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Breaking of resonance for elliptic problems with strong degeneration at infinity

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  • 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.

    Mathematics Subject Classification: Primary: 35D05, 35J20; Secondary: 35J25, 35J70.


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