Variational Methods for Potential Operator Equations

Abstract

Part 1 Constrained minimization: preliminaries constrained minimization dual method minimizers with the least energy application of dual method multiple solutions of nonhomogeneous equation sets of constraints constrained minimization for Ff subcritical problem application to the p-Laplacian critical problem. Part 2 Applications of Lusternik-Schnirelman theory: Palais-Smale condition, case p not equal to q duality mapping Palais-Smale condition, case p=q the Lusternik-Schnirelman theory case p>q case p q set of constraints V application to a critical case p=n technical lemmas existence result for problem (3.34). Part 4 Potentials with covariance condition: preliminaries and constrained minimization dual method minimization subject to constraint V Sobol inequality mountain pass theorem and constrained minimization minimization problem for a system of equations. Part 5 Eigenvalues and level sets: level sets continuity and monotonicity of delta the differentiability properties of delta Schechters's version of the mountain pass theorem general condition for solvability of (5.11) properties of the function K(t) Hilbert space case application to elliptic equations. Part 6 Generalizations of the mountain pass theorem: version of a deformation lemma mountain pass alternative consequences of mountain pass alternative Hampwile alternative applicability of the mountain pass theorem mountain pass and Hampwile alternative. Part 7 Nondifferentiable functionals: concept of a generalized gradient generalized gradients in function spaces mountain pass theorem for locally Lipschitz functionals consequences of theorem 7.3.1 application to boundary value problems with discontinuous nonlinearity lower semicontinuous perturbation deformation lemma for functionals satisfying condition (L) application to variational inequalities. Part 8 Concentration-compactness principle - subcritical case: concentration-compactness principle at infinity - subcritical case constrained minimization - subcritical case constrained minimization b not equal const, subcritical case behaviour of the Palais-Smale sequences the exterior Dirichlet problem the Palais-Smale condition concentration-compactness principle 1. Part 9 Concentration-compactness principle - critical case: critical Sobolev exponent concentration-compactness principle 2 - loss of mass at infinity constrained minimization - critical case - Palais-Smale sequences in critica case symmetric solutions remarks on compact embeddings into L2*(Q) and L2*(R) appendix.