Abstract
Infinitely many solutions for elliptic problems in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math>involving the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-Laplacian
Highlights
Introduction and main resultsLet us consider the following nonlinear elliptic problem:. where p : RN → R is Lipschitz continuous and 1 < p− := infRN p(x) ≤ supRN p(x) := p+ < N, V is the new potential function, and the nonlinear term f is sublinear with some precise assumptions that we state below
Introduction and main resultsLet us consider the following nonlinear elliptic problem:− p(x)u + V(x)|u|p(x)−2u = f (x, u), in RN, (P)where p : RN → R is Lipschitz continuous and 1 < p− := infRN p(x) ≤ supRN p(x) := p+ < N, V is the new potential function, and the nonlinear term f is sublinear with some precise assumptions that we state below.We emphasize that the operator −∆p(x)u = div(|∇u|p(x)−2∇u) is said to be p(x)-Laplacian, which becomes p-Laplacian when p(x) ≡ p
The study of various mathematical problems with variable exponent growth condition has received considerable attention in recent years. These problems appear in a lot of applications, such as image processing models, stationary thermorheological viscous flows and the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium
Summary
Let us consider the following nonlinear elliptic problem:. where p : RN → R is Lipschitz continuous and 1 < p− := infRN p(x) ≤ supRN p(x) := p+ < N, V is the new potential function, and the nonlinear term f is sublinear with some precise assumptions that we state below. The study of various mathematical problems with variable exponent growth condition has received considerable attention in recent years. These problems appear in a lot of applications, such as image processing models We refer to [4, 14,15,16,17,18, 23,24,25] for the study of the p(x)-Laplacian equations and the corresponding variational problems It is well known, the main difficulty in treating problem (P) in RN arises from the lack of compactness of the Sobolev embeddings, which prevents from checking directly that the energy functional associated with (P) satisfies the C-condition.
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