Multiplicity results for p(x)-biharmonic equations with nonlinear boundary conditions

In this paper, we are interested in the existence of multiple weak solutions of the following fourth-order nonlinear elliptic problem with a p(x)biharmonic operator {Delta(2)(p(x)) u = lambda f(x) vertical bar u vertical bar(q(x)-2) u, x is an element of Omega, partial derivative vertical bar Delta u vertical bar(p(x)-2) Delta u)/partial derivative n = g(x)vertical bar u vertical bar(r(x)-2)u, x is an element of partial derivative Omega, where Omega subset of R-N is a bounded domain, Delta(2)(p(x))u = Delta(vertical bar Delta u vertical bar(p(x)-2) Delta u) is the operator of fourth order called the p(x)-biharmonic operator, p(x), q(x), r(x) is an element of C((Omega) over bar), and f is an element of C((Omega) over bar), g is an element of C(partial derivative(Omega) over bar) are non-negative weight functions with compact support in (Omega) over bar. Our analysis mainly relies on variational arguments and some recent theory on the generalized Lebesgue-Sobolev spaces L-p(x)(Omega) and W-m,W-p(x)(Omega).