Nassif Ghoussoub

MULTIPLE SOLUTIONS FOR QUASI-LINEAR PDES INVOLVING THE CRITICAL SOBOLEV AND HARDY EXPONENTS

We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: ( −4pu = λ|u|r−2u+ μ |u| q−2 |x|s u in Ω, u|∂Ω = 0, where λ and μ are two positive parameters and Ω is a smooth bounded domain in Rn containing 0 in its interior. The variational approach requires that 1 < p < n, p ≤ q ≤ p∗(s) ≡ n−s n−pp and p ≤ r ≤ p ∗ ≡ p∗(0) = np n−p , which we assume throughout. However, the situations differ widely with q and r, and the interesting cases occur either at the critical Sobolev exponent (r = p∗) or in the Hardy-critical setting (s = p = q) or in the more general Hardy-Sobolev setting when q = n−s n−pp. In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen dual sets. Many of the results are new even in the case p = 2, especially those corresponding to singularities (i.e., when 0 < s ≤ p).

Selected new aspects of the calculus of variations in the large

We discuss some of the recent developments in variational methods while emphasizing new applications to nonlinear problems. We touch on several issues: (i) the formulation of variational set-ups which provide more information on the location of critical points and therefore on the qualitative properties of the solutions of corresponding Euler-Lagrange equations; (ii) the relationships between the energy of variationally generated solutions, their Morse indices, and the Hausdorff measure of their nodal sets; (iii) the gluing of several topological obstructions; (iv) the preservation of critical levels after deformation of functionals; (v) and the various ways to recover compactness in certain borderline variational problems.

A general mountain pass principle for locating and classifying critical points

Abstract A general “Mountain Pass” principle that extends the theorem of Ambrosetti-Rabinowitz and which gives more information about the location of critical points, is established. This theorem also covers the problem of the “limiting case”, i.e. when “the separating mountain range has zero altitude”. It is also shown how this principle yields localized versions of recent results of Hofer and Pucci-Serrin concerning the structure of the critical set.

On the Partial Differential Equations of Electrostatic MEMS Devices: Stationary Case

We analyze the nonlinear elliptic problem

Gδ-embeddings in Hilbert space

\Delta u =\frac{\lambda f(x)}{(1+u)^2}

On the best possible remaining term in the Hardy inequality

on a bounded domain Ω of

Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth

R^N

The Conley index and the critical groups via an extension of Gromoll-Meyer theory

with Dirichlet boundary conditions. This equation models a simple electrostatic micro‐electromechanical system (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at ‐1. When a voltage—represented here by λ—is applied, the membrane deflects towards the ground plate, and a snap‐through may occur when it exceeds a certain critical value

A lattice renorming theorem and applications to vector-valued processes

\lambda^*

On a Fourth Order Elliptic Problem with a Singular Nonlinearity

(pull‐in voltage). This creates a so‐called pull‐in instability, which greatly affects the design of many devices. The mathematical model leads to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane, which will be considered in a forthcoming paper. Here, we focus on the stationary equation and on estimates for