An Elliptic Problem with a Lower Order Term Having Singular Behaviour

Giachetti, Daniela   ; Murat, François

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 2, pp. 349-370

Résumé

We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution $u$ in a singular way and on its gradient $Du$ with quadratic growth. The prototype of the problem under consideration is $$\begin{cases} - \Delta u + \lambda u = \pm \frac{|Du|^{2}}{|u|^{k}} + f \quad \text{in} \, \Omega, \\ u=0 \text{on} \, \partial \Omega, \end{cases}$$ where $\lambda > 0$, $k > 0$; $f(x) \in L^{\infty}(\Omega)$, $f(x) \ge 0$ (and so $u \ge 0$). If $0 k 1$, we prove the existence of a solution for both the "+" and the "-" signs, while if $k \ge 1$, we prove the existence of a solution for the "+" sign only.