Bifurcation for Quasilinear Elliptic Singular BVP

Abstract

Abstract. For a continuous function g ≥ 0 on (0, +∞) (which may be singular at zero), we confront a quasilinear elliptic differential operator with natural growth in ∇u, −∆u + g(u)|∇u| 2 , with a power type nonlinearity, λu p + f 0 (x). The range of values of the parameter λ for which the associated homogeneous Dirichlet boundary value problem admits positive solutions depends on the behavior of g and on the exponent p. Using bifurcations techniques we deduce sufficient conditions for the boundedness or unboundedness of the cited range.

Existence and non-existence of positive solutions for nonlinear elliptic singular equations with natural growth

“…For f (λ, s) = λs q and g(s) = k/s γ with 0 < γ < 1 and γ + q < 2 the existence of positive solution of (1.4) for every λ > 0 was proved in [10], see also [3].…”

“…This kind of equations (with quadratic gradient terms) has attracted much interest since the pioneering works [8,9]. In the last years, attention has been paid in singular terms in front of the gradient terms [2,3,4,5,6,11]. In fact, in most of these papers the right hand side of the equation is not identically zero, i.e.…”

“…In this case, it is proved in [3] the existence of positive solution for every λ > 0 if f (λ, s) = λs q with q < 1. If q = 1 and g is a continuous function satisfying (1.5), in [3] is obtained existence of solution for λ > λ 1 , where λ 1 denotes the first eigenvalue of −∆ under homogeneous Dirichelt boundary conditions.…”

“…In this case, it is proved in [3] the existence of positive solution for every λ > 0 if f (λ, s) = λs q with q < 1. If q = 1 and g is a continuous function satisfying (1.5), in [3] is obtained existence of solution for λ > λ 1 , where λ 1 denotes the first eigenvalue of −∆ under homogeneous Dirichelt boundary conditions. In [14], for f (λ, s) = λs q with q > 1 and given a continuous nonnegative function g satisfying (1.5), it is proved existence of positive solution if and only if λ ≥ λ * for some λ * > 0.…”

“…In [3] it was proved that this regularizing effect remains true while g(s) ≥ k/s γ , γ < 1, s large.…”