$${W^{1,1}_0}$$ -solutions for elliptic problems having gradient quadratic lower order terms

Abstract

In this paper we deal with solutions of problems of the type $$\left\{\begin{array}{ll}-{\rm div} \Big(\frac{a(x)Du}{(1+|u|)^2} \Big)+u = \frac{b(x)|Du|^2}{(1+|u|)^3} +f \quad &{\rm in} \, u=0 &{\rm on} \partial \, \Omega, \end{array} \right.$$ where \({0 0, f \in L^2 (\Omega)}\) and Ω is a bounded subset of \({\mathbb{R}^N}\) with N ≥ 3. We prove the existence of at least one solution for such a problem in the space \({W_{0}^{1, 1}(\Omega) \cap L^{2}(\Omega)}\) if the size of the lower order term satisfies a smallness condition when compared with the principal part of the operator. This kind of problems naturally appears when one looks for positive minima of a functional whose model is: $$J (v) = \frac{\alpha}{2} \int_{\Omega}\frac{|D v|^2}{(1 + |v|)^{2}} + \frac{12}{\int_{\Omega}|v|^2} - \int_{\Omega}f\,v , \quad f \in L^2(\Omega),$$ where in this case a(x) ≡ b(x) = α > 0.