Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system

Abstract

Abstract We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system { - Δ λ ⁢ u = | v | p - 1 ⁢ v in ⁢ Ω , - Δ λ ⁢ v = | u | q - 1 ⁢ u in ⁢ Ω , u = v = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} &\displaystyle{-}\Delta_{\lambda}u=\lvert v\rvert^{p-1}% v&&\displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle{-}\Delta_{\lambda}v=\lvert u\rvert^{q-1}u&&\displaystyle% \phantom{}\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\phantom{}\text{on }\partial\Omega,\end{% aligned}\right. in a bounded domain Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} with smooth boundary ∂ ⁡ Ω {\partial\Omega} . Here p , q > 1 {p,q>1} , and Δ λ {\Delta_{\lambda}} is the strongly degenerate operator of the form Δ λ ⁢ u = ∑ j = 1 N ∂ ∂ ⁡ x j ⁢ ( λ j 2 ⁢ ( x ) ⁢ ∂ ⁡ u ∂ ⁡ x j ) , \Delta_{\lambda}u=\sum^{N}_{j=1}\frac{\partial}{\partial x_{j}}\Bigl{(}\lambda% _{j}^{2}(x)\frac{\partial u}{\partial x_{j}}\Bigr{)}, where λ ⁢ ( x ) = ( λ 1 ⁢ ( x ) , … , λ N ⁢ ( x ) ) {\lambda(x)=(\lambda_{1}(x),\dots,\lambda_{N}(x))} satisfies certain conditions.