The effect of a singular term in a quadratic quasi-linear problem

Abstract

For an open bounded set \(\Omega \subset \mathbb {R}^N\), \(N\ge 3\), \(0\lvertneqq f\in L^m(\Omega )\) with \(m> 1\), \(0\le \mu (x)\in L^\infty (\Omega )\) and assuming in addition \(\Vert \mu \Vert _\infty < \frac{N(m-1)}{N-2m}\) if \(m<\frac{N}{2}\), we prove the existence of a positive solution for the singular b.v.p $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda u + \mu (x)\,\frac{|\nabla u |^2}{u}\, + f(x), &{} \text { in } \Omega , \\ u=0, &{} \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$ provided that \(\lambda <\lambda _1/(1+\Vert \mu \Vert _{ L^\infty (\Omega ) })\) (extending the previous results in [3] for \(\lambda =0\)). The model case \(\mu (x)\equiv B<1\) is studied in more detail obtaining in addition the uniqueness (resp. nonexistence) of positive solution if the parameter \(\lambda <\frac{\lambda _1}{B+1}\) (resp. \(\lambda \ge \frac{\lambda _1}{B+1}\)). Even more, the solutions constitute a continuum of solutions bifurcating from infinity at \(\lambda =\frac{\lambda _1}{B+1}\). This is in contrast with [5], where the multiplicity of solutions of the nonsingular problem (\(\frac{1}{u}\) do not appear in the equation) is deduced due to the bifurcation from infinity at \(\lambda =0\).