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

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