On the existence of a weak solution for some singular p(x)-biharmonic equation with Navier boundary conditions

In the present paper, we investigate the existence of solutions for the following inhomogeneous singular equation involving the p(x)-biharmonic operator: {Delta(vertical bar Delta u vertical bar(p(x)-2)Delta u) = g(x)u(-y(x)) -/+ lambda f(x,u) in Omega, Delta u = u = 0 on partial derivative Omega, where Omega subset of R-N (N >= 3) is a bounded domain with C-2 boundary, lambda is a positive parameter, gamma : (Omega) over bar -> (0, 1) is a continuous function, p is an element of C((Omega) over bar) with 1 < p(-) := inf(x is an element of Omega) p(x) <= p(+) :=sup(x is an element of Omega) p(x) < N/2, as usual, p* (x) = Np(x)/N-2p(x), g is an element of Lp*(x)/p*(x)+y(x)-1 (Omega), and f(x, u) is assumed to satisfy assumptions (f1)-(f6) in Section 3. In the proofs of our results, we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. In addition, an example to illustrate our result is given.