Nonlocal eigenvalue problems with variable exponent

Abstract

Abstract We consider the nonlocal eigenvalue problem of the following form ( 𝒫 k ) { 𝒧 K p ( x ) u ( x ) + | u ( x ) | p ¯ ( x ) - 2 u ( x ) = λ | u ( x ) | r ( x ) - 2 u ( x ) i n Ω , u = 0 i n 𝕉 N \ Ω , $$(\mathcal{P}k)\left\{ {\matrix{ {\mathcal{L}_K^{p(x)}u(x) + {{\left| {u(x)} \right|}^{\bar p(x) - 2}}u(x)} \hfill & = \hfill & {\lambda {{\left| {u(x)} \right|}^{r(x) - 2}}u(x)} \hfill & {in} \hfill & {\Omega ,} \hfill \cr u \hfill & = \hfill & 0 \hfill & {in} \hfill & {{{\rm\mathbb{R}}^N}\backslash \Omega ,} \hfill \cr } } \right.$$ where Ω is a smooth open and bounded set in 𝕉 N (N ⩾ 3), λ > 0 is a real number, K</