Existence of a continuum of solutions for a quasilinear elliptic singular problem

Abstract

In this paper we study the existence of positive solution u ∈ H1 0 (Ω) for some quasilinear elliptic equations, having lower order terms with quadratic growth in the gradient and singularities, whose model is −∆u + µ(x) |∇u| 2 u γ + u β = λup + f0(x), x ∈ Ω, 0 < γ ≤ β, 0 < p. Using topological methods we obtain the existence of an unbounded continuum of solutions. In the case µ(x) constant we derive the existence of solution for every λ > 0 if 1 < γ < 2 for any β and p < 1. Even more for µ ∈ L∞(Ω) we prove this result if β ≤ 1 and p < 2 − β.