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Semilinear elliptic equations with sublinear indefinite nonlinearities. (English) Zbl 0952.35052
It is considered the Neumann problem for $-\Delta u - \lambda u = a(x)u^q + \gamma u^p, \quad u\geq 0,$ in a bounded, smooth domain $$\Omega\subset\mathbb{R}^N$$. Here are $$0<q<1<p$$, $$\lambda\in\mathbb{R}$$, $$\gamma\geq 0$$, and $$a\in C^{\beta}$$, $$\beta\in (0,1]$$. The novelty in the problem is some combined effect of $$\text{sign} a(x)$$ indefiniteness and the non-Lipschitz character of $$u^q$$ near zero. That sort of problems includes some population ecology equations.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B32 Bifurcations in context of PDEs 47J30 Variational methods involving nonlinear operators 58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces 92D40 Ecology