Existence of solution for an indefinite weight quasilinear problem with variable exponent

Abstract

We study the nonlinear boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝ N with smooth boundary; λ, μ are positive real numbers; p 1, p 2, q, α are a continuous functions on ; V 1 and V 2 are weight functions in the generalized Lebesgues spaces and respectively, such that V 1 > 0 in an open set Ω0 ⊂ Ω and V 2 ≥ 0 on Ω. We prove, under appropriate conditions that for any μ > 0, there exists a λ* large enough, such that for any λ ≥ λ*, the above nonhomogeneous quasilinear problem has a non-trivial positive weak solution. Moreover, under supplementary conditions on these functions, we establish that for any μ, λ > 0, the problem has a non-trivial solution. The proof relies on some variational method.