On a Kirchhoff Singular p(x)-Biharmonic Problem with Navier Boundary Conditions

The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular Kirchhoff problem involving the p(x)-biharmonic operator: {M(t)(Delta(2)(p(x))u+a(x)vertical bar u vertical bar(p(x)-2)u) = g(x)u(-gamma(x)) -/+ lambda f(x,u), in Omega, Delta u = u = 0, on partial derivative Omega, where Omega subset of R-N , (N >= 3) be a bounded domain with C-2 boundary, lambda is a positive parameter, gamma: (Omega) over bar -> (0,1) be 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)+gamma(x)-1(Omega). We assume that M(t) is a continuous function with t := integral(Omega)1/p(x)(vertical bar Delta u vertical bar(p(x)) + a(x)vertical bar u vertical bar(p(x)))dx, and assumed to verify assertions (M1)-(M3) in Sect. 3, moreover f (x, u) are assumed to satisfy assumptions (f1)-(f6). In the proofs of our results we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces.