Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem

Abstract

We consider a general nonlinear elliptic problem of the second order whose associated functional presents two linking structures and we prove the existence of three nontrivial solutions to the problem.2000 Mathematics Subject Classification: 35J65, 35J20, 49J40.

Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue

“…Wang proved that, if Ω is a bounded and smooth domain of R N and g : R −→ R is a C 1 superlinear and subcritical function such that g(0) = g (0) = 0, then the nonlinear Dirichlet problem −∆u = g (u) in Ω, u = 0 on ∂Ω, (1.1) has at least three nontrivial solutions (see [23]). In the same spirit, in [19] it was proved that, if g : Ω × R −→ R is a superlinear and subcritical Carathéodory's function, then there exists δ i > 0 such that ∀λ ∈ (λ i − δ i , λ i ), i ≥ 2, the problem −∆u − λu = g (x, u) in Ω,…”

“…In particular in [19] it was proved that two of such solutions have positive energy, approaching 0 as λ → λ − i . In this paper we want to prove that such solutions have some finer properties, inherited by the fact that λ is near an eigenvalue and by the fact that their energy approaches 0.…”

Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3, 379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition

“…Not to be unfair, we only quote our [3], were such a mistake was done assuming (3) with R = 0 and deducing (6) with c 2 = 0. Luckily such a mistake was not done in [2], a natural development of [3].…”

“…Luckily such a mistake was not done in [2], a natural development of [3]. However, this deduction is false.…”

“…Therefore, in [3], where condition (3) was assumed with R = 0, one must ignore Remark 1, while condition (6) with c 2 = 0 must be taken as part of hypothesis (g 4 ).…”