Variational perturbative methods and bifurcation of bound states from the essential spectrum
01 Jan 1998-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (Cambridge University Press)-Vol. 128, Iss: 6, pp 1131-1161
TL;DR: In this article, a perturbative method in critical point theory is proposed for the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum.
Abstract: This paper consists of two main parts. The first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results. The second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum.
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TL;DR: In this paper, the existence of a nontrivial solution via variational methods for a system of weakly coupled nonlinear Schrodinger equations is established, and the main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero.
341 citations
TL;DR: In this article, the existence of ground states in weighted Sobolev spaces is proved under the assumption that the ground states satisfy the condition that the potentials of the Schr\"odinger equations satisfy a certain property.
Abstract: We deal with a class on nonlinear Schr\"odinger equations \eqref{eq:1} with potentials $V(x)\sim |x|^{-\a}$, $0 0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\e}$ belonging to $W^{1,2}(\Rn)$ is proved under the assumption that $p$ satisfies \eqref{eq:p}. Furthermore, it is shown that $v_{\e}$ are {\em spikes} concentrating at a minimum of ${\cal A}=V^{\theta}K^{-2/(p-1)}$, where $\theta= (p+1)/(p-1)-1/2$.
261 citations
TL;DR: In this article, multiple semiclassical standing waves for a class of nonlinear Schrodinger equations with potentials were found by means of a perturbative variational method.
Abstract: Multiple semiclassical standing waves for a class of nonlinear Schrodinger equations with potentials are found by means of a perturbative variational method.
252 citations
Cites methods from "Variational perturbative methods an..."
...Our proof relies on perturbation arguments, variational in nature, as in [1] [2]....
[...]
TL;DR: In this paper, the existence of positive radial solutions concentrating on spheres to singularly perturbed elliptic problems was studied and necessary and sufficient conditions for concentration as well as the bifurcation of non-radial solutions were provided.
Abstract: We deal with the existence of positive radial solutions concentrating on spheres to a class of singularly perturbed elliptic problems like −ɛ2Δu+V(|x|)u=u
p
,uH
1
(ℝ
n
). Under suitable assumptions on the auxiliary potential M(r)=r
n−1
V
θ
(r), θ(p+1)/(p−1)−1/2, we provide necessary and sufficient conditions for concentration as well as the bifurcation of non-radial solutions.
234 citations
Cites methods from "Variational perturbative methods an..."
...We will use a finite dimensional reduction discussed in [1, 2]....
[...]
TL;DR: Some nonlinear elliptic equations on R N which arise perturbing the scalar curvature problem with the critical Sobolev exponent are studied in this article, where some results dealing with scalar curve curvature in R N are given.
178 citations
References
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TL;DR: In this paper, the authors proposed a nonlinear eigenvalue problem for constrained constrained bifurcation with Sobolev spaces, and showed that the problem is NP-hard.
Abstract: Keywords: $Lsp p$-bifurcation ; constrained ; minimization ; Sobolev spaces ; nonlinear eigenvalue problem ; semilinear ; spectrum ; linearisation ; variational ; structure Reference ANA-ARTICLE-1988-001doi:10.1112/plms/s3-57.3.511 Record created on 2008-12-10, modified on 2016-08-08
79 citations
TL;DR: In this article, non-compact nonlinearities, essential spectrum, bifurcation, and semilinear elliptic equation are studied. But the authors focus on the essential spectrum.
Abstract: Keywords: non-compact nonlinearities ; essential spectrum ; bifurcation ; semilinear elliptic equation Reference ANA-ARTICLE-1989-001doi:10.1002/mma.1670110408View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08
40 citations
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TL;DR: In this article, the conditions générales d'utilisation (http://www.numdam.unipd.org/legal. php) of the agreement with the Rendiconti del Seminario Matematico della Università di Padova are discussed.
Abstract: L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
23 citations
TL;DR: In this article, a resealing approach is proposed to solve the problem of finding simple zeros of a function analogous to Melnikov's function in a continuous spectrum, where the perturbations are not periodic and are not smooth with respect to uniform convergence.
Abstract: Bifurcation for nonlinear eigenvalue problems involving a second-order ordinary differential equation on the line is considered. Solutions are required to vanish at infinity in both directions and so correspond to homoclinic orbits. When posed in function spaces, the problem concerns bifurcation from the continuous spectrum. The present approach is based on a resealing that reduces the problem to that of continuing a nontrivial homoclinic orbit in a context where the perturbations are not periodic and are not smooth with respect to uniform convergence. Nonetheless, the nondegeneracy required for continuationamounts to finding simple zeros of a function analogous to Melnikov’s function.
17 citations
01 Jan 1996
TL;DR: On prouve des resultats du type Poincare-Melnikov, concernant les homoclines, par une methode variationnelle as discussed by the authors, par une femme variationnellette.
Abstract: On prouve des resultats du type Poincare-Melnikov, concernant les homoclines, par une methode variationnelle.
12 citations