A compression type mountain pass theorem in conical shells (original) (raw)

A Mountain Pass Theorem for a Suitable Class of Functions

Rocky Mountain Journal of Mathematics, 2009

The main purpose of this paper is to establish a three critical points result without assuming the coercivity of the involved functional. To this end, a mountain-pass theorem, where the usual Palais-Smale condition is not requested, is presented. These results are then applied to prove the existence of three solutions for a two-point boundary value problem with no asymptotic conditions.

Mountain Pass Theorems for Non-differentiable

We present some versions of the Mountain Pass Theorem of Ambrosetti and Rabinowitz for locally Lipschitz functionals. A multivalued elliptic problem is solved as an application of these results.

Mountain pass solutions to equations of p-Laplacian type

Nonlinear Analysis: Theory, Methods & Applications, 2003

This work is devoted to study the existence of solutions to equations of p-Laplacian type. We prove the existence of at least one solution, and under further assumptions, the existence of inÿnitely many solutions. In order to apply mountain pass results, we introduce a notion of uniformly convex functional that generalizes the notion of uniformly convex norm. ?

A Mountain Pass-type Theorem for Vector-valued Functions

Set-valued and Variational Analysis

The mountain pass theorem for scalar functionals is a fundamental result of the minimax methods in variational analysis. In this work we extend this theorem to the class of mathcalC1\mathcal{C}^{1}mathcalC1 functions f:mathbbRnrightarrowmathbbRmf:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}f:mathbbRnrightarrowmathbbRm , where the image space is ordered by the nonnegative orthant mathbbR+m\mathbb{R}_{+}^{m}mathbbR+m . Under suitable geometrical assumptions, we prove the existence of a critical point of f and we localize this point as a solution of a minimax problem. We remark that the considered minimax problem consists of an inner vector maximization problem and of an outer set-valued minimization problem. To deal with the outer set-valued problem we use an ordering relation among subsets of mathbbRm\mathbb{R}^{m}mathbbRm introduced by Kuroiwa. In order to prove our result, we develop an Ekeland-type principle for set-valued maps and we extensively use the notion of vector pseudogradient.

Asymptotics of solutions of second order parabolic equations near conical points and edges

Boundary Value Problems, 2014

The authors consider the first boundary value problem for a second order parabolic equation with variable coefficients in a domain with conical points or edges. In the first part of the paper, they study the Green function for this problem in the domain K × R n-m , where K is an infinite cone in R m , 2 ≤ m ≤ n. They obtain the asymptotics of the Green function near the vertex (n = m) and edge (n > m), respectively. This result is applied in the second part of the paper, which deals with the initial-boundary value problem in this domain. Here, the right-hand side f of the differential equation belongs to a weighted L p space. At the end of the paper, the initial-boundary value problem in a bounded domain with conical points or edges is studied. http://www.boundaryvalueproblems.com/content/2014/1/252

The intrinsic mountain pass principle

Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 1999

In this Note we present a general mountain pass principle dropping any smoothness or even continuity assumptions on the functional. As a corollary, we prove existence of a point with an arbitrarily fixed small slope if the "low" part of the "barrier" is sufficiently far from the "boundary", obtaining in addition estimates for the location of the point. © 1999 Academie des Sciences/Editions scientifiques et medicales Elsevier SAS