On the multiplicity of radial solutions to superlinear Dirichlet problems in bounded domains

Abstract

In this paper we are concerned with the existence and multiplicity of nodal solutions to the Dirichlet problem associated to the elliptic equation Du þ qðjxjÞgðuÞ ¼ 0 in a ball or in an annulus in R N :The nonlinearity g has a superlinear and subcritical growth at infinity, while the weight function q is nonnegative in ½0; 1 and strictly positive in some interval ½r 1 ; r 2 C½0; 1:By means of a shooting approach, together with a phase-plane analysis, we are able to prove the existence of infinitely many solutions with prescribed nodal properties. r

Multiple positive solutions to elliptic boundary blow-up problems

“…On the other hand, some complementary results not requiring (3.5) are also available. For instance, a shooting approach on the lines of [17] (see also [15,25] Notice that s µ is well defined and strictly positive by continuous dependence arguments (u ≡ 0 is a solution of the equation); moreover the map…”

Nodal radial solutions for a superlinear problem

“…The superlinear Dirichlet problem was considered in many papers in which existence of positive or nodal solutions was proved [3,2], with the necessary condition…”

Radial solutions of Dirichlet problems with concave–convex nonlinearities

“…In the superlinear context, interesting multiplicity results have been also achieved under relaxed sign conditions on the weight q. In particular, we refer to [13], dealing with a possibly vanishing coefficient q and to [5], [39] and reference therein, dealing with nonradial and indefinite (changing sign) weights. More in detalis, in [5] q is supposed to have a "thick" zero set, while in [39] the authors assume that ∇q(x) = 0 for every x ∈ Ω with q(x) = 0.…”

“…In particular, we assume that q ≻ 0, p ≻ 0, (1.2) where, given a continuous function ϕ : [0, 1] → R, by ϕ ≻ 0 we mean that ϕ(r) ≥ 0 for every r ∈ [0, 1] and ϕ ≡ 0. According to [13], we also require that both q and p satisfy some regularity conditions in a neighbourhood of r = 0 and of their zeros. More precisely, we first suppose that one of the following alternatives holds q(0) > 0 (1.…”