Blow-up theory for symmetric critical equations involving the p-Laplacian

Abstract

Abstract. We describe in this paper the asymptotic behaviour in Sobolev spaces of sequences of solutions of critical equations involving the p-Laplacian (see equations (Eα) below) on a compact Riemannian manifold (M, g) which are invariant by a subgroup of the group of isometries of (M, g). We also prove pointwise estimates.2000 Mathematics Subject Classification: 35B40, 35J20, 35B33

Multiplicity of nodal solutions to the Yamabe problem

“…Anisotropic critical problems with symmetries were treated in [10]. Palais-Smale sequences of positive functions for some Yamabe-type problems on a closed manifold were described by Druet, Hebey and Robert in [17], and symmetric ones were treated in [31].…”

Compactness results for divergence type nonlinear elliptic equations

“…Compactness and existence problems for second order nonlinear elliptic equations on compact Riemannian manifolds have been extensively studied by several authors and important advances have been recently obtained, see for example [12,14,15,21,[23][24][25]27,28]. These works were concerned with equations involving critical growth of the kind −div g |∇ g u| p−2 ∇ g u + a α (x)|u| p−2 u = |u| p * −2 u in M for a family of continuous coefficients a α and a metric g on a compact manifold M. Part of them are strongly connected with the first-order Riemannian Sobolev best constant theory which have been developed during the past decades, see [3,13,18] for a complete survey on this theory.…”

“…Using (27), (28) and the part (b) of Lemma 4.2, we obtain the C 1 regularity of u 0 from a well-known elliptic regularity result due to Tolksdorf and its positivity from a strong maximum principle for p-regular equations due to Pucci and Serrin (Theorem 11.1 of [26]). …”

Best constant and Mountain-Pass solutions for a supercritical Hardy-Sobolev problem in the presence of symmetries

“…In the present work, we consider the more intricate case where G does not act freely. We make the fundamental Assumption 8 on G inspired by the work of Saintier [16] (more details are in Section 2): note that under this assumption, we cannot consider problem (1) on the quotient M/G since it is not necessarily a manifold.…”