Abstract
The purpose of this work is to study the existence and multiplicity of positive solutions for a class of singular elliptic systems involving the p(x)- Laplace operator and nonlinear boundary conditions.
Highlights
This paper is concerned with the multiplicity of positive solutions for the following singular elliptic system involving the p(x)-Laplace operator and sub-linear Neumann nonlinearities:
We prove the following crucial lemma
Existence of minimizer on Nλ+,μ we will show that the minimum of Eλ,μ is achieved in Nλ+,μ. We show that this minimizer is the first solution of problem (1.1)
Summary
For any continuous and bounded function a we define a+ := ess sup a(x) and a− := ess inf a(x). Fibering map; singular system; Neumann boundary conditions; p(x)Laplace operator; generalized Lebesgue Sobolev spaces. The inclusion between Lebesgue spaces generalizes naturally: if 0 < |Ω| < ∞ and p1, p2 are variable exponents so that p1(x) ≤ p2(x) almost everywhere in Ω, there exists the continuous embedding Lp2(x)(Ω) → Lp1(x)(Ω).
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