On a nonlocal p(x)-Laplacian Dirichlet problem involving several critical Sobolev-Hardy exponents

Abstract

The aim of this work is to present a result of multiplicity of solutions, in generalized Sobolev spaces, for a non-local elliptic problem with \(p(x)\)-Laplace operator containing \(k\) distinct critical Sobolev-Hardy exponents combined with singularity points \[ \begin{cases} M(\psi(u)) (- \Delta_{p(x)} u + |u|^{p(x)-2} u) = \sum_{i=1}^{k} h_i(x) \dfrac{|u|^{p^*_{s_i}(x)-2} u}{|x|^{s_i(x)}} + f(x,u) &\text{in }\Omega, \\ u=0 &\text{on }\partial \Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^N\) is a bounded domain with \(0 \in \Omega\) and \(1 \lt p^- \leq p(x) \leq p^+ \lt N\). The real function \(M\) is bounded in \([0, +\infty)\) and the functions \(h_i\) \((i=1, \ldots, k)\) and \(f\) are functions whose properties will be given later. To obtain the result we use the Lions' concentration-compactness principle for critical Sobolev-Hardy exponent in the space \(W^{1,p(x)}_{0}(\Omega)\) due to Yu, Fu and Li, and the Fountain Theorem.

Keywords

• generalized lebesgue-sobolev spaces
• \(p(x)\)-laplacian nonlocal operator
• sobolev-hardy critical exponents
• concentration-compactness principle for critical sobolev-hardy exponent
• fountain theorem