Mostafa Allaoui, Omar Darhouche, Abderrachid El amrouss, Najib Tsouli, Existence of solutions for a class of nonlinear type problems involving the p(x)-Laplacian operator, Vol. 2020 (2020), Article ID 1, pp. 1-11

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DOI: 10.23952/jnfa.2020.1

Received April 16, 2019; Accepted January 5, 2020; Published January 16, 2020

 

Abstract. The aim of this paper is to establish the existence of at least one nontrivial weak solution of the following problem
-\Delta_{p(x)}u+\alpha(x)|u|^{p(x)-2}u=\lambda a(x)|u|^{q(x)-2}u\ \text{in}\ \Omega,
|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}+\beta(x)|u|^{p(x)-2}u=\lambda b(x)|u|^{r(x)-2} u\ \text{on}\ \partial\Omega.
Our technical approach is based on the direct variational method combined with the mountain pass theorem and the theory of Lebesgue and Sobolev spaces.

 

How to Cite this Article:
Mostafa Allaoui, Omar Darhouche, Abderrachid El amrouss, Najib Tsouli, Existence of solutions for a class of nonlinear type problems involving the p(x)-Laplacian operator, Journal of Nonlinear Functional Analysis, Vol. 2020 (2020), Article ID 1, pp. 1-11.