CRITICAL POINT THEOREMS AND APPLICATIONS TO SOME NONLINEAR PROBLEMS WITH -STRONG” RESONANCE AT INFINITY + P. BARTOLO and V. BENCI Istituto di Matematica Applicata. Via Re David. X0-Bari. Ital) and Istituto di Analisi Matematica. Palazzo Ateneo. L’ia Scolai. 2-Bari. Italy (Received in revised form 20 December 1952)

Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity

Progress in Nonlinear Differential Equations and their Applications

The Schrödinger–Poisson equation under the effect of a nonlinear local term

The existence of semiclassical states to some nonlinear Schrodinger equations that concentrate near the critical points of the potential V is studied by means of a local approach, variational in nature. We also discuss stability and necessary conditions for concentration. The same method is used to find multiple homoclinic orbits to a class of second‐order Hamiltonian systems.

Semiclassical States of Nonlinear Schrödinger Equations

Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3

Examples and Motivations.- Pertubation in Critical Point Theory.- Bifurcation from the Essential Spectrum.- Elliptic Problems on ?n with Subcritical Growth.- Elliptic Problems with Critical Exponent.- The Yamabe Problem.- Other Problems in Conformal Geometry.- Nonlinear Schrodinger Equations.- Singularly Perturbed Neumann Problems.- Concentration at Spheres for Radial Problems.

Perturbation Methods and Semilinear Elliptic Problems on R^n

We discuss some recent results dealing with the existence of bound states of the nonlinear Schrodinger-Poisson system

 $$\left\{ \begin{gathered} - \Delta u + V(x)u + \lambda K(x)\phi (x)u = |u|^{{p - 1}} u, \hfill \\ - \Delta \phi = K(x)u^{2}, \hfill \\ \end{gathered} \right.$$
 as well as of the corresponding semiclassical limits. The proofs are based upon Critical Point theory and Perturbation Methods.

On Schrödinger-Poisson Systems

We introduce a variational approach to obtain some Poincare-Melnikov type results on the existence and multiplicity of homoclinics.

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Antonio Ambrosetti

Papers

Semiclassical States of Nonlinear Schrödinger Equations

Perturbation Methods and Semilinear Elliptic Problems on R^n

On Schrödinger-Poisson Systems

Homoclinics: Poincaré-Melnikov type results via a variational approach

Perturbation of Δu+u(N+2)/(N−2)=0, the Scalar Curvature Problem in RN, and Related Topics