Three weak solutions for a Neumann elliptic equations involving the p→(x)\vec p\left( x \right)-Laplacian operator
| Autor: | Ahmed Ahmed, Mohamed Saad Bouh Elemine Vall |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
variational principle 35d30 Numerical Analysis Pure mathematics Applied Mathematics 34b15 anisotropic sobolev spaces with variable exponents 35j20 35a15 35j60 critical point theory 35j57 QA1-939 Laplace operator Analysis p→(x)-laplacian operator neumann elliptic equations Mathematics |
| Zdroj: | Nonautonomous Dynamical Systems, Vol 7, Iss 1, Pp 224-236 (2020) |
| ISSN: | 2353-0626 |
| Popis: | The aim of this paper is to establish the existence of at least three weak solutions for the following elliptic Neumann problem{-Δp→(x)u+α(x)|u|p0(x)-2u=λf(x,u)inΩ,∑i=1N|∂u∂xi|pi(x)-2∂u∂xiγi=0on∂Ω,\left\{ {\matrix{ { - {\Delta _{\vec p\left( x \right)}}u + \alpha \left( x \right){{\left| u \right|}^{{p_0}\left( x \right) - 2}}u = \lambda f\left( {x,u} \right)} \hfill & {in} \hfill & {\Omega ,} \hfill \cr {\sum\limits_{i = 1}^N {{{\left| {{{\partial u} \over {\partial {x_i}}}} \right|}^{{p_i}\left( x \right) - 2}}{{\partial u} \over {\partial {x_i}}}{\gamma _i} = 0} } \hfill & {on} \hfill & {\partial \Omega ,} \hfill \cr } } \right.in the anisotropic variable exponent Sobolev spacesW1,p→(⋅)(Ω)\vec p\left( \cdot \right)\left( \Omega \right)where λ > 0 andf(x,t) = |t|q(x)−2t− |t|s(x)−2t,x∈ Ω,t∈ andq(·),s(⋅)∈𝒞+(Ω¯)s\left( \cdot \right) \in {\mathcal{C}_ + }\left( {\bar \Omega } \right). |
| Databáze: | OpenAIRE |
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