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A population biological model with a singular nonlinearity. (English) Zbl 1340.35056

The following quasilinear elliptic problem is considered
\[ -{\text{div}}(|x|^{-\alpha p}|\nabla u|^{p-2}\nabla u)=|x|^{-(\alpha+1)p+\beta}\Big(au^{p-1}-f(u)-cu^{-\gamma} \Big) \]
in a smooth and bounded domain \(\Omega\subset{\mathbb R}^N\) which contains the origin. Here \(1<p<N\), \(0\leq \alpha<(N-p)/p\), \(\gamma\in (0,1)\), and \(a,\alpha, \beta,c\) are positive constants. The solution is assumed to satisfy \(u=0\) on \(\partial\Omega\). Under some additional assumptions on \(f(u)\) and on the contants \(\gamma, a\) and \(c>0\) it is obtained the existence of a positive solution. The approach relies on the upper and lower solution method.

MSC:

35J62 Quasilinear elliptic equations
92D25 Population dynamics (general)
35B09 Positive solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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