Nonexistence results for semilinear systems in unbounded domains

Abstract

This paper concerns the non-existence of nontrivial solutions for the semi-linear system of gradient type $$displaylines{ lambda frac{partial ^{2}u_{k}}{partial t^{2}} -sum_{i=1}^n frac{partial }{partial x_{i}}(p_{i}(x)frac{ partial u_{k}}{partial x_{i}})+f_{k}(x,u_{1},dots ,u_{m}) =0quad ext{in }Omega ,; k=1,dots ,m }$$ with Dirichlet, Neumann or Robin boundary conditions. The functions $f_{k}:mathcal{D}imes mathbb{R}^{m}o mathbb{R}$ $(k=1,dots ,m)$ are locally Lipschitz continuous and satisfy $$ 2H(x,u_{1},dots ,u_{m})-sum_{k=1}^m u_{k}f_{k}(x,u_{1},dots ,u_{m})geq 0quad (ext{resp.}leq 0) $$ for $lambda >0$ (resp. $lambda <0$). We establish the non-existence of nontrivial solutions using Pohozaev-type identities. Here $u_{1},dots ,u_{m}$ are in $H^{2}(Omega )cap L^{infty }(Omega )$, $Omega =mathbb{R}imes mathcal{D}$ with $mathcal{D}=prod_{i=1}^n (alpha _{i},eta _{i})$ and $Hin mathcal{C}^{1}(overline{mathcal{D}}imes mathbb{R}^{m})$ such that $frac{partial H}{partial u_{k}}=f_{k}$, $k=1,dots ,m $.

Keywords

• Semi linear systems
• Pohozaev identity
• trivial solution
• Robin boundary condition