A New Approach to Inspect Weakly Coupled Logistic Systems and their Asymptotic Behavior

Abstract

We consider the weakly coupled elliptic system of logistic type, \begin{equation}\label{LS} \begin{cases} -\Delta u &=\lambda_1 u- |u|^{p-2}u+ \beta |u|^{\frac{p}{2}-2}u |v|{^{\frac{p}{2}-1}}v\mbox{ in }\Omega, -\Delta v & =\lambda_2 v- |v|^{p-2}v+\beta |u|^{\frac{p}{2}-1}u|v|^{\frac{p}{2}-2}v \mbox{ in }\Omega, \ \ u,v &\in H_0^1(\Omega), \end{cases} \tag{$LS$} \end{equation} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with $N\geq 2$, $2< p < 2^*$, and $\lambda_1(\Omega)< \lambda_1 \leq \lambda_2$. We say the system is competitive if $\beta<0$ and cooperative if $\beta>0$, for $\beta \in \mathbb{R}$. We prove the existence and multiplicity of solutions to the problem \eqref{LS} in alternative variational frameworks, depending on the range of the parameter $\beta.$ We do not rely on bifurcation or degree theory, which have been used in the literature for logistic-type problems. Instead, the novelty is to obtain min-max type solutions by exploiting the different geometry of the functional associated with the logistic problem. In case $N\geq 2$ and suitable values of $p$, we extend the existence results, for all $\beta$ in the whole line, and possibly for the classical case $N=3$ and $p=4$. Furthermore, we analyze the asymptotic behavior of such solutions as $\beta \to 0$ or $\beta \to \pm \infty.$} \bigskip \newline \textsc{Key words: Logistic System, Ground State Solution, Linking structure, seminodal Solution.}{\small}