On subhomogeneous indefinite p-Laplace equations in the supercritical spectral interval

Abstract

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-\Delta _p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $\Omega \subset {\mathbb {R}}^N$, where $1<q<p$, $\lambda \in {\mathbb {R}}$, and a is a sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in $\{x\in \Omega : a(x)>0\}$, when the parameter $\lambda$ lies in a neighborhood of the critical value $\lambda ^* := \inf \left\{ \int _\Omega |\nabla u|^p \, dx/\int _\Omega |u|^p \, dx: u\in W_0^{1,p}(\Omega ) {\setminus } \{0\},\ \int _\Omega a|u|^q\,dx \ge 0\,\right\}$. Among main results, we show that if $p>2q$ and either $\int _\Omega a\varphi _p^q\,dx=0$ or $\int _\Omega a\varphi _p^q\,dx>0$ is sufficiently small, then such solutions do exist in a right neighborhood of $\lambda ^*$. Here $\varphi _p$ is the first eigenfunction of the Dirichlet p-Laplacian in $\Omega$. This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case $q>p$ or in the sublinear case $q<p=2$ the nonexistence takes place for any $\lambda \ge \lambda ^*$. Moreover, we prove that if $p>2q$ and $\int _\Omega a\varphi _p^q\,dx>0$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of $\lambda ^*$, two of which are strictly positive in $\{x\in \Omega : a(x)>0\}$.