Heriot-Watt Mathematics Report Series
HWM06-15, 21 Apr 2006

$p$-Laplacian problems with jumping nonlinearities,

B P Rynne

Abstract

We consider the $p$-Laplacian boundary value problem \begin{gather*} \tag{1} -(\phi_p(u'(x))' = f(x,u(x),u'(x)) , \quad \text{a.e. $x \in (0,1)$}, \\ c_{00} u(0) = c_{01} u'(0) , \quad c_{10} u(1) = c_{11} u'(1) , \tag{2} \end{gather*} where $p > 1$ is a fixed number, $\phi_p(s) = |s|^{p-2} s$, $s \in R$, and for each $j=0,1,$ $|c_{j0}| + |c_{j1}| > 0$. The function $f : [0,1] \X R^2 \to R$ is a Carath{é}odory function satisfying, for $(x,s,t) \in [0,1] \X R^2,$ $$ \psi_\pm(x) \phi_p(s) - E(x,s,t) \le f(x,s,t) \le \Psi_\pm(x) \phi_p(s) + E(x,s,t), \quad \pm s \ge 0, $$ where $\psi_\pm,\,\Psi_\pm \in L^1(0,1),$ and $E$ has the form $E(x,s,t) = \zeta(x) e(|s|+|t|)$, with $\zeta \in L^1(0,1)$, $\zeta \ge 0$, $e \ge 0$ and $\lim_{r \to \infty} e(r) r^{1-p} = 0$. This condition allows the nonlinearity in (1) to behave differently as $u \to \pm\infty$. Such a nonlinearity is often termed jumping}.

Related to (1), (2) is the problem \begin{equation*} \tag{3} -(\phi_p(u')' = a \phi_p(u^+) - b \phi_p(u^-) +\lambda \phi_p(u) , \quad \mbox{in } (0,1), \end{equation*} together with (2), where $a,\,b \in L^1(0,1)$, $\lambda \in R$, and $u^{\pm}(x) =\max\{\pm u(x),0\}$ for $x \in [0,1]$. This problem is `positively-homogeneous' and jumping. Values of $\lambda$ for which (2), (3) has a non-trivial solution $u$ will be called half-eigenvalues}, while the corresponding solutions $u$ will be called half-eigenfunctions}.

We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to existence and non-existence results for the problem (1), (2). We also consider a related bifurcation problem, and obtain a global bifurcation result similar to the well-known Rabinowitz global bifurcation theorem. This then leads to a multiplicity result for (1), (2).

When the functions $a$ and $b$ are constant the set of half-eigenvalues is closely related to the `Fucik spectrum' of the problem, and equivalent solvability results are obtained using the two approaches. However, when $a$ and $b$ are not constant the half-eigenvalue approach yields stronger results.

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