Uniqueness and nondegeneracy of positive solutions to an elliptic system in ecology

Abstract

<p style='text-indent:20px;'>In this paper, we study the follwing important elliptic system which arises from the Lotka-Volterra ecological model in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+\lambda u = \mu_1u^2+\beta uv, &amp; x\in\mathbb{R}^N,\\ -\Delta v+\lambda v = \mu_2v^2+\beta uv, &amp; x\in \mathbb{R}^N,\\ u, v&gt;0, u, v\in H^1(\mathbb{R}^N), \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ N\leq 5, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M3">\begin{document}$ \lambda, \mu_1, \mu_2 $\end{document}</tex-math></inline-formula> are positive constants, <inline-formula><tex-math id="M4">\begin{document}$ \beta\geq 0 $\end{document}</tex-math></inline-formula> is a coupling constant. Firstly, we prove the uniqueness of positive solutions under general conditions, then we show the nondegeneracy of the positive solution and the degeneracy of semi-trivial solutions. Finally, we give a complete classification of positive solutions when <inline-formula><tex-math id="M5">\begin{document}$ \mu_1 = \mu_2 = \beta. $\end{document}</tex-math></inline-formula></p>