Nonhomogeneous boundary value problems in Orlicz-Sobolev spaces

We study the nonlinear Dirichlet problem -div(log(1 + vertical bar del u vertical bar(q))vertical bar del u vertical bar(p-2)del u) = -lambda vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(r-2)u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain in R-N with smooth boundary, while p, q and r are real numbers satisfying p, q > 1, p + q < min{N, r}, r < (Np - N + p)/(N - p). The main result of this Note establishes that for any lambda > 0 this boundary value problem has infinitely many solutions in the Orlicz-Sobolev space W-0(1) L-Phi (Omega), where Phi(t) = integral(t)(0) log(1 + vertical bar s vertical bar(q))(.)vertical bar s vertical bar(p-2)s ds.