The effect of the graph topology on the existence ofmultipeak solutions for nonlinear Schrödinger equations

Dancer, E. N.; Lam, Kee Yuen; Yan, Shusen

doi:10.1155/s1085337501000276

Cited by 14 publications

(16 citation statements)

References 22 publications

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“…On the other hand, the multiplicity of positive and nodal solutions of classical elliptic problems are actively studied. More specifically, for the singularly perturbed problem (Equation ) with s =1, the number of the critical points of V ( x )(cf literature), the type of the critical points of V ( x )(cf other works), and the topology of the level set of V ( x ) (cf previous studies), can affect the number of solutions.…”

Section: Introductionmentioning

confidence: 99%

Nodal bound state of nonlinear problems involving the fractional Laplacian

Wei

2017

Math Methods in App Sciences

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Section: Introductionmentioning

confidence: 99%

Nodal bound state of nonlinear problems involving the fractional Laplacian

Wei

2017

Math Methods in App Sciences

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“…The number of critical points has been related to the number and/or type of critical points of a.x/ and also to the topology of sublevel sets of a.x/ (see, for example, [1,13,14,15,16,19,22] and references therein). The number of critical points has been related to the number and/or type of critical points of a.x/ and also to the topology of sublevel sets of a.x/ (see, for example, [1,13,14,15,16,19,22] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Infinitely Many Positive Solutions to Some Scalar Field Equations with Nonsymmetric Coefficients

Cerami

Passaseo

Solimini

2012

Comm Pure Appl Math

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In this paper the equation $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}} - \Delta u + a(x)u = |u|^{p - 1} u\;{\rm in }\;{\R}^N $ is considered, when N ≥ 2, p > 1, and $p < {{N + 2} \over {N - 2}}$ if N ≥ 3. Assuming that the potential a(x) is a positive function belonging to $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}L_{{\rm loc}}^{N/2} ({\R}^N )$ such that a(x) → a∞ > 0 as |x|→∞ and satisfies slow decay assumptions but does not need to fulfill any symmetry property, the existence of infinitely many positive solutions, by purely variational methods, is proved. The shape of the solutions is described as is, and furthermore, their asymptotic behavior when $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}|a(x) - a_\infty |_{L_{{\rm loc}}^{N/2} ({\R}^N )} \to 0$. © 2012 Wiley Periodicals, Inc.

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“…Oh extended the result in a higher dimension and proved the existence of multi‐peak solutions, which concentrate around any finite subsets of the nondegenerate critical points of V . For more results, we refer to and the references therein.…”

Section: Introductionmentioning

confidence: 99%