Soliton solutions for quasilinear Schrödinger equations: The critical exponential case

Quasilinear elliptic equations in R 2 of second order with critical exponential growth are considered. By using a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H 1 ( R 2 ) and satisfy the geometric hypot...

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Bibliographic Details
Published in:Nonlinear analysis 2007-12, Vol.67 (12), p.3357-3372
Main Authors: do Ó, João M.B., Miyagaki, Olímpio H., Soares, Sérgio H.M.
Format: Article
Language:English
Subjects:
Calculus of variations and optimal control
Critical exponents
Elliptic equations
Exact sciences and technology
Global analysis, analysis on manifolds
Mathematical analysis
Mathematics
Partial differential equations
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Trudinger–Moser inequality
Variational methods
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Summary:Quasilinear elliptic equations in R 2 of second order with critical exponential growth are considered. By using a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H 1 ( R 2 ) and satisfy the geometric hypotheses of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v . In the proof that v is nontrivial, the main tool is the concentration–compactness principle [P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. H. Poincaré Anal. Non. Linéaire 1 (1984) 109–145, 223–283] combined with test functions connected with optimal Trudinger–Moser inequality.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2006.10.018