Manuel del Pino

Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions

Abstract In this paper, we find optimal constants of a special class of Gagliardo–Nirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of the intermediate asymptotics of solutions to nonlinear diffusion equations.

Multi-peak bound states for nonlinear Schrödinger equations

Abstract In this paper we consider the study of standing wave solutions for a nonlinear Schrodinger equation. This problem reduces to that of finding nonnegative solutions of ϵ 2 δu − V(x)u+ƒ(u)− in ω with finite energy. Here ϵ is a small parameter, ω is a smooth, possibly unbounded domain, f is an appropriate superlinear function, and V is a positive potential, bounded away from zero. It is the purpose of this article to obtain multi-peak solutions in the “multiple well case”. We find solutions exhibiting concentration at any prescribed finite set of local minima, possibly degenerate, of the potential. The proof relies on variational arguments, where a penalization-type method is developed for the identification of the desired solutions.

Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition

We show existence for a nonlinear fourth-order boundary value problem under a nonresonance condition involving a two-parameter linear eigenvalue problem. We also state extensions of this result to certain higher-order P.D.E. cases

Existence and multiplicity of solutions with prescribed period for a second order quasilinear O.D.E.

MANUEL A. DEL PINO School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. RAIL F. MANASEVICH Departamento de Matematicas, F.C.F.M., Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile and ALEJANDRO E. MUR~A Division of Applied Mathematics, Brown University, Providence, RI 02912, U.S.A. (Received 20 September 1990; received for publication 13 March 1991)

ON THE ROLE OF MEAN CURVATURE IN SOME SINGULARLY PERTURBED NEUMANN PROBLEMS

We construct solutions exhibiting a single spike-layer shape around some point of the boundary as

The optimal Euclidean Lp-Sobolev logarithmic inequality☆

\var \to 0

A global estimate for the gradient in a singular elliptic boundary value problem

for the problem \left\{ \begin{array}{l} \var^2 \tri u - u + u^p = 0 \quad \mbox{in} \ \Omega, \\ u > 0 \quad \mbox{in} \ \Omega, \\ \frac{\partial u}{\partial \nu} = 0 \quad \mbox{on} \ \partial \Omega, \end{array} \right.\label {1.1} where

Bubble-tower radial solutions in the slightly supercritical Brezis–Nirenberg problem

\Omega

ASYMPTOTIC BEHAVIOR OF BEST CONSTANTS AND EXTREMALS FOR TRACE EMBEDDINGS IN EXPANDING DOMAINS1

is a bounded domain with smooth boundary in

Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking

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