Abstract
We study the nonlinear boundary value problem
-div ((vertical bar del u(x)vertical bar(p1(x)-2) + vertical bar del u(x)vertical bar(p2(x)-2))del u(x)) =
= lambda vertical bar u vertical bar(q(x)-2)u - vertical bar u vertical bar(alpha(x)-2u)
in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain in R-N with smooth boundary, lambda, mu are positive real numbers, p1, p2, q and alpha are a continuous functions on (Omega) over bar satisfying appropriate conditions. First result we show the existence of infinitely many weak solutions for any lambda, mu > 0. Second we prove that for any mu > 0, there exists lambda(*) sufficiently small, and lambda* large enough such that for any lambda is an element of, (0, lambda(*)) boolean OR (lambda*, infinity), the above nonhomogeneous quasilinear problem has a non-trivial weak solution. The proof relies on some variational arguments based on a Z(2)-symmetric version for even functionals of the mountain pass theorem, the Ekelands variational principle and some adequate variational methods .