Existence and uniqueness results for a class of p(x)-Kirchhoff-type problems with convection term and Neumann boundary data

摘要

We establish an existence and uniqueness results for a homogeneous Neumann boundary value problem involving the p(x)-Kirchhoff-Laplace operator of the following form {-M(integral(Omega)1/p(x)(vertical bar del u vertical bar(p(x)) + vertical bar u vertical bar(p(x)))dx) (div(vertical bar del u vertical bar(p(x)-2)del u) - vertical bar u vertical bar(p(x)-2)u) =f(x, u, del u) in, vertical bar del u vertical bar(p(x)-2)partial derivative u/partial derivative eta = 0 on partial derivative Omega. where Omega is a smooth bounded domain in R-N, partial derivative u/partial derivative eta is the exterior normal derivative, p(x) is an element of C+ ((Omega) over bar) with p(x) >= 2. By means of a topological degree of Berkovits for a class of demicontinuous operators of generalized (S+) type and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on f and M, we obtain a results on the existence and uniqueness of weak solution to the considered problem.