Global Compactness Properties of Semilinear Elliptic Equations with Critical Exponential Growth

Abstract

Sequences of positive solutions to semilinear elliptic equations of critical exponential growth in the plane either are precompact in the Sobolev H 1 -topology or concentrate at isolated points of the domain. For energies allowing at most single-point blow-up, we establish a universal blow-up pattern near the concentration point and uniquely characterize the blow-up energy in terms of a geometric limiting problem.

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

“…We also prove a structural theorem for bounded sequences in W 1,N 0 (B), where B = B 1 (0) is a unit ball in R N , which is similar to Struwe's "global compactness" [12] and its subsequent refinements known in the case N > p. (Note that global compactness results in [3,11] while providing asymptotic behavior of bounded sequences in W 1,N 0 , use the different blowup transformations than this paper and end up with asymptotic profiles supported in the whole R N .) Our starting point for this result is the following version of global compactness, see [14,Theorem 5.1]:…”

“…[1][2][3]7])) as it deals with general bounded sequences rather than with critical sequences of specific functionals. Provided that a suitable extension of transformations (1.6) to non-radial functions is found, an analysis based such transformations appears to be more promising than the study of blow-ups based on the ination on the linear scale.…”

A global compactness result for singular elliptic problems involving critical Sobolev exponent

“…To do this, following the analogous argument in [14], for any x ∈ R N , we can find ρ ∈ [1,2] such that the solution φ m of the Dirichlet problems…”

“…In [16] S.Yan generalizes this global compactness result to the case of p-Laplacian successfully. Very recently, Adimurthi and M.Struwe in [1] have studied the convergence of (P.S.) sequences of the energy functional associated with a semilinear elliptic problem on a bounded domain in R 2 with critical nonlinearity f (s) growing like exp(4πs 2 ) as s → ∞.…”

Critical and subcritical elliptic systems in dimension two

“…The case of a single semilinear elliptic equation in bounded subsets of R 2 has been investigated by several authors, see for example [2,3,5,6]. It has been observed that criticality in dimension two is connected with the imbedding of…”