Michel Willem

On An Elliptic Equation With Concave and Convex Nonlinearities

We study the semilinear elliptic equation -Delta u=lambdau(q-2)u+muu(p-2)u in an open bounded domain Omega subset of R(N) with Dirichlet boundary conditions; here 1 0 and mu is an element of R arbitrary there exists a sequence (upsilon(k)) of solutions with negative energy converging to 0 as k --> infinity. Moreover, for mu > 0 and lambda arbitrary there exists a sequence of solutions with unbounded energy. This answers a question of Ambrosetti, Brezis and Cerami. The main ingredient is a new critical point theorem, which guarantees the existence of infinitely many critical values of an even functional in a bounded range. We can also treat strongly indefinite functionals and obtain similar results for first-order Hamiltonian systems.

Partial symmetry of least energy nodal solutions to some variational problems

We investigate the symmetry properties of several radially symmetric minimization problems. The minimizers which we obtain are nodal solutions of superlinear elliptic problems, or eigenfunctions of weighted asymmetric eigenvalue problems, or they lie on the first curve in the Fucik spectrum. In all instances, we prove that the minimizers are foliated Schwarz symmetric. We give examples showing that the minimizers are in general not radially symmetric. The basic tool which we use is polarization, a concept going back to Ahlfors. We develop this method of symmetrization for sign changing functions.

Non-radial ground states for the Henon equation

We analyse symmetry breaking for ground states of the Henon equation [7] in a ball. Asymptotic estimates of the transition are also given when p is close to either 2 or 2*.

Nontrivial solution of a semilinear Schrodinger equation

This paper deals with strongly indefinite functionals whose gradients are Fredholm operators of index 0 and map weakly convergent sequences to weakly convergent sequences. We show bow these results apply to a Z(N)-invariant semilinear Schrodinger equation on R(N).

Elliptic problems with critical exponents and Hardy potentials

This paper is devoted to the existence of positive solutions of a Dirichlet problem with critical exponent and a singular potential. Under various assumption on the domain Omega, which include some kinds of unbounded domains, we prove the existence of ground states and of symmetric solutions

The Dirichlet problem for superlinear elliptic equations

Boundary value problems for nonlinear elliptic partial differential equations have been a major focus of research in nonlinear analysis for decades. This chapter focuses on some basic ideas in a simple setting and presents survey selected results on the Dirichlet problem for the equation given in this chapter with superlinear nonlinearity. The chapter consists of three sections: (1) discusses the positive solutions of the equation, (2) focuses on sign-changing solutions on bounded domains, and (3) treats the unbounded domain Ω =ℝ N . No effort is being made to be as general as possible. This chapter does not present results on the bifurcation of solutions nor for the p-Laplace operator, nor does it treat singularly perturbed equations in detail.

Critical points of convex perturbations of some indefinite quadratic forms and semi-linear boundary value problems at resonance

Abstract Critical points of convex perturbations of indefinite quadratic forms are obtained from the dual least action principle. The main result leads to necessary and sufficient conditions for the existence of a critical point when the corresponding Euler equation is scalar or when the perturbation is strictly convex. Applications are given to periodic solution of hamiltonian systems and to systems of semi-linear beam equations. A global version of the averaging method is given.

Nonlinear Schrodinger equations with unbounded and decaying radial potentials

We establish some embedding results of weighted Sobolev spaces of radially symmetric functions. The results are then used to obtain ground state solutions of nonlinear Schrodinger equations with unbounded and decaying radial potentials. Our work unifies and generalizes many existing partial results in the literature.

Remarks on a Hardy-Sobolev inequality

We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics

Subharmonic oscillations of forced pendulum-type equations

where f is periodic with minimal period T and mean value zero. We have in mind as a particular case the pendulum equation, where g(x) = A sin x. First results on the existence of subharmonic orbits in a neighborhood of a given periodic motion were obtained by Birkhoff and Lewis (cf. [3] and [ 143) by perturbation-type techniques. Rabinowitz [ 151 was able to prove the existence of subharmonic solutions for Hamiltonian systems by the use of variational methods. His approach is not of local type like the one in [3], and enables one to obtain a sequence of solutions whose minimal period tends toward infinity in the case when the Hamiltonian function has subquadratic or superquadratic growth. These results have been extended in various directions, cf. [2, 5, 6, 8, 13, 16-181. Local results on subharmonies for the forced pendulum equation can be found in [19]. Hamiltonian systems with periodic nonlinearity were studied by Conley and Zehnder [6]. They proved the existence of subharmonic solutions under some assumptions on the nondegenerateness of the solutions, by the use of Morse-Conley theory. In this paper we will prove the existence of subharmonic oscillations of a pendulum-type equation by the use of classical Morse theory together with an iteration formula for the index due to Bott [4] and developed in [7] and [l].