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.

Keywords:

• p(x)-Laplace operator
• generalized Sobolev spaces
• weak solution

AMS Subject Classifications::

• 35D05
• 35J60
• 35J70
• 58E05
• 76A02