On subhomogeneous indefinite $p$-Laplace equations in 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 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 ${\lambda }^{\ast }=\inf \left\{{\int }_{\Omega }|\nabla u{|}^{p}dx/{\int }_{\Omega }|u{|}^{p}dx:u\in W_0^{1,p}(\Omega )\setminus \{0\},{\int }_{\Omega }a|u{|}^{q}dx\ge 0\right\}$. Among main results, we show that if $p>2q$ and either ${\int }_{\Omega }a{\phi }_p^{q}dx=0$ or ${\int }_{\Omega }a{\phi }_p^{q}dx>0$ is sufficiently small, then such solutions do exist in a right neighborhood of ${\lambda }^{\ast }$. Here ${\phi }_p$ 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 $\lambda \ge {\lambda }^{\ast }$. Moreover, we prove that if $p>2q$ and ${\int }_{\Omega }a{\phi }_p^{q}dx>0$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of ${\lambda }^{\ast }$, two of which are strictly positive in $\{x\in \Omega :a(x)>0\}$.

39 pages, 4 figures

References in corpus (1)

• An overview on constrained critical points of Dirichlet integrals