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The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain. (English summary) Nonlinearity 16 (2003), no. 2, 579–590. The authors consider the semilinear elliptic problem ∆u + λu + |u | 2 ∗ −2 u = 0 in Ω, with the Dirichlet boundary condition u = 0 on ∂Ω. The purpose of the work is to make precise the influence of the invariance of the domain Ω under some orthogonal involution on the number of nontrivial solutions of this problem which change sign only once in Ω. Here Ω is a smooth and bounded domain in R N, N ≥ 4, 2 ∗ is the critical Sobolev exponent 2 ∗ = 2N/(N − 2) and λ is a positive real which has to be chosen small enough (0 < λ < λ1(Ω)) in order to ensure the existence of a nontrivial solution of this problem [see H. Brézis and L. Nirenberg, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477; MR0709644 (84h:35059)]. The authors consider a nontrivial orthogonal involution τ of R N; assume that Ω is invariant under τ and add the condition u(τx) = −u(x) for every x in Ω to the above problem. The authors prove that this problem has at least one pair of solutions which change sign exactly once. Morever, for every λ less than some λ ∗ , they express the number of such pairs of solutions in terms of the equivariant Lyusternik-Shnirel ′ man category for the group {I, τ}. Furthermore, they specify how these special solutions concentrate when λ goes to 0. The proof is based on the study of the Palais-Smale sequences of the energy functional associated to this problem and is restricted to the so-called Nehari manifold containing the nontrivial critical points. The authors also quote results obtained by Struwe in his book [Variational methods

Year: 2013
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