*,p)} \Big(\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)} \Big)^{(p-1)/p} \end{align*} Thus $\| G_p'(u_n)-G_p'(u)\|\leq \|g^+\|_{(N/p,\infty)}^{1/p} \Big(\int_\Omega g^+|( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)} \Big)^{(p-1)/p}$ Now it is sufficient to show that $$\Big(\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u ) |^{p/(p-1)}\Big)^{(p-1)/p}\to 0, \quad \text{as } n\to \infty.$$ Let $\varepsilon>0$ and $g_\varepsilon\in C_c^\infty(\Omega)$ be arbitrary. $$\label{eq10} \begin{split} &\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)} \\ &= \int_\Omega g_\varepsilon |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)} + \int_\Omega (g^+-g_\varepsilon ) |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)} \end{split}$$ First we estimate the second integral. Observe that $\big |( |u_n|^{p-2}u_n-|u|^{p-2}u )\big |^{p/(p-1)}$ is bounded in $L(\frac{p^*}{p},1)$. Let \begin{gather*} m= \sup_n \| \big |\big (|u_n|^{p-2}u_n-|u|^{p-2}u \big ) \big |^{p/(p-1)}\|_{\,(\frac{p^*}{p},1)},\\ \int_\Omega |(g^+-g_\varepsilon)| |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)} \leq C m \big \| \big (g^+-g_\varepsilon \big ) \big \|_{(N/p,\infty)} \end{gather*} where the constant C includes all the constants that appear in the H\"{o}lder inequality and the Lorentz-Sobolev embedding. Now since $g^+\in F_{N/p}$, from the definition of $F_{N/p}$, we can choose $g_\varepsilon \in C^\infty_c(\Omega)$ such that $$m \| (g^+-g_\varepsilon) \|_{(N/p,\infty)} <\frac{\varepsilon}{2 C}$$ Thus we can make the second integral in \eqref{eq10} smaller than $\frac{\varepsilon}{2}$ for a suitable choice of $g_\varepsilon$. Since $X$ is embedded compactly into $L_{\rm loc}^p(\Omega)$, the first integral converges to zero up to a subsequence $\{u_{n_k}\}$ of $\{u_n\}$. Hence we obtain $k_0\in \mathbb{N}$ so that, $\int_\Omega g^+ |( |u_{n_k}|^{p-2}u_n-|u|^{p-2}u ) |^{p/(p-1)}< \varepsilon,\quad \forall\, k>k_0.$ Now the uniqueness of limit of subsequence helps us to conclude, as in Lemma \ref{compactness}, that $\big(\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)} \big)^{(p-1)/p}\to 0$ as $n\to \infty$. Hence the proof. \end{proof} \begin{definition} \label{def5.7} \rm For $\lambda\in \mathbb{R}^+$, we define $A_\lambda:X\to X'$ as $A_\lambda={J_p}'+\lambda\, {G_p^-}'.$ \end{definition} In the next proposition we show that the map $J_p$ indeed satisfies P.S. condition on the $M$. \begin{proposition} \label{prop5.8} $J_p$ satisfies P.S. condition on $M$. \end{proposition} \begin{proof} Let $\{u_n\}$ be a sequence in $M$, such that $J_p(u_n)\to \lambda$ and $dJ_p(u_n)\to 0$. Thus there exists a sequence $\{\lambda_n\}$ such that $$\label{ps} J_p'(u_n)-\lambda_n G_p'(u_n)\to 0 \quad \text{as } n\to \infty,$$ Since $J_p(u_n)$ is bounded, using the estimate \eqref{bound}, we see that $\{ G_p^{-}(u_n) \}$ is bounded. Thus the sequence $\{u_n\}$ is bounded in $X$ and hence by the reflexivity we may assume passing to a subsequence that $u_n\rightharpoonup u$. Since $G_p^+$ is weakly continuous, we obtain $G_p^+(u_n)\to G_p^+(u)$. Now by Fatou's lemma, $$\label{weaklimit} \int_\Omega g^-|u|^p \leq \liminf \int_\Omega g^+|u_n|^p -1 =\int_\Omega g^+|u|^p -1 .$$ Thus $\int_\Omega g|u|^p \geq 1$ and hence $u\neq 0$. Further, $\lambda_n \to \lambda$ as $n\to \infty$, since $$p(J_p(u_n)-\lambda_n)=\langle J_p'(u_n)-\lambda_n G_p'(u_n),u_n\rangle \to 0.$$ Now we write \eqref{ps} as $A_{\lambda_n}(u_n)-\lambda_n {G_p^+}{'}(u_n)\to 0.$ Since $\lambda_n\to \lambda$, we obtains $A_{\lambda_n}(u_n)- A_{\lambda}(u_n)\to 0$. Now the compactness of ${G_p^+}'$ yields the strong convergence of $A_\lambda(u_n)$ and hence $\langle A_\lambda(u_n),u_n-u\rangle\to 0$. Since $u_n\rightharpoonup u$, using \cite[Lemma 4.3]{SWi} one obtain $u_n\to u$. \end{proof} We borrow an idea from \cite[Proposition 4.2]{KPL}, for the proof of the following lemma. \begin{lemma} For each $n\in\mathbb{N}$, the set $\Gamma_n\neq \emptyset$. \end{lemma} \begin{proof} The idea is to construct odd continuous maps from $S^{n-1}\to M$, for each $n\in \mathbb{N}$. Let $\Omega^+= \{x: g^+(x)>0\}$. Since $|\Omega^+|>0$, using the Lebesgue-Besicovitch differentiation theorem, one can choose $n$ points $x_1, x_2, \dots x_n$ in $\Omega^+$ such that $$\lim_{r\to 0}\frac{1}{|B_r(x_i)|}\int_{B_r(x_i)}g(y)dy=g(x_i)>0.$$ Thus there exists $R>0$, such that $B_R(x_i)\cap B_R(x_j)=\emptyset$ and $\int_{B_r(x_i)}g(y)dy>0,\,\text{ for } 00  Thus we obtain v_i= u_i/(\int_{\Omega}g|u_i|^p)^{1/p} \in M. Note that the support of v_is are disjoint. Now for \alpha=(\alpha_1, \alpha_2, \dots \alpha_n)\in \mathbb{R}^n with \sum|\alpha_i|^p=1, we have \sum \alpha_i v_i \in C_c^\infty(\Omega) and \int_{\Omega}g|\sum \alpha_i v_i |^p =1. It is easy to see that the map \phi(\alpha)= \sum \alpha_i u_i is an odd continuous map from S^{n-1} into M. Thus \phi(S^{n-1}) is compact and symmetric about origin. Now from the definition of genus it follows that \gamma(\phi(S^{n-1}))\geq \gamma(S^{n-1})=n. \end{proof} Now we are in a position to adapt the Ljusternik-Schnirelmann theorem available for \mathcal{C}^1 manifold in our situation and prove the existence of infinitely many eigenvalues for \eqref{eq1}. \begin{proof}[Proof of Theorem \ref{infinite}] Since J and M satisfy all the requirements of Theorem \ref{Cuesta}, for each j\in \mathbb{N}, we have \gamma(K_{c_j})\geq 1. Thus K_{c_j}\neq\emptyset and hence there exist u_j\in M such that dJ(u_j)=0 and J(u_j)=c_j. Therefore c_j is an eigenvalue of \eqref{eq1} and u_j is an eigenfunction corresponding to c_j. A proof for the unboundedness of the sequence \{c_n\} is given in \cite{Huang}(see Theorem 2). For the sake of completeness we adapt their idea in our situation. Recall that the space X is separable (see \cite[(3.5)]{Adams}) and hence X admits a biorthogonal system \{e_m,e_m^*\}, (see \cite[Proposition 1.f.3]{Linden}) such that \begin{gather*} \overline{\{e_m,m:\in \mathbb{N}\}}= X,\quad e_m^*\in X',\quad \langle e_m^*,e_n\rangle=\delta_{n,m}, \\ \langle e_m^*,x\rangle=0,\quad \forall m \,\Rightarrow x=0. \end{gather*} Let E_n=\operatorname{span} \{e_1,e_2,\dots,e_n\} and let \[ E_n^{\bot}= \overline{\operatorname{span} \{e_{n+1},e_{n+2},\dots\}}.$ Since $E_{n-1}^{\bot}$ is of codimension $n-1$, for any $A\in \Gamma_{n}$ we have $A\cap E_{n-1}^{\bot}\neq \emptyset$ (see \cite[Proposition 7.8]{Rabinowitz}). Let $\mu_n=\inf_{A\in \Gamma_{n}}\sup_{A\cap E_{n-1}^{\bot}}J(u),\quad n=1,2, \dots$ Now we show that $\mu_n\to \infty$. If possible let $\{\mu_n\}$ be bounded, then there exists $u_n\in E_{n-1}^{\bot}\cap M$ such that $\mu_n\leq J(u_n)0$. Since $u_n\in M$, the estimate \eqref{bound} shows that $u_n$ is indeed bounded in $X$. Thus $u_n\rightharpoonup u$ for some $u\in X$. Now by the choice of biorthogonal system, for each $m$, $\langle e_m^*,u_n \rangle\to 0$ as $n\to \infty$. Thus $u_n\rightharpoonup 0$, in $X$ and hence $u=0$, a contradiction to $\int_{\Omega}g|u|^p\geq 1$ (see the conclusion followed estimate \eqref{weaklimit}). Therefore, $\mu_n\to \infty$ and hence $c_n\to \infty$ as $\mu_n\leq c_n$. \end{proof} \begin{remark} \label{rmk5.10} \rm If $g^-\in \mathcal{F}_{N/p} \setminus \{0\}$, then there exists a sequence $\mu_n$ of negative eigenvalues of \eqref{eq1} tending to $-\infty$. Further, $\mu_1$ is simple and it is the unique negative principal eigenvalue of \eqref{eq1}. \end{remark} \section{Remarks} In this section we remark about possible extensions and applications of weighted eigenvalue problems for the $p$-Laplacian. One can study the existence of ground states for the $\Delta_p$ operator with a more general subcritical nonlinearities in the right hand side. More precisely, for given locally integrable functions $V, g$ on a domain $\Omega \subset \mathbb{R}^N$ with $V \geq 0$ but $g$ allowed to change sign, we look for the positive solutions in $\mathcal{D}^{1,p}_0(\Omega)$ for the problem $$\label{extn1} \Delta_p u + V |u|^{p-2}u = \lambda g |u|^{q-2}u , \quad u \in \mathcal{D}^{1,p}_0(\Omega),$$ where $q\in [p,p^*)$ and $10 \}$ and hence a positive solution of \eqref{extn1}. For the positivity of this minimizer one can use \cite[Proposition 5.3]{KPL}. \end{remark} \begin{remark} \label{rmk6.2} \rm Let $g$ be as in the above remark. Then the following generalized Hardy-Sobolev inequality holds $$\label{Hardy-Sobolev} \Big(\int_\Omega g|u|^q \Big)^{q/p} \leq \frac{1}{\lambda_1}\int_\Omega \{ |{\nabla }u|^p + V |u|^p \} , \quad \forall \, u \in \mathcal{D}^{1,p}_0(\Omega) ,\, \int_{\Omega}g|u|^q>0$$ where $\lambda_1$ is the minimum of ${\int_{\Omega}\{|{\nabla }u|^p+V|u|^p\}}$ on $M_q$. Further the best constant is attained. This extends the results of Visciglia \cite{Visciglia} for $p\neq 2$. \end{remark} \begin{remark} \label{rmk6.3} \rm The existence of a simple eigenvalue for \eqref{eq1} can be applied to study the bifurcation phenomena of the solutions for the semilinear problem of the type $$\label{OurProblem} -\Delta_p u= \lambda \big( a(x) u + b(x) r(u) \big), \quad u \in \mathcal{D}^{1,p}_0(\Omega)$$ for a real parameter $\lambda$ when $a, b$ are in certain sub class of weak Lebesgue space with a suitable growth condition on $r$. Such a result is available for $p=2$ see in \cite{ALM}. 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