Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions

AbstractIn this article, we study the nonlocal p(x)-Laplacian problem of the following form a(∫Ω1p(x)(|∇u|p(x)+|u|p(x))dx)(-div(|∇u|p(x)-2∇u)+|u|p(x)-2u)=b(∫ΩF(x,u)dx)f(x,u)inΩa∫Ω1p(x)(|∇u|p(x)+|u|p(x))dx|∇u|p(x)-2∂u∂ν=g(x,u)on∂Ω, where Ω is a smooth bounded domain and ν is the outward normal vector on the boundary ∂Ω, and F(x,u)=∫0uf(x,t)dt. By using the variational method and the theory of the variable exponent Sobolev space, under appropriate assumptions on f, g, a and b, we obtain some results on existence and multiplicity of solutions of the problem.Mathematics Subject Classification (2000): 35B38; 35D05; 35J20.

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