Breaking of resonance for elliptic problems with strong degeneration at infinity (Q651985)

In this paper the authors consider the competition between the strong degeneration at infinity of an elliptic operator, the regularizing effect of a power gradient and the effect of a linear reaction term by analyzing the following problem \[ \begin{cases} -\text{div}\left( \frac{Du}{\left( 1+u\right) ^{\theta}}\right) +\left| Du\right| ^{q}=\lambda g\left( x\right) u+f&\text{in }\Omega\\ u=0&\text{on }\partial\Omega\\ u\geq0&\text{in }\Omega,\end{cases} \] where \(\Omega\subset R^{N}\) is bounded and open, \(1<q\leq2\), \(\theta\geq0\), \(f\in L^{1}\left( \Omega\right) \), \(f>0\). Under some additonal assumptions on \(g\), the problem under consideration has at least one (entropy) solution for all \(\lambda>0\).