Minimizers and symmetric minimizers for problems with critical Sobolev exponents. (English) Zbl 1200.35128

Waliullah, Shoyeb

Topol. Methods Nonlinear Anal. 34, No. 2, 291-326 (2009).

The paper is concerned with elliptic partial differential equations arising from the problem of finding mappings realizing the best Sobolev constant fo the embedding \(D^{k,p}(\mathbb R^n)\hookrightarrow L^{p^*}(\mathbb R^n,Q)\), where the latter is the weighted space with weight function \(Q\). If \(Q\) is not constant, it has been known that in most cases no minimzer exists.
However, if one asks the same question for the corresponding spaces of \(G\)-invariant mappings with a compact group \(G\) acting isometrically on \((\mathbb R^n,Q)\), the situation may be different. By symmetric criticality, minimizers will still solve the Euler-Lagrange equation of the non-invariant problem. But in the symmetric case, under certain (slightly technical) assumptions, a concentration-compactness lemma can be proven which enables the author to conclude the existence of \(G\)-minimizers. The equations thus solved are of the form \(\Delta_p u=Q|u|^{p^*-2}u\), for suitable \(Q\), and generalizations thereof.

Reviewer: Andreas Gastel (Duisburg)

MSC:

35J60	Nonlinear elliptic equations
58E15	Variational problems concerning extremal problems in several variables; Yang-Mills functionals
35J20	Variational methods for second-order elliptic equations
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

concentration-compactness principle; critical Sobolev exponent; symmetric solutions of elliptic equations; Sobolev embeddings in weighted spaces