Search results for "Mountain pass"

Three solutions for a two-point boundary value problem with the prescribed mean curvature equation

2015

The existence of at least three classical solutions for a parametric ordinary Dirichlet problem involving the mean curvature operator are established. In particular, a variational approach is proposed and the main results are obtained simply requiring the sublinearity at zero of the considered nonlinearity.

A Mountain Pass Theorem for a Suitable Class of Functions

2009

Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles

2012

Abstract We study a quasilinear elliptic problem depending on a parameter λ of the form − Δ p u = λ f ( u ) in  Ω , u = 0 on  ∂ Ω . We present a novel variational approach that allows us to obtain multiplicity, regularity and a priori estimate of solutions by assuming certain growth and sign conditions on f prescribed only near zero. More precisely, we describe an interval of parameters λ for which the problem under consideration admits at least three nontrivial solutions: two extremal constant-sign solutions and one sign-changing solution. Our approach is based on an abstract localization principle of critical points of functionals of the form E = Φ − λ Ψ on open sublevels Φ − 1 ( ] − ∞ , …

Bounded Palais–Smale sequences for non-differentiable functions

2011

The existence of bounded Palais-Smale sequences (briefly BPS) for functionals depending on a parameter belonging to a real interval and which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function, is obtained when the parameter runs in a full measure subset of the given interval. Specifically, for this class of non-smooth functions, we obtain BPS related to mountain pass and to global infima levels. This is done by developing a unifying approach, which applies to both cases and relies on a suitable deformation lemma. © 2011 Elsevier Ltd. All rights reserved.

Superlinear (p(z), q(z))-equations

2017

AbstractWe consider Dirichlet boundary value problems for equations involving the (p(z), q(z))-Laplacian operator in the principal part and prove the existence of one and three nontrivial weak solutions, respectively. Here, the nonlinearity in the reaction term is allowed to depend on the solution, but does not satisfy the Ambrosetti–Rabinowitz condition. The hypotheses on the reaction term ensure that the Euler–Lagrange functional, associated to the problem, satisfies both the -condition and a mountain pass geometry.

Homoclinic Solutions of Nonlinear Laplacian Difference Equations Without Ambrosetti-Rabinowitz Condition

2021

The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using Ambrosetti-Rabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals.

An elliptic equation on n-dimensional manifolds

2020

We consider an elliptic equation driven by a p-Laplacian-like operator, on an n-dimensional Riemannian manifold. The growth condition on the right-hand side of the equation depends on the geometry of the manifold. We produce a nontrivial solution by using a Palais–Smale compactness condition and a mountain pass geometry.

Bifurcation phenomena for the positive solutions of semilinear elliptic problems with mixed boundary conditions

2016

We consider a parametric semilinear elliptic equation with a Cara-theodory reaction which exhibits competing nonlinearities. It is "concave" (sub-linear) near the origin and "convex" (superlinear) or linear near $+\infty$. Using variational methods based on the critical point theory, coupled with suitable truncation and comparison techniques, we prove a bifurcation-type theorem, describing the set of positive solutions as the parameter varies.

A min-max principle for non-differentiable functions with a weak compactness condition

2009

A general critical point result established by Ghoussoub is extended to the case of locally Lipschitz continuous functions satisfying a weak Palais-Smale hypothesis, which includes the so-called non-smooth Cerami condition. Some special cases are then pointed out.

Nonlinear scalar field equations with general nonlinearity

2018

Consider the nonlinear scalar field equation \begin{equation} \label{a1} -\Delta{u}= f(u)\quad\text{in}~\mathbb{R}^N,\qquad u\in H^1(\mathbb{R}^N), \end{equation} where $N\geq3$ and $f$ satisfies the general Berestycki-Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski [30] made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski's results on the scalar field equation. T…