Concentration-compactness at the mountain pass level in semilinear elliptic problems

2007/03/28 by Kyril Tintarev, TIntarev, Kyril
Computer Science · Mathematics · #35J20 #35J60 #49J35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.math/0703851

openalex publication_date 2007/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The concentration compactness framework for semilinear elliptic equations without compactness, set originally by P.-L.Lions for constrained minimization in the case of homogeneous nonlinearity, is extended here to the case of general nonlinearities in the standard mountain pass setting of Ambrosetti-Rabinowitz. In these setting, existence of solutions at the mountain pass level is verified under a single penalty condition analogous to that in the Lions' case. Problems on the whole space and problems with critical nonlinearity are considered. Particular attention is given to nonhomogeneous critical nonlinearities that oscillate about the "critical stem".

Citations

• A relation between pointwise convergence of functions and convergence of functionals

Related

• Concentration-compactness principle for mountain pass problems
• An infinite-dimensional mountain pass theorem with applications to nonlinear elliptic systems
• The concentration-compactness principle for Orlicz spaces and applications
• Normalized concentrating solutions to nonlinear elliptic problems
• Fractional elliptic problems with critical growth in the whole of \Rⁿ