Solutions for a quasilinear elliptic p(x)-Kirchhoff type problem with weight and nonlinear Robin boundary conditions

This paper deals with the existence and multiplicity of weak solutions to a class of quasilinear elliptic p(x)-Kirchhoff type problems with weight and a nonlinear Robin boundary condition such as {a + bK(Sigma(N)(i=1) integral(Omega) 1/p(i)(x) vertical bar partial derivative u/partial derivative x(i vertical bar)vertical bar(pi(x))dx)) (-Delta((p) over right arrow (x)) u) + Sigma(N)(i=1) V-i (x)vertical bar u vertical bar(pi(x)-2) u = theta(x)vertical bar u vertical bar(m(x)-2) u + f (x, u) in Omega, Sigma(N)(i+1) vertical bar partial derivative u/partial derivative x(i)vertical bar(pi(x)-2) partial derivative u/partial derivative x(i) v(i) = eta vertical bar u vertical bar(q(x)-2) u on partial derivative Omega, where Omega is a smooth bounded domain. Under suitable conditions on the data, we show the existence and multiplicity of weak solutions by means of a variational approach in the framework of anisotropic Sobolev spaces with variable exponents.