Abstract
AbstractWe show the existence of a week solution in "Equation missing" to a Dirichlet problem for "Equation missing" in "Equation missing", and its localization. This approach is based on the nonlinear alternative of Leray-Schauder.
Highlights
IntroductionWe consider the boundary value problem −Δp x u f x, u in Ω, P u 0 on ∂Ω, where Ω ⊂ RN, N ≥ 2, is a nonempty bounded open set with smooth boundary ∂Ω, Δp x u div |∇u|p x −2∇u is the so-called p x -Laplacian operator, and CAR : f : Ω × R → R is a Caratheodory function which satisfies the growth condition f x, s ≤ a x C|s|q x /q x for a.e. x ∈ Ω and all s ∈ R, 1.1 with C const. > 0, 1/q x 1/q x 1 for a.e. x ∈ Ω, and a ∈ Lq x Ω , a x ≥ 0 for a.e
Either i the equation λKu u has a solution in B 0, R for λ 1 or ii there exists an element u ∈ E with u R satisfying λKu u for some λ ∈ 0, 1
We present new existence and localization results for X-solutions to problem P, under CAR condition on f
Summary
We consider the boundary value problem −Δp x u f x, u in Ω, P u 0 on ∂Ω, where Ω ⊂ RN, N ≥ 2, is a nonempty bounded open set with smooth boundary ∂Ω, Δp x u div |∇u|p x −2∇u is the so-called p x -Laplacian operator, and CAR : f : Ω × R → R is a Caratheodory function which satisfies the growth condition f x, s ≤ a x C|s|q x /q x for a.e. x ∈ Ω and all s ∈ R, 1.1 with C const. > 0, 1/q x 1/q x 1 for a.e. x ∈ Ω, and a ∈ Lq x Ω , a x ≥ 0 for a.e. We refer to 1, 2 for the fundamental properties of these spaces
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