A Multiplicity Result for Elliptic Equations at Critical Growth in Low Dimension

Abstract

We consider the problem -Δu = |u|2*-2u + λu in Ω, u = 0 on ∂Ω, where Ω is an open regular subset of ℝN (N ≥ 3), [Formula: see text] is the critical Sobolev exponent and λ is a constant in ]0, λ1[ where λ1 is the first eigenvalue of -Δ. In this paper we show that, when N ≥ 4, the problem has at least [Formula: see text] (pairs of) solutions, improving a result obtained in [4] for N ≥ 6.

The Ljapunov–Schmidt Reduction for Some Critical Problems

“…The existence of infinitely many solutions to (BN )ǫ for any ǫ > 0 was established by Devillanova-Solimini in [22] when n ≥ 7. Moreover, for low dimensions n = 4, 5, 6, in [23] they proved the existence of at least n + 1 pairs of solutions provided ǫ is small enough. Ben Ayed-El Mehdi-Pacella in [7,8] studied the blow up of the low energy sign-changing solutions of problems (AC)ǫ and (BN )ǫ as ǫ goes to zero and they classified these solutions according to the concentration speeds of the positive and negative part.…”

Energy bounds for entire nodal solutions of autonomous superlinear equations

“…This bound has been used in some papers to obtain existence and multiplicity results for solutions of (1.3), see e.g. [16,17,24]. The proof of Lemma 1.1 is easy, and it is given for instance in [39, p.185].…”

“…For λ > 0, the existence of nontrivial solutions of (1.3) was proved in several cases in a celebrated paper of Brezis and Nirenberg [12]. Further existence and multiplicity results were derived in [13][14][15]45], whereas the most recent results can be found in [16,23,24]. For λ = 0, Pohožaev's identity (see [34] or [39, p. 171]) implies that (1.3) has no nontrivial solutions on strictly starshaped domains.…”

Minimal nodal solutions of the pure critical exponent problem on a symmetric domain

“…This gap is common in the literature (see e.g. [5,6,12,21,23] and [8,Chapter 8]). Basically, it consists in assuming that the natural constraint…”