The Nehari manifold approach for singular equations involving the p(x)-Laplace operator

Abstract

We study the following singular problem involving the p(x)-Laplace operator ${\mathrm{\Delta }}_{p(x)}u=div(|\mathrm{\nabla }u{|}^{p(x)-2}\mathrm{\nabla }u)$, where $p(x)$ is a nonconstant continuous function, $({\mathrm{P}}_{\lambda })\{\begin{array}{ll}{\displaystyle -{\mathrm{\Delta }}_{p(x)}u=a(x)|u{|}^{q(x)-2}u(x)+\frac{\lambda b(x)}{u^{\delta (x)}}}& \text{in}\mathrm{\Omega },\\ {\displaystyle u>0}& \text{in}\mathrm{\Omega },\\ {\displaystyle u=0}& \text{on}\mathrm{\partial }\mathrm{\Omega }.\end{array}$ Here, Ω is a bounded domain in ${\mathbb{R}}^{N\ge 2}$ with $C^2$-boundary, λ is a positive parameter, $a(x),b(x)\in C(\overline{\mathrm{\Omega }})$ are positive weight functions with compact support in Ω, and $\delta (x),$ $p(x),$ $q(x)\in C(\overline{\mathrm{\Omega }})$ satisfy certain hypotheses ($A_0$) and ($A_1$). We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem $({\mathrm{P}}_{\lambda })$.

Keywords:

• Nehari manifold
• generalized Lebesgue–Sobolev space
• topological method
• singular equation
• p(x)-Laplace operator
• multiplicity

AMS Subject Classifications:

• 35J20
• 35J60
• 35J70
• 47J10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by the Slovenian Research Agency program P1-0292, N1-0114,N1-0083.