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

Description

1 Constrained minimization 1.1 Preliminaries 1.2 Constrained minimization 1.3 Dual method 1.4 Minimizers with the least energy 1.5 Application of dual method 1.6 Multiple solutions of nonhomogeneous equation 1.7 Sets of constraints 1.8 Constrained minimization for Ff 1.9 Subcritical problem 1.10 Application to the p-Laplacian 1.11 Critical problem 1.12 Bibliographical notes 2 Applications of Lusternik-Schnirelman theory 2.1 Palais-Smale condition, case ρ ≠ q 2.2 Duality mapping 2.3 Palais-Smale condition, case ? = q 2.4 The Lusternik-Schnirelman theory 2.5 Case ρ > q 2.6 Case ρ q 3.8 Set of constraints V 3.9 Application to a critical case ρ = n 3.10 Technical lemmas 3.11 Existence result for problem (3.34) 3.12 Bibliographical notes 4 Potentials with covariance condition 4.1 Preliminaries and constrained minimization 4.2 Dual method 4.3 Minimization subject to constraint V 4.4 Sobolev inequality 4.5 Mountain pass theorem and constrained minimization 4.6 Minimization problem for a system of equations 4.7 Bibliographical notes 5 Eigenvalues and level sets 5.1 Level sets 5.2 Continuity and monotonicity of σ 5.3 The differentiability properties of σ 5.4 Schechter's version of the mountain pass theorem 5.5 General condition for solvability of (5.11) 5.6 Properties of the function κ(t) 5.7 Hilbert space case 5.8 Application to elliptic equations 5.9 Bibliographical notes 6 Generalizations of the mountain pass theorem 6.1 Version of a deformation lemma 6.2 Mountain pass alternative 6.3 Consequences of mountain pass alternative 6.4 Hampwile alternative 6.5 Applicability of the mountain pass theorem 6.6 Mountain pass and Hampwile alternative 6.7 Bibliographical notes 7 Nondifferentiable functionals 7.1 Concept of a generalized gradient 7.2 Generalized gradients in function spaces 7.3 Mountain pass theorem for locally Lipschitz functionals 7.4 Consequences of Theorem 7.3.1 7.5 Application to boundary value problem with discontinuous nonlinearity 7.6 Lower semicontinuous perturbation 7.7 Deformation lemma for functionals satisfying condition (L) 7.8 Application to variational inequalities 7.9 Bibliographical notes 8 Concentration-compactness principle — subcritical case 8.1 Concentration-compactness principle at infinity — subcritical case 8.2 Constrained minimization — subcritical case 8.3 Constrained minimization with b ≢ φ const, subcritical case 8.4 Behaviour of the Palais-Smale sequences 8.5 The exterior Dirichlet problem 8.6 The Palais-Smale condition 8.7 Concentration-compactness principle I 8.8 Bibliographical notes 9 Concentration-compactness principle — critical case 9.1 Critical Sobolev exponent 9.2 Concentration-compactness principle II 9.3 Loss of mass at infinity 9.4 Constrained minimization — critical case 9.5 Palais-Smale sequences in critical case 9.6 Symmetric solutions 9.7 Remarks on compact embeddings into L2*(Q) and L2*K (R)n 9.8 Bibliographical notes Appendix A.1 Sobolev spaces A.2 Embedding theorems A.3 Compact embeddings of spaces W1,p(Rn) and D1,p(Rn) A.4 Conditions of concentration and uniform decay at infinity A.5 Compact embedding for H1R (Rn) A.6 Schwarz symmetrization A.7 Pointwise convergence A.8 Gâteaux derivatives Bibliography Glossary Index