GENERAL QUASILINEAR PROBLEMS INVOLVING \(p(x)\)-LAPLACIAN WITH ROBIN BOUNDARY CONDITION

Hassan Belaouidel     (Department of Mathematics and Computer Science, Faculty of Sciences, University Mohamed I, Mohammed V av., P.O. Box 524, Oujda 60000, Morocco)
Anass Ourraoui     (Department of Mathematics and Computer Science, Faculty of Sciences, University Mohamed I, Mohammed V av., P.O. Box 524, Oujda 60000, Morocco)
Najib Tsouli     (Department of Mathematics and Computer Science, Faculty of Sciences, University Mohamed I, Mohammed V av., P.O. Box 524, Oujda 60000, Morocco)

Abstract


This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving \(p(x)\)-Laplace type equation, namely $$
\left\{\begin{array}{lll}
-\mathrm{div}\, (a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u)= \lambda f(x,u)&\text{in}&\Omega,\\
n\cdot a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u +b(x)|u|^{p(x)-2}u=g(x,u) &\text{on}&\partial\Omega.
\end{array}\right.
$$
Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.


Keywords


\(p(x)\)-Laplacian, Mountain pass theorem, Multiple solutions, Critical point theory.

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References


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DOI: http://dx.doi.org/10.15826/umj.2020.1.003

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