On the location and profile of spike‐layer solutions to singularly perturbed semilinear dirichlet problems

Ni, Wei Ming; Wei, Juncheng

doi:10.1002/cpa.3160480704

Cited by 283 publications

(305 citation statements)

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“…For p < n+2 n−2 , and ε small, problem (1.2) admits solutions with spike layers concentrating at (local or global) maximum points of the distance function. See [10], [11], [18], [22], [24], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Multiple Clustered Layer Solutions for Semilinear Elliptic Problems onSⁿ

Bandle¹,

Wei²

2008

Communications in Partial Differential Equations

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Abstract. We consider the following superlinear elliptic equation onwhere D is a geodesic ball on S n with geodesic radius θ 1 , and ∆ S n is the Laplace-Beltrami operator on S n . We prove that for any θ ∈ ( π 2 , π) and for any positive integer N ≥ 1, there exist at least 2N + 1 solutions to the above problem for ε sufficiently small. Moreover, the asymptotic behavior of such solutions is also characterized. We then apply this result to the Brezis-Nirenberg problem and establish the existence of solutions which are not minimizers of the associated energy.

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Section: Introductionmentioning

confidence: 99%

Multiple Clustered Layer Solutions for Semilinear Elliptic Problems onSⁿ

Bandle¹,

Wei²

2008

Communications in Partial Differential Equations

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“…Here we have used that f (0) ≤ 0. Hence as in [30] or [33], we can find a mountain pass solution u 1 between 0 and u for small positive . By well known results (cf.…”

Section: Solutions On Bounded Domains Of Bounded Morse Indexmentioning

confidence: 82%

Stable and finite Morse index solutions on 𝐑ⁿ or on bounded domains with small diffusion

Dancer

2004

Trans. Amer. Math. Soc.

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Abstract. In this paper, we study bounded solutions of −∆u = f (u) on R n (where n = 2 and sometimes n = 3) and show that, for most f 's, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of − 2 ∆u = f (u) on Ω with Dirichlet or Neumann boundary conditions for small .The purpose of the present paper is to study the equationon R n (or a half space T ), where we are interested in solutions which are weakly stable or have finite Morse index (that is, have only finitely many negative eigenvalues in some suitable generalized sense). Usually, we study positive solutions (but not always). For n = 2 and sometimes for n = 3 we prove that these solutions are very simple and easy to understand. (For example, if n = 2, the weakly stable solutions usually have to be constant.) This is an interesting contrast to the results in [8] where for certain nonlinearities there are a great many positive bounded solutions which are periodic in some variables and decay in others. In addition, for many f , it is easy to use variational methods (or bifurcation methods) to construct many solutions periodic (and non-constant) in all variables. Thus it seems that the structure of all solutions of (1) may be quite complicated. Our results show that the finite Morse index solutions are usually quite simple. We also prove closely related results on half spaces.As an application of these ideas, we have a number of results on the solutions (usually but not always positive) ofwhere Ω is a bounded open set in R 2 (sometimes in R 3 ) and we have homogeneous Neumann or Dirichlet boundary conditions. For many f 's we obtain the exact number of stable positive solutions for small positive and show it is independent of the shape of the domain Ω (and easy to calculate). This contrasts strongly with the results for all positive solutions in [9], [10] and [11] for small positive and results for the stable positive solutions when is not small. We also prove that stable positive solutions are constant in the case of Neumann boundary conditions, and for either boundary condition there are no stable sign-changing positive solutions for small positive (for many f 's). These two results contrast with results in [31] and [16].

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“…Subsequently, Ni and Wey ( [14]) considered equation (1.5) together with Dirichlet boundary conditions and showed that in this case the least energy solutions concentrate at points of Ω which maximize the distance to the boundary of Ω. Related results were proved e.g.…”

Section: Introductionmentioning

confidence: 94%

“…in [12], [18], [9] (where a "local" version of Ni-Wei's result is proved) and in [2], [6], [11] (where a linear term V (x)u is added to equation (1.5) while Ω = R N ). The subject was revisited by Del Pino and Felmer, for both Neumann and Dirichlet boundary conditions, in [5], where shorter and more elementay arguments were introduced, with respect to those in [13,14]. We refer the reader to the nice Introduction in [5] for further references and a developed discussion on the subject.…”

Section: Introductionmentioning

confidence: 99%

Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions

Pistoia

Ramos

2008

Nonlinear differ. equ. appl.

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Abstract. We consider a system of the form −ε 2 ∆u + u = g(v), −ε 2 ∆v + v = f (u) in Ω with Dirichlet boundary condition on ∂Ω, where Ω is a smooth bounded domain in R N , N 3 and f, g are power-type nonlinearities having superlinear and subcritical growth at infinity. We prove that the least energy solutions to such a system concentrate, as ε goes to zero, at a point of Ω which maximizes the distance to the boundary of Ω.2000 Mathematics Subject Classification: 35J50, 35J55, 38E05.

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On the location and profile of spike‐layer solutions to singularly perturbed semilinear dirichlet problems

Cited by 283 publications

References 12 publications

Multiple Clustered Layer Solutions for Semilinear Elliptic Problems onSⁿ

Multiple Clustered Layer Solutions for Semilinear Elliptic Problems onSⁿ

Stable and finite Morse index solutions on 𝐑ⁿ or on bounded domains with small diffusion

Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions

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On the location and profile of spike‐layer solutions to singularly perturbed semilinear dirichlet problems

Cited by 283 publications

References 12 publications

Multiple Clustered Layer Solutions for Semilinear Elliptic Problems onSn

Multiple Clustered Layer Solutions for Semilinear Elliptic Problems onSn

Stable and finite Morse index solutions on 𝐑ⁿ or on bounded domains with small diffusion

Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions

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Multiple Clustered Layer Solutions for Semilinear Elliptic Problems onSⁿ

Multiple Clustered Layer Solutions for Semilinear Elliptic Problems onSⁿ