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 V-1(x)vertical bar u vertical bar(q(x)-2)u - mu V-2(x)vertical bar u vertical bar(alpha(x)-2)u 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; p(1), p(2), q, alpha are a continuous functions on (Omega) over bar; V-1 and V2 are weight functions in the generalized Lebesgues spaces L-s1(x)(Omega) and L-s2(x)(Omega), respectively, such that V-1 > 0 in an open set Omega(0) subset of Omega and V-2 >= 0 on Omega. We prove, under appropriate conditions that for any mu > 0, there exists a lambda* large enough, such that for any lambda >= lambda*, the above nonhomogeneous quasilinear problem has a non-trivial positive weak solution. Moreover, under supplementary conditions on these functions, we establish that for any mu, lambda > 0, the problem has a non-trivial solution. The proof relies on some variational method.