Existence and nonexistence of positive solutions of some elliptic problems with variable exponents

摘要

In this paper, we study the existence and the nonexistence of some positive solutions for nonlinear equations involving variable exponent Laplace operator and concave–convex second term: −Δp(x)v=λk(x)vq(x)±h(x)vr(x),$$ -{Delta}_{p(x)}v=lambda k(x){v}^{q(x)}pm h(x){v}^{r(x)}, $$ under Robin boundary condition in a regular open‐bounded domain Ω$$ Omega $$ of ℝN,N≥2$$ {mathbb{R}}^N,Nge 2 $$. The p(x)$$ p(x) $$‐Laplacian is given by Δp(x)v=div|∇v|p(x)−2∇v,$$ {Delta}_{p(x)}v=operatorname{div}left({left|nabla vright|}^{p(x)-2}nabla vright), $$ where p∈C1(Ω‾)$$ pin {C}^1left(overline{Omega}right) $$ and p≥1$$ pge 1 $$. Our proofs are based on the subsupersolutions method mixed with variational arguments.