Nehari’s problem and competing species systems. (English) Zbl 1090.35076

Conti, M.; Terracini, S.; Verzini, G.

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19, No. 6, 871-888 (2002).

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 [T. Bartsch and 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.

Reviewer: Yuri V. Rogovchenko (Famagusta)

MSC:

35J60	Nonlinear elliptic equations
35J45	Systems of elliptic equations, general (MSC2000)
35J50	Variational methods for elliptic systems
92D25	Population dynamics (general)

Keywords:

optimal partition; extremality conditions; superlinear elliptic equations; variational problems; Lotka-Volterra systems; existence; sign changing solutions; Nehari’s method

Citations:

Zbl 0790.35020

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