Search results for "Mountain pass theorem"

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.

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 ( ] − ∞ , …

Multiple nodal solutions for semilinear robin problems with indefinite linear part and concave terms

2017

We consider a semilinear Robin problem driven by Laplacian plus an indefinite and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal solutions which converge to zero in $C^1(\overline{\Omega})$. If the coefficient of the concave term is sign changing, then again we produce a sequence of smooth solutions converging to zero in $C^1(\overline{\Omega})$, but we cannot claim that they are nodal.

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.

A Mountain Pass Theorem for a Suitable Class of Functions

2009

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.