Abstract
Weak solutions to Dirichlet boundary value problem driven by p(x)-Laplacian-like operator
Highlights
In this article we consider the following Dirichlet boundary value problem:−∆lp(x)u(x) + |u(x)|p(x)−2u(x) = λ g(x, u(x)) in Ω, u = 0 on ∂Ω, (Pλ) where∆lp(x)u := div 1 +|∇u|p(x) |∇u|p(x)−2∇u1 + |∇u|2p(x) is the p(x)-Laplacian-like, Ω ⊂ Rn is an open bounded domain with smooth boundary, p ∈ C(Ω) is a function with some regularity satisfying1 < p− := inf p(x) ≤ p(x) ≤ p+ := sup p(x) < +∞. x∈ΩThe function g : Ω × R → R is Carathéodory (that is, for all z ∈ R, x → g(x, z) is measurable and for a.a. x ∈ Ω, z → g(x, z) is continuous) and λ is a real positive parameter
In the sequel of this article, we assume that the reaction term g(x, z) satisfies the hypothesis: Email: calogero.vetro@unipa.it
Existence and multiplicity results for problems involving the p(x)-Laplacian-like were obtained by Rodrigues [13] (Dirichlet boundary condition), Afrouzi–Kirane–Shokooh [1] (Neumann boundary condition)
Summary
In this article we consider the following Dirichlet boundary value problem:. 1 + |∇u|2p(x) is the p(x)-Laplacian-like, Ω ⊂ Rn is an open bounded domain with smooth boundary, p ∈ C(Ω) is a function with some regularity satisfying. For a weak solution of problem (Pλ), we mean a function u ∈ W01,p(x)(Ω) such that. Existence and multiplicity results for problems involving the p(x)-Laplacian-like were obtained by Rodrigues [13] (Dirichlet boundary condition), Afrouzi–Kirane–Shokooh [1] (Neumann boundary condition). We prove the existence of weak solutions to the Dirichlet boundary value problem (Pλ), by using variational methods and critical point theory. We apply a result of Bonanno [2] for functionals satisfying the Palais–Smale condition cut off upper at r (the (PS)[r]-condition for short), to obtain the existence of at least one nontrivial weak solution. We use a result of Bonanno–Marano [4] to obtain the existence of three weak solutions. The motivation of this study comes from the use of such problems to model the behaviour of electrorheological fluids in physics (as discussed in Diening–Harjulehto–Hästö–Ružicka [8]) and, in particular, the phenomenon of capillarity which depends on solid and liquid interfacial properties such as surface tension, contact angle, and solid surface geometry
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