Variational Methods for Potential Operator Equations

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 pq; 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.