Infinitely many radial solutions for the p(x)-Kirchhoff-type equation with oscillatory nonlinearities in RN

In this paper we consider the p(x)-Kirchhoff-type equation in R-N of the form {a(integral(RN) vertical bar del u vertical bar(p(x)) + vertical bar u vertical bar(p(x))/p(x) dx) (-Delta(p(x))u + vertical bar u vertical bar(p(x)-2)u) = Q(x) f(u), u >= 0, x is an element of R-N, u(x) -> 0, as vertical bar x vertical bar -> +infinity, where 1 < p(x) < N for x is an element of R-N, Q : R-N -> R+ is a radial potential, f: [0, +infinity) -> R is a continuous nonlinearity which oscillates near the origin or at infinity and a is allowed to be singular at zero. By means of a direct variational method and the principle of symmetric criticality for non-smooth Szulkin-type functionals. the existence of infinitely many radially symmetric solutions of the problem is established. Meanwhile, the sequence of solutions in L-infinity-norm tends to 0 (resp., to +infinity) whenever f oscillates at the origin (reasp., at infinity). (C) 2011 Elsevier Inc. All rights reserved.