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  1. Calculus of Variations and Partial Differential Equations
  2. Calculus of Variations and Partial Differential Equations : Volume 55
  3. Calculus of Variations and Partial Differential Equations : Volume 55, Issue 4, August 2016
  4. A generalized mountain-pass theorem: the existence of multiple critical points
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Calculus of Variations and Partial Differential Equations : Volume 56
Calculus of Variations and Partial Differential Equations : Volume 55
Calculus of Variations and Partial Differential Equations : Volume 55, Issue 6, December 2016
Calculus of Variations and Partial Differential Equations : Volume 55, Issue 5, October 2016
Calculus of Variations and Partial Differential Equations : Volume 55, Issue 4, August 2016
A Minkowski type inequality in space forms
Hölder regularity of the solution to the complex Monge-Ampère equation with $$L^p$$ density
Kazdan–Warner equation on graph
Uniqueness of weak solutions for the normalised Ricci flow in two dimensions
Variational mean curvatures in abstract measure spaces
Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space
Bifurcation on compact spin manifold
On supercritical Sobolev type inequalities and related elliptic equations
Regularity of harmonic discs in spaces with quadratic isoperimetric inequality
A generalized mountain-pass theorem: the existence of multiple critical points
$$C^{1,\alpha }$$ estimates for the fully nonlinear Signorini problem
Regularity and the behavior of eigenvalues for minimizers of a constrained Q-tensor energy for liquid crystals
A doubly nonlinear evolution for the optimal Poincaré inequality
An overdetermined problem for the anisotropic capacity
On a Lotka–Volterra competition model: the effects of advection and spatial variation
Refined asymptotics of the Teichmüller harmonic map flow into general targets
Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group
Graph properties for nonlocal minimal surfaces
The Brezis–Nirenberg problem for the fractional p-Laplacian
Best constants and existence of maximizers for weighted Trudinger–Moser inequalities
Reduced measures for semilinear elliptic equations involving Dirichlet operators
Integrable boundaries and fractals for Hölder classes; the Gauss–Green theorem
Two Whyburn type topological theorems and its applications to Monge–Ampère equations
Mean curvature flow of contractions between Euclidean spaces
The interior regularity of the Calabi flow on a toric surface
Deformations of Q-curvature I
Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential
Local Li–Yau’s estimates on RCD $$^*(K,N)$$ metric measure spaces
The total variation flow with time dependent boundary values
Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations
Nonlinear Schrödinger equation: concentration on circles driven by an external magnetic field
Long time behavior for solutions of the diffusive logistic equation with advection and free boundary
Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities
Solutions to the Reifenberg Plateau problem with cohomological spanning conditions
Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities
Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities
Positive steady states of an indefinite equation with a nonlinear boundary condition: existence, multiplicity, stability and asymptotic profiles
Calculus of Variations and Partial Differential Equations : Volume 55, Issue 3, June 2016
Calculus of Variations and Partial Differential Equations : Volume 55, Issue 2, April 2016
Calculus of Variations and Partial Differential Equations : Volume 55, Issue 1, February 2016
Calculus of Variations and Partial Differential Equations : Volume 54
Calculus of Variations and Partial Differential Equations : Volume 53
Calculus of Variations and Partial Differential Equations : Volume 52
Calculus of Variations and Partial Differential Equations : Volume 51
Calculus of Variations and Partial Differential Equations : Volume 50
Calculus of Variations and Partial Differential Equations : Volume 49
Calculus of Variations and Partial Differential Equations : Volume 48
Calculus of Variations and Partial Differential Equations : Volume 47
Calculus of Variations and Partial Differential Equations : Volume 46
Calculus of Variations and Partial Differential Equations : Volume 45
Calculus of Variations and Partial Differential Equations : Volume 44
Calculus of Variations and Partial Differential Equations : Volume 43
Calculus of Variations and Partial Differential Equations : Volume 42
Calculus of Variations and Partial Differential Equations : Volume 41
Calculus of Variations and Partial Differential Equations : Volume 40
Calculus of Variations and Partial Differential Equations : Volume 39
Calculus of Variations and Partial Differential Equations : Volume 38
Calculus of Variations and Partial Differential Equations : Volume 37
Calculus of Variations and Partial Differential Equations : Volume 36
Calculus of Variations and Partial Differential Equations : Volume 35
Calculus of Variations and Partial Differential Equations : Volume 34
Calculus of Variations and Partial Differential Equations : Volume 33
Calculus of Variations and Partial Differential Equations : Volume 32
Calculus of Variations and Partial Differential Equations : Volume 31
Calculus of Variations and Partial Differential Equations : Volume 30
Calculus of Variations and Partial Differential Equations : Volume 29
Calculus of Variations and Partial Differential Equations : Volume 28
Calculus of Variations and Partial Differential Equations : Volume 27
Calculus of Variations and Partial Differential Equations : Volume 26
Calculus of Variations and Partial Differential Equations : Volume 25
Calculus of Variations and Partial Differential Equations : Volume 24
Calculus of Variations and Partial Differential Equations : Volume 23
Calculus of Variations and Partial Differential Equations : Volume 22
Calculus of Variations and Partial Differential Equations : Volume 21
Calculus of Variations and Partial Differential Equations : Volume 20
Calculus of Variations and Partial Differential Equations : Volume 19
Calculus of Variations and Partial Differential Equations : Volume 18
Calculus of Variations and Partial Differential Equations : Volume 17
Calculus of Variations and Partial Differential Equations : Volume 16
Calculus of Variations and Partial Differential Equations : Volume 15
Calculus of Variations and Partial Differential Equations : Volume 14
Calculus of Variations and Partial Differential Equations : Volume 13
Calculus of Variations and Partial Differential Equations : Volume 12
Calculus of Variations and Partial Differential Equations : Volume 11
Calculus of Variations and Partial Differential Equations : Volume 10
Calculus of Variations and Partial Differential Equations : Volume 9
Calculus of Variations and Partial Differential Equations : Volume 8
Calculus of Variations and Partial Differential Equations : Volume 7
Calculus of Variations and Partial Differential Equations : Volume 6
Calculus of Variations and Partial Differential Equations : Volume 5

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A generalized mountain-pass theorem: the existence of multiple critical points

Content Provider SpringerLink
Author Ruppen, Hans Jörg
Copyright Year 2016
Abstract We analyse the existence of multiple critical points for an even functional $$\begin{aligned} J:H\rightarrow {\mathbb {R}}\end{aligned}$$ in the following context: the Hilbert space H can be split into an orthogonal sum $$H = Y \oplus Z$$ in such a way that $$\begin{aligned} \inf \{ J(u) : u \in Z \text { and } \Vert u\Vert = \rho \} \ge \alpha > J(0) \end{aligned}$$ and that here exits a point $$b \in H$$ with $$\Vert b\Vert > \rho $$ and with $$J(b) \le J(0)$$ . We develop a new variational characterization of multiple critical levels without an assumption on the dimension of Y. Our characterization is simple and natural: we can for example avoid the notion of pseudo-index and the definition of the activated levels does not substantially differs from the one used for the lowest critical level, giving us in this way a unified view of critical levels. We apply our results to a semi-linear Schrödinger equation of the form $$\begin{aligned} \left\{ \begin{array}{ll} &{} -\Delta u + V (x) u - q (x) |u|^ \sigma u = \lambda u, \qquad x\in {\mathbb {R}}^ N\\ &{} u \in {H^1({\mathbb {R}}^N)}\setminus \{ 0 \} \end{array} \right. \end{aligned}$$ where $$\lambda $$ is inside a spectral gap bounded on both sides by parts of the essential spectrum.
Ending Page 20
Page Count 20
Starting Page 1
File Format PDF
ISSN 09442669
e-ISSN 14320835
Journal Calculus of Variations and Partial Differential Equations
Issue Number 4
Volume Number 55
Language English
Publisher Springer Berlin Heidelberg
Publisher Date 2016-07-15
Publisher Place Berlin, Heidelberg
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Non-linear eigenvalue problems Partial differential equations Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics Abstract bifurcation theory Variational characterization of critical values Non-linear Schrödinger equations Bifurcation Nonlinear eigenvalue problems, nonlinear spectral theory Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Content Type Text
Resource Type Article
Subject Applied Mathematics Analysis
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