Solutions to indefinite weakly coupled cooperative elliptic systems

Abstract

We study the elliptic system \begin{equation*} \begin{cases} -Δu_1 - κ_1u_1 = μ_1|u_1|^{p-2}u_1 + λα|u_1|^{α-2}|u_2|^βu_1, \\ -Δu_2 - κ_2u_2 = μ_2|u_2|^{p-2}u_2 + λβ|u_1|^α|u_2|^{β-2}u_2, \\ u_1,u_2\in D^{1,2}_0(Ω), \end{cases} \end{equation*} where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$, $N\ge 3$, ${\kappa }_1,{\kappa }_2\in \mathbb{R}$, ${\mu }_1,{\mu }_2,\lambda >0$, $\alpha ,\beta >1$, and $\alpha +\beta =p\le 2^{\ast }:=\frac{2N}{N-2}$. For $p\in (2,2^{\ast })$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $\lambda$ sufficiently large. If $p=2^{\ast }$ and ${\kappa }_1,{\kappa }_2>0$ we establish the existence of a ground state for $\lambda$ sufficiently large if, either $N\ge 5$, or $N=4$ and neither ${\kappa }_1$ nor ${\kappa }_2$ are Dirichlet eigenvalues of $-\Delta$ in $\Omega$.

18 pages