Nehari's problem and competing species systems.
Let \(\Omega\subset\mathbb{R}^{N},\) \(N\geq1,\) be a smooth bounded domain and \(f\in C([0,\infty))\) be superquadratic in the sense that there exists a \(\gamma>0\) such that, for all \(s\neq0\) and almost all \(x\in\Omega,\) \[ f_{s}(x,s)s^{2}-(1+\gamma)f(x,s)s>0. \] Let \( F(s)=\int_{0}^{s}f(t)\,dt \) and consider, for all \(u\in H_{0}^{1}(\Omega),\) the functional \[ J^{\ast}(u)=\int_{\Omega}\left( \frac{1}{2}\left| \nabla u(x)\right| ^{2}-F(u(x))\right)\, dx. \] Further, let \(\omega\subset\Omega\) be an open subset of \(\Omega\) and \[ \varphi(\omega)=\inf_{\underset{w\in H^{1}(\omega)}{w>0}}\sup_{\lambda >0}J^{\ast}(\lambda w). \] The goal of the paper is to find a partition of \(\Omega\) that achieves \[ \inf\left\{ \varphi(\omega_{1})+\varphi(\omega_{2})| \overline{\omega_{1} \cup\omega_{2}}=\overline{\Omega},\quad\omega_{1}\cap\omega_{2}=\varnothing \right\} .\tag{a} \] The existence of the optimal partition can be easily established in the case \(N=1\) and it is known [\textit{T. Bartsch} and \textit{M. Willem}, Arch. Ration. Mech. Anal. 124, 261--276 (1993; Zbl 0790.35020)] that the minimal pair provides the supports of the positive and negative parts of a changing sign solution of a differential equation associated with \(J^{\ast}.\) In this paper, the authors are concerned with the \(N\)-dimensional case discussing solvability of ({a}) for \(N\geq2\) and determination of the extremality conditions provided by the minimization procedure. Interesting connections between a variational problem, a model for the competition between two species with large interaction, and changing sign solutions to elliptic superlinear equations are thoroughly explored.
scientific article; zbMATH DE number 1881300
scientific article