Weak Convergence Methods for Semilinear Elliptic EquationsThis book deals with nonlinear boundary value problems for semilinear elliptic equations on unbounded domains with nonlinearities involving the subcritical Sobolev exponent. The variational problems investigated in the book originate in many branches of applied science. A typical example is the nonlinear Schr dinger equation which appears in mathematical modeling phenomena arising in nonlinear optics and plasma physics. Solutions to these problems are found as critical points of variational functionals. The main difficulty in examining the compactness of Palais-Smale sequences arises from the fact that the Sobolev compact embedding theorems are no longer true on unbounded domains. In this book we develop the concentration-compactness principle at infinity, which is used to obtain the relative compactness of minimizing sequences. This tool, combined with some basic methods from the Lusternik-Schnirelman theory of critical points, is to investigate the existence of positive, symmetric and nodal solutions. The book also emphasizes the effect of the graph topology of coefficients on the existence of multiple solutions. |
Contents
Preface | 1 |
Constrained minimization | 21 |
10 | 32 |
23 | 41 |
Nonlinear eigenvalue problem | 67 |
Artificial constraints | 89 |
3 | 103 |
Inverse power method | 119 |
3 | 129 |
Effect of topology | 135 |
5 | 149 |
Multipeak solutions | 159 |
Multiple positive and nodal solutions | 205 |
| 225 | |
| 233 | |
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Common terms and phrases
Arguing indirectly assume bounded in H¹(RN coefficient compact in H¹(RN completes the proof concentration-compactness principle contradiction critical point deduce define Dirichlet problem dx RN dy RN dz RN eigenvalue Ekeland's variational principle elliptic equations exist constants existence result exists a sequence F(um Hardy inequality Hence Hölder Hölder inequality I(um implies large-interior domain Lemma lim lim sup minimization problem minimizing sequence multi-peak solution nodal solutions nonlinear Schrödinger equation Palais-Smale condition Palais-Smale sequence PJRN positive solutions proof of Proposition Proposition 1.2 relatively compact result follows RN RN satisfying Sobolev embedding theorem Sobolev inequalities Sobolev spaces solvable sufficiently large Suppose u² dx um(x unbounded domains uniformly variational functional Vu² Vul² weakly convergent μη نه
References to this book
Concentration Compactness: Functional-analytic Grounds and Applications Kyril Tintarev,Karl-Heinz Fieseler No preview available - 2007 |