Multiplicity results for nonlinear elliptic equations involving critical Sobolev exponent

Abstract

SynopsisIn this paper we study the following boundary value problemwhere Ω is a bounded domain in Rn, n≧3, x ∈Rn, p* = 2n/(n – 2) is the critical exponent for the Sobolev embedding is a real parameter and f(x, t) increases, at infinity, more slowly than .By using variational techniques, we prove the existence of multiple solutions to the equations (0.1), in the case when λ belongs to a suitable left neighbourhood of an arbitrary eigenvalue of −Δ, and the existence of at least one solution for any λ sufficiently large.

An existence result for nonlinear elliptic problems involving critical Sobolev exponent

“…In [4 ] Cerami, Fortunato and Struwe have obtained multiplicity results for (0.1) in the case in which £ belongs to a suitable left neighbourhood of an arbitrary eigenvalue of -0394 (cf. also [3 ]).…”

Positive solutions of critical quasilinear elliptic problems ingeneral domains

“…For bounded domains, the case where p attains its maximum at a boundary point has been studied in [19]. Note also that by reasoning as in [12] one could extend our results to the case where the function p(x) in (1) is replaced by a suitable function p(x, u).…”

“…First observe that if S denotes the best Sobolev constant of the embedding D 1,2 0 ⊂ L 2 * , then by (12) and Sobolev inequality we get (14) next note that (3) and (6) yield…”

“…Lemma 2.3(i) in [11] and (I 6 ) in [12]. To this end, when a ij (x, s) ≡ δ ij , one may assume that (see e.g.…”

“…For certain problems having critical growth one may still prove that the Palais-Smale condition holds at certain energy levels, see [5,8,12,19]; for the functional J in (2), even in the nontrivial energy range determined by Lemma 2 below, the Palais-Smale condition (in the sense of [14,15]) does not hold because J may have negative critical levels, see Section 6. We overcome this problem by means of Lemma 3 below which is, somehow, the heart of our proofs: it states that the compactness is recovered if the loss of energy between the Palais-Smale sequence and its weak limit is less than a suitable threshold.…”