General quasilinear problems involving \(p(x)\)-Laplacian with Robin boundary condition. (English) Zbl 1448.35255

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


35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
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