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

TL;DR

This paper investigates the existence and multiplicity of solutions to a nonlinear p-Laplace equation with sign-changing weights near a critical spectral parameter, revealing unique subhomogeneous phenomena.

Contribution

It establishes new existence and multiplicity results for solutions in the supercritical spectral interval, highlighting phenomena specific to subhomogeneous nonlinearities.

Findings

Existence of solutions near the critical parameter under certain conditions.

Presence of three nonzero nonnegative solutions in specific parameter regimes.

Nonexistence results for superhomogeneous and sublinear cases.

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 continuous 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 $\lambda^* = \inf\left\{\int_\Omega |\nabla u|^p \, dx/\int_\Omega |u|^p \, dx: u\in W_0^{1,p}(\Omega) \setminus \{0\},\ \int_\Omega a|u|^q\,dx \geq 0\,\right\}$. 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 $\lambda \geq \lambda^*$. 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 $\{x\in \Omega: a(x)>0\}$.