Soit Ω un domaine borne dans R n avec n≥3 On etudie l'existence d'une fonction u satisfaisant l'equation elliptique non lineaire -Δu=u P +f(x,u) sur Ω, u>0 sur Ω, u=0 sur ∂Ω, ou p=(n+2)/(n−2), f(x,0)=0 et f(x,u) est une perturbation de u P de bas ordre au sens ou lim u→+α f(x,u)/u P =0

Positive solutions of nonlinear elliptic equations involving critical sobolev exponents

This paper concerns the existence of standing wave solutions of nonlinear Schrodinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation: (*)
 
$$ - \Delta u + b(x)u = f(x, u), x \in \mathbb{R}^n .$$

 The functionf is assumed to be “superlinear”. A special case is the power nonlinearityf(x, z)=∥z∥
s−1
z 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 ∈W
1,2(ℝ
 n
 ) are established.

On a class of nonlinear Schro¨dinger equations

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

The ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications. However, in observational studies, the variables of interest (e.g., education, prices) are often endogenous, making conventional quantile regression inconsistent and hence inappropriate for recovering the causal effects of these variables on the quantiles of economic outcomes. In order to address this problem, we develop a model of quantile treatment effects (QTE) in the presence of endogeneity and obtain conditions for identification of the QTE without functional form assumptions. The principal feature of the model is the imposition of conditions that restrict the evolution of ranks across treatment states. This feature allows us to overcome the endogeneity problem and recover the true QTE through the use of instrumental variables. The proposed model can also be equivalently viewed as a structural simultaneous equation model with nonadditive errors, where QTE can be interpreted as the structural quantile effects (SQE).

/pdf/an-iv-model-of-quantile-treatment-effects-2j8id40hla.pdf

An IV Model of Quantile Treatment Effects

A Model Problem.- Semiconcave Functions.- Generalized Gradients and Semiconcavity.- Singularities of Semiconcave Functions.- Hamilton-Jacobi Equations.- Calculus of Variations.- Optimal Control Problems.- Control Problems with Exit Time.

/pdf/semiconcave-functions-hamilton-jacobi-equations-and-optimal-1vuumifya9.pdf

Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control

Dual variational methods in critical point theory and applications

This is an introduction to nonlinear functional analysis, in particular to those methods based on differential calculus in Banach spaces. It is in two parts; the first deals with the geometry of Banach spaces and includes a discussion of local and global inversion theorems for differentiable mappings. In the second part, the authors are more concerned with bifurcation theory, including the Hopf bifurcation. They include plenty of motivational and illustrative applications, which indeed provide much of the justification of nonlinear analysis. In particular, they discuss bifurcation problems arising from such areas as mechanics and fluid dynamics. The book is intended to accompany upper division courses for students of pure and applied mathematics and physics; exercises are consequently included.

A primer of nonlinear analysis

In this paper, we study the problem \[ \left\{\begin{array}{@{}l} -\Delta u + u + V(x)u = u^p,\\[3pt] -\Delta V = \lambda u^2, \quad \displaystyle\lim_{|x| \to +\infty} V(x)=0, \end{array}\right. \] where u, V : ℝ3 → ℝ are radial functions, λ > 0 and 1 < p < 5. We give multiplicity results, depending on p and on the parameter λ.

Multiple bound states for the Schroedinger-Poisson problem

We deal with a class of systems of nonlinear Schrodinger equations, proving the existence of bound and ground states provided the coupling parameter is small and large, respectively.

http://icm2006.org/v_f/AbsDef/Shorts/abs_0731.pdf

Standing waves of some coupled nonlinear Schrödinger equations

Preface 1. Preliminaries Part I. Topological Methods: 2. A primer on bifurcation theory 3. Topological degree, I 4. Topological degree, II: global properties Part II. Variational Methods, I: 5. Critical points: extrema 6. Constrained critical points 7. Deformations and the Palais-Smale condition 8. Saddle points and min-max methods Part III. Variational Methods, II: 9. Lusternik-Schnirelman theory 10. Critical points of even functionals on symmetric manifolds 11. Further results on Elliptic Dirichlet problems 12. Morse theory Part IV. Appendices: Appendix 1. Qualitative results Appendix 2. The concentration compactness principle Appendix 3. Bifurcation for problems on Rn Appendix 4. Vortex rings in an ideal fluid Appendix 5. Perturbation methods Appendix 6. Some problems arising in differential geometry Bibliography Index.

/pdf/nonlinear-analysis-and-semilinear-elliptic-problems-23z6q620rj.pdf

Antonio Ambrosetti

Papers

Dual variational methods in critical point theory and applications

A primer of nonlinear analysis

Multiple bound states for the Schroedinger-Poisson problem

Standing waves of some coupled nonlinear Schrödinger equations

Nonlinear Analysis and Semilinear Elliptic Problems