Semilinear elliptic equations with sublinear indefinite nonlinearities. (English) Zbl 0952.35052

Alama, Stanley

Adv. Differ. Equ. 4, No. 6, 813-842 (1999).

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.

Reviewer: Serghey G.Suvorov (Donetsk)

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

Keywords:

nonlinear elliptic Neumann problem; sublinear indefinite nonlinearity; bifurcation analysis; variational methods; sub-solutions; super-solutions