Existence and Regularity Results for Some Elliptic Equations with Degenerate Coercivity and Singular Quadratic Lower-Order Terms

Abstract

In this paper, we study the existence and regularity results for some elliptic equations with degenerate coercivity and singular quadratic lower-order terms with natural growth with respect to the gradient. The model problem is 0.1 $$\begin{aligned} \left\{ \begin{array}{ll} - \mathrm {div}\left( \frac{\nabla u}{(1+|u|)^{\gamma }}\right) +\frac{\vert \nabla u\vert ^{2}}{u^{\theta }}=f+u^{r} &{} \quad \text{ in }\ \Omega ,\\ u=0&{} \quad \text{ on }\ \partial \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega $$ is a bounded open subset in $$\mathbb {R}^{N}$$ , $$0<\theta <1$$ , $$\gamma >0$$ and $$0<r<2-\theta $$ . We will prove existence results for solutions under various assumptions on the summability of the source f.