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

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.

### MSC:

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

Zbl 0790.35020
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### References:

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