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E-raamat: Variational Methods for Potential Operator Equations: With Applications to Nonlinear Elliptic Equations

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Variational Methods for Potential Operator Equations: With Applications to Nonlinear Elliptic EquationsSuurem pilt
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The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob. Titles in planning includeFlavia Smarazzo and Alberto Tesei, Measure Theory: Radon Measures, Young Measures, and Applications to Parabolic Problems (2019)Elena Cordero and Luigi Rodino, Time-Frequency Analysis of Operators (2019)Mark M. Meerschaert, Alla Sikorskii, and Mohsen Zayernouri, Stochastic and Computational Models for Fractional Calculus, second edition (2020)Mariusz Lemaczyk, Ergodic Theory: Spectral Theory, Joinings, and Their Applications (2020)Marco Abate, Holomorphic Dynamics on Hyperbolic Complex Manifolds (2021)Miroslava Anti, Joeri Van der Veken, and Luc Vrancken, Differential Geometry of Submanifolds: Submanifolds of Almost Complex Spaces and Almost Product Spaces (2021)Kai Liu, Ilpo Laine, and Lianzhong Yang, Complex Differential-Difference Equations (2021)Rajendra Vasant Gurjar, Kayo Masuda, and Masayoshi Miyanishi, Affine Space Fibrations (2022)
1 Constrained minimization
1(38)
1.1 Preliminaries
1(7)
1.2 Constrained minimization
8(5)
1.3 Dual method
13(1)
1.4 Minimizers with the least energy
14(1)
1.5 Application of dual method
15(2)
1.6 Multiple solutions of nonhomogeneous equation
17(2)
1.7 Sets of constraints
19(5)
1.8 Constrained minimization for F(f)
24(5)
1.9 Subscritical problem
29(1)
1.10 Application to the p-Laplacian
30(5)
1.11 Critical problem
35(2)
1.12 Bibliographical notes
37(2)
2 Applications of Lusternik-Schnirelman theory
39(35)
2.1 Palais-Smale condition, case p XXX q
39(1)
2.2 Duality mapping
40(3)
2.3 Palais-Smale condition, case p = q
43(4)
2.4 The Lusternik-Schnirelman theory
47(8)
2.5 Case p > q
55(1)
2.6 Case p less than q
56(4)
2.7 Case p = q
60(3)
2.8 The p-Laplacian in bounded domain
63(4)
2.9 Iterative construction of eigenvectors
67(3)
2.10 Critical points of higher order
70(3)
2.11 Bibliographical notes
73(1)
3 Nonhomogeneous potentials
74(41)
3.1 Preliminaries and assumptions
74(2)
3.2 Constrained minimization
76(3)
3.3 Application -- compact case
79(2)
3.4 Perturbation theorems -- noncompact case
81(4)
3.5 Perturbation of the functional a -- noncompact case
85(3)
3.6 Existence of infinitely many solutions
88(2)
3.7 General minimization -- case p > q
90(9)
3.8 Set of constraints V
99(2)
3.9 Application to a critical case p = n
101(2)
3.10 Technical lemmas
103(9)
3.11 Existence result for problem (3.34)
112(1)
3.12 Bibliographical notes
113(2)
4 Potentials with covariance condition
115(13)
4.1 Preliminaries and constrained minimization
115(5)
4.2 Dual method
120(1)
4.3 Minimization subject to constraint V
120(1)
4.4 Sobolev inequality
121(1)
4.5 Mountain pass theorem and constrained minimization
122(3)
4.6 Minimization problem for a system of equations
125(2)
4.7 Bibliographical notes
127(1)
5 Eigenvalues and level sets
128(21)
5.1 Level sets
128(2)
5.2 Continuity and monotonicity of XXX
130(2)
5.3 The differentiability properties of XXX
132(3)
5.4 Schechter's version of the mountain pass theorem
135(3)
5.5 General condition for solvability of (5.11)
138(2)
5.6 Properties of the function k(t)
140(2)
5.7 Hilbert space case
142(1)
5.8 Application to elliptic equations
143(5)
5.9 Bibliographical notes
148(1)
6 Generalizations of the mountain pass theorem
149(18)
6.1 Version of a deformation lemma
149(4)
6.2 Mountain pass alternative
153(2)
6.3 Consequences of mountain pass alternative
155(2)
6.4 Hampwile alternative
157(3)
6.5 Applicability of the mountain pass theorem
160(3)
6.6 Mountain pass and Hampwile alternative
163(3)
6.7 Bibliographical notes
166(1)
7 Nondifferentiable functionals
167(31)
7.1 Concept of a generalized gradient
167(5)
7.2 Generalized gradients in function spaces
172(2)
7.3 Mountain pass theorem for locally Lipschitz functionals
174(7)
7.4 Consequences of Theorem 7.3.1
181(2)
7.5 Application to boundary value problem with discontinuous nonlinearity
183(2)
7.6 Lower semicontinuous perturbation
185(3)
7.7 Deformation lemma for functionals satisfying condition (L)
188(7)
7.8 Application to variational inequalities
195(2)
7.9 Bibliographical notes
197(1)
8 Concentration-compactness principle -- subcritical case
198(26)
8.1 Concentration-compactness principle at infinity -- subcritical case
198(2)
8.2 Constrained minimization -- subcritical case
200(5)
8.3 Constrained minimization with b XXX const, subcritical case
205(6)
8.4 Behaviour of the Palais-Smale sequences
211(4)
8.5 The exterior Dirichlet problem
215(3)
8.6 The Palais-Smale condition
218(3)
8.7 Concentration-compactness principle I
221(2)
8.8 Bibliographical notes
223(1)
9 Concentration-compactness principle -- critical case
224(29)
9.1 Critical Sobolev exponent
224(4)
9.2 Concentration-compactness principle II
228(1)
9.3 Loss of mass at infinity
229(4)
9.4 Constrained minimization -- critical case
233(4)
9.5 Palais-Smale sequences in critical case
237(7)
9.6 Symmetric solutions
244(6)
9.7 Remarks on compact embeddings into L(2*) (Q) and L(2*K)(R(n))
250(2)
9.8 Bibliographical notes
252(1)
Appendix 253(17)
A.1 Sobolev spaces 253(1)
A.2 Embedding theorems 254(1)
A.3 Compact embeddings of spaces W(1, p)(R(n)) and D(1, p)(R(n)) 255(4)
A.4 Conditions of concentration and uniform decay at infinity 259(2)
A.5 Compact embedding for H(1r)(R(n)) 261(3)
A.6 Schwarz symmetrization 264(1)
A.7 Pointwise convergence 264(2)
A.8 Gateaux derivatives 266(4)
Bibliography 270(17)
Glossary 287(2)
Index 289


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