### B P Rynne

#### Abstract

We consider the boundary-value problem \begin{gather*} - \phi_p (u'(x))' = \lambda f(x,u(x)) , \quad x \in (0,1),\\ u(0) = u(1) = 0, \end{gather*} where $p>1$ ($p \ne 2$), $\phi_p(s) := |s|^{p-1} \mathop{\rm sign} s$, $s \in \mathbb{R}$, $\lambda \ge 0$, and the function $f : [0,1] \times \mathbb{R} \to \mathbb{R}$ is $C^1$ and satisfies \begin{gather*} %\tag{3} f(x,\xi) > 0, \quad (x,\xi) \in [0,1] \times \mathbb{R} ,\\ (p-1)f(x,\xi) \ge f_\xi(x,\xi) \xi , \quad (x,\xi) \in [0,1] \times (0,\infty) . \end{gather*} These assumptions on $f$ imply that the trivial solution $(\lambda,u)=(0,0)$ is the only solution with $\lambda=0$ or $u=0$, and if $\lambda > 0$ then any solution $u$ is positive}, that is, $u > 0$ on $(0,1)$.

We prove that the set of nontrivial solutions consists of a $C^1$ curve of positive solutions in $(0,\lambda_{\rm max}) \times C^0[0,1]$, with a parametrisation of the form $\lambda \to (\lambda,u(\lambda))$, where $u$ is a $C^1$ function defined on $(0,\lambda_{\rm max})$, and $\lambda_{\rm max}$ is a suitable weighted eigenvalue of the $p$-Laplacian ($\lambda_{\rm max}$ may be finite or $\infty$), and $u$ satisfies $$\lim_{\lambda\to 0} u(\lambda) = 0, \quad \lim_{\lambda \to \lambda_{\rm max}} |u(\lambda)|_0 = \infty .$$ We also show that for each $\lambda \in (0,\lambda_{\rm max})$ the solution $u(\lambda)$ is globally asymptotically stable, with respect to positive solutions (in a suitable sense).

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