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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
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