Global multiplicity of solutions for a singular p-Laplacian quasilinear Schrödinger equation

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Abstract

We consider a class of p-Laplace quasilinear Schrödinger Equations

$\left\{\begin{array}{c} -\Delta_{p} u-\frac{p}{2^{p-1}} u \Delta_{p}\left(u^{2}\right)=\lambda u^{-\gamma}+u^{q} \text { in } \Omega, \\ u>0 \text { in } \Omega, \quad u=0 \text { on } \partial \Omega, \end{array}\right.$

where $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with regular boundary, 1 < p < ∞, 0 < γ < 1, 2 p−1 < q ≤ 2·p^*−1 for p ≤ N, 2p − 1 < q < ∞ for p > N, where $p^{*}=\frac{N p}{N-p}$ if 1 < p < N,p^* ∈ (p, ∞) is arbitrarily large if p =N, and p^* = ∞ if p > N. We establish global existence and multiplicity of positive solutions via a new strong comparison principle and a regularity result for weak solutions.

Keywords

global multiplicity / quasilinear Schrödinger equations / singular term

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