Multiplicity of solutions for a class of (p1,p2,…,pn) $(p_{1},p_{2},\ldots,p_{n})$-Laplacian elliptic systems with a nonsmooth potential

Abstract

Abstract In this paper, we prove that the following (p1,p2,…,pn) $(p_{1},p_{2},\ldots,p_{n})$-Laplacian elliptic system with a nonsmooth potential has at least three weak solutions: {−Δp1u1+b1(x)|u1|p1−2u1∈λ∂u1F(x,u1,…,un)in Ω,⋯−Δpnun+bn(x)|un|pn−2un∈λ∂unF(x,u1,…,un)in Ω,ui=0for 1≤i≤n on ∂Ω. $$\begin{aligned} \textstyle\begin{cases} -\Delta _{p_{1}}u_{1}+b_{1}(x) \vert u_{1} \vert ^{p_{1}-2}u_{1}\in \lambda \partial _{u_{1}}F(x,u_{1},\ldots,u_{n})& \mbox{in } \varOmega,\\ \cdots \\ -\Delta _{p_{n}}u_{n}+b_{n}(x) \vert u_{n} \vert ^{p_{n}-2}u _{n}\in \lambda \partial _{u_{n}}F(x,u_{1},\ldots,u_{n})& \mbox{in } \varOmega,\\ u_{i}=0& \mbox{for }1\leq i\leq n \mbox{ on }\partial \varOmega. \end{cases}\displaystyle \end{aligned}$$ The proof is based on a three critical points theorem for nondifferentiable functionals. Some recent results in the literature are generalized and improved.

Keywords

• Nonsmooth critical point
• Locally Lipschitz
• ( p 1 , p 2 , … , p n ) $(p_{1},p_{2},\ldots,p_{n})$ -Laplacian
• Multiple solutions
• Variational methods