A New Approach to Inspect Weakly Coupled Logistic Systems and their Asymptotic Behavior
Haoyu Li, Liliane Maia, Mayra Soares
TL;DR
The paper develops a rigorous variational framework for a weakly coupled logistic system on bounded domains, handling competitive and cooperative coupling through the parameter $\beta$. By designing energy functionals $J_\beta$ and a constrained Nehari manifold $\mathcal{N}_\beta$, it establishes existence and multiplicity results via min–max and mountain-pass methods, avoiding bifurcation/degree theory. It proves the existence of ground-state vectorial solutions for $\beta\in(-\delta_0,1)$ with energies $m_\beta<\min\{c_1,c_2\}$ and characterizes asymptotics as $\beta\to1^-$; for $\beta<0$ with large $|\beta|$ it constructs nontrivial vectorial solutions through a mountain-pass structure and demonstrates a segregation phenomenon in the limit $\beta\to-\infty$, leading to sign-changing scalar limits. In the cooperative regime $\beta\ge1$, positive vectorial solutions do not exist, but a sign-changing vectorial solution is obtained via a linking argument, underpinned by a Cerami compactness framework; the work also discusses a synchronized profile in the symmetric case $\lambda_1=\lambda_2$ and analyzes asymptotics as $\beta\to0$ and $\beta\to\pm\infty$, broadening the variational toolkit for logistic-type systems.
Abstract
We consider the weakly coupled elliptic system of logistic type, \begin{equation}\label{LS} \begin{cases} -Δu &=λ_1 u- |u|^{p-2}u+ β|u|^{\frac{p}{2}-2}u |v|{^{\frac{p}{2}-1}}v\mbox{ in }Ω, -Δv & =λ_2 v- |v|^{p-2}v+β|u|^{\frac{p}{2}-1}u|v|^{\frac{p}{2}-2}v \mbox{ in }Ω, \ \ u,v &\in H_0^1(Ω), \end{cases} \tag{$LS$} \end{equation} where $Ω\subset\mathbb{R}^N$ is a bounded domain with $N\geq 2$, $2< p < 2^*$, and $λ_1(Ω)< λ_1 \leq λ_2$. We say the system is competitive if $β<0$ and cooperative if $β>0$, for $β\in \mathbb{R}$. We prove the existence and multiplicity of solutions to the problem \eqref{LS} in alternative variational frameworks, depending on the range of the parameter $β.$ 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 $β$ 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 $β\to 0$ or $β\to \pm \infty.$} \bigskip \newline \textsc{Key words: Logistic System, Ground State Solution, Linking structure, seminodal Solution.}{\small}