Multiple solutions with constant sign for a ( p,q $p,q$)-elliptic system Dirichlet problem with product nonlinear term

Abstract

Abstract In this paper, we consider the existence of multiple solutions of the homogeneous Dirichlet problem for a ( p,q $p,q$)-elliptic system with nonlinear product term as follows: {−Δpu=λα(x)|u|α(x)−2u|v|β(x)+Fu(x,u,v)in Ω,−Δqv=λβ(x)|u|α(x)|v|β(x)−2v+Fv(x,u,v)in Ω,u=0=von ∂Ω. $$ \textstyle\begin{cases} {-}\Delta_{p}u=\lambda \alpha (x)\vert u\vert ^{ \alpha (x)-2}u\vert v\vert ^{\beta (x)}+F_{u}(x,u,v)&\text{in }\Omega, \\ {-}\Delta_{q}v=\lambda \beta (x)\vert u\vert ^{ \alpha (x)}\vert v\vert ^{\beta (x)-2}v+F_{v}(x,u,v)&\text{in }\Omega, \\ u=0=v&\text{on }\partial \Omega . \end{cases} $$ We emphasize that the potential F(x,u,v) $F(x,u,v)$ might contain a nonlinear product term which includes F(x,u,v)=|u|θ1(x)|v|θ2(x)ln(1+|u|)ln(1+|v|) $F(x,u,v)=\vert u\vert ^{\theta_{1}(x)} \vert v\vert ^{\theta_{2}(x)}\ln (1+\vert u\vert ) \ln (1+\vert v\vert )$ as a prototype, and does not require F(x,u,v)→+∞ $F(x,u,v)\rightarrow +\infty $ as |u|+|v|→+∞ $\vert u\vert +\vert v\vert \rightarrow +\infty $. With novel growth conditions on F(x,u,v) $F(x,u,v)$, we develop a new method to check the Cerami compactness condition. Through arguments of critical point theory, we prove the existence of multiple constant-sign solutions for our elliptic system without requiring the well-known Ambrosetti–Rabinowitz condition. Moreover, we also give a result guaranteeing the existence of infinitely many solutions.

Keywords

• Elliptic system
• Dirichlet problem
• Cerami condition
• Without Ambrosetti–Rabinowitz condition
• Constant-sign solution