Paul H. Rabinowitz

Dual variational methods in critical point theory and applications

This paper contains some general existence theorems for critical points of a continuously differentiable functional I on a real Banach space. The strongest results are for the case in which I is even. Applications are given to partial differential and integral equations.

Minimax methods in critical point theory with applications to differential equations

An overview The mountain pass theorem and some applications Some variants of the mountain pass theorem The saddle point theorem Some generalizations of the mountain pass theorem Applications to Hamiltonian systems Functionals with symmetries and index theorems Multiple critical points of symmetric functionals: problems with constraints Multiple critical points of symmetric functionals: the unconstrained case Pertubations from symmetry Variational methods in bifurcation theory.

Some global results for nonlinear eigenvalue problems

In this paper we investigate the structure of the solution set for a large class of nonlinear eigenvalue problems in a Banach space. Our main results demonstrate the existence of continua, i.e., closed connected sets, of solutions of these equations. Although the emphasis is on the case when bifurcation occurs, the nonbifurcation situation is also treated. Applications are given to ordinary and partial differential equations and to integral equations.

On a class of nonlinear Schro¨dinger equations

AbstractThis paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation: (*)

Critical point theorems for indefinite functionals

- \Delta u + b(x)u = f(x, u), x \in \mathbb{R}^n .

On bifurcation from infinity

The functionf is assumed to be “superlinear”. A special case is the power nonlinearityf(x, z)=∥z∥s−1z where 1<s<(n+2)(n−2)−1. Making different assumptions onb(x), mainly at infinity, various sufficient conditions for the existence of nontrivial solutionsu ∈W1,2(ℝn) are established.

Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations

Abstract : Elliptic boundary value problems of the form Lu + g(x,u) in omega and u = 0 on the boundary of omega are studied where g is singular in that g(x,r) goes to infinity uniformly as r goes to zero from above. Existence of classical and generalized solutions is established and an associated nonlinear eigenvalue problem is treated. A detailed study is made of the behaviour of the solutions and their gradients near the boundary of omega. This leads to global estimates for the modulus of continuity of solutions. (Author)

Periodic and heteroclinic orbits for a periodic hamiltonian system

A variational principle of a minimax nature is developed and used to prove the existence of critical points for certain variational problems which are indefinite. The proofs are carried out directly in an infinite dimensional Hilbert space. Special cases of these problems previously had been tractable only by an elaborate finite dimensional approximation procedure. The main applications given here are to Hamiltonian systems of ordinary differential equations where the existence of time periodic solutions is established for several classes of Hamiltonians.