A New Approach to Inspect Weakly Coupled Logistic Systems and their Asymptotic Behavior
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| Format: | Preprint |
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2025
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| _version_ | 1866916707914743808 |
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| author | Li, Haoyu Maia, Liliane Soares, Mayra |
| author_facet | Li, Haoyu Maia, Liliane Soares, Mayra |
| contents | 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} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18750 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A New Approach to Inspect Weakly Coupled Logistic Systems and their Asymptotic Behavior Li, Haoyu Maia, Liliane Soares, Mayra Analysis of PDEs 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} |
| title | A New Approach to Inspect Weakly Coupled Logistic Systems and their Asymptotic Behavior |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.18750 |