Existence Result for Solutions to Some Noncoercive Elliptic Equations

In this work, we study a class of degenerate Dirichlet problems, whose prototype is

$$ \left \{ \begin{aligned} &-{\mathrm{div}}\Big(\frac{\nabla u}{(1+|u|)^{\gamma }}+c(x)|u|^{\theta -1}u \log ^{\beta }(1+|u|)\Big)= f\ \ {\mathrm{in}}\ \Omega , \\ & u=0\ \ {\mathrm{on}}\ {\partial \Omega }, \end{aligned} \right . $$

where \(\Omega \) is a bounded open subset of \(\mathbb{R}^{N}\). \(0<\gamma <1\), \(0<\theta \leq 1\) and \(0\leq \beta <1\). We prove existence of bounded solutions when \(f\) and \(c\) belong to suitable Lebesgue spaces. Moreover, we investegate the existence of renormalized solutions when the function \(f\) belongs only to \(L^{1}(\Omega )\).