Breaking of resonance for elliptic problems with strong degeneration at infinity
Francesco Della Pietra Ireneo Peral
Communications on Pure & Applied Analysis 2011, 10(2): 593-612 doi: 10.3934/cpaa.2011.10.593
In this paper we study the problem

-div$(\frac{Du}{(1+u)^\theta})+|Du|^q =\lambda g(x)u +f$ in $\Omega,$

$u=0$ on $\partial \Omega, $

$u\geq 0$ in $\Omega,$

where $\Omega$ is a bounded open set of $R^n$, $1 < q \leq 2$, $\theta\geq 0$, $f\in L^1(\Omega)$, and $f>0$. The main feature is to show that even for large values of $\theta$ there is solution for all $\lambda>0$.
The problem could be seen as a reaction-diffusion model which produces a saturation effect, that is, the diffusion goes to zero when $u$ go to infinity.

keywords: Non-coercive non-linear elliptic equations degeneration at infinity existence and nonexistence.

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