Three weak solutions for a Neumann elliptic equations involving the p→(x)\vec p\left( x \right)-Laplacian operator

Abstract

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 spaces W1,p→(⋅)(Ω)\vec p\left( \cdot \right)\left( \Omega \right) where λ > 0 and f (x, t) = |t|q(x)−2 t − |t|s(x)−2 t, x ∈ Ω, t ∈ 𝕉 and q(·), s(⋅)∈𝒞+(Ω¯)s\left( \cdot \right) \in {\mathcal{C}_ + }\left( {\bar \Omega } \right).

Keywords

• p→(x)-laplacian operator
• neumann elliptic equations
• variational principle
• critical point theory
• anisotropic sobolev spaces with variable exponents
• 35j57
• 35d30
• 35j20
• 35j60
• 34b15
• 35a15