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 $-Δ_p u = λ|u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$, where $1 0\}$, when the parameter $λ$ lies in a neighborhood of the critical value $λ^* = \inf\left\{\int_Ω|\nabla u|^p \, dx/\int_Ω|u|^p \, dx: u\in W_0^{1,p}(Ω) \setminus \{0\},\ \int_Ωa|u|^q\,dx \geq 0\,\right\}$. Among main results, we show that if $p>2q$ and either $\int_Ωaφ_p^q\,dx=0$ or $\int_Ωaφ_p^q\,dx>0$ is sufficiently small, then such solutions do exist in a right neighborhood of $λ^*$. Here $φ_p$ is the first eigenfunction of the Dirichlet $p$-Laplacian in $Ω$. This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case $q>p$ or in the sublinear case $q 2q$ and $\int_Ωaφ_p^q\,dx>0$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of $λ^*$, two of which are strictly positive in $\{x\in Ω: a(x)>0\}$.