Deformation retracts to the fat diagonal and applications to the existence of peak solutions of nonlinear elliptic equations

Dancer, E. N.; Hillman, Jonathan A.; Pistoia, Angela

doi:10.2140/pjm.2012.256.67

Cited by 5 publications

(2 citation statements)

References 28 publications

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“…In particular, Dancer and Yan ( [10]) proved that if the domain has a nontrivial topology, then there always exists a k-peak positive solution for any k ≥ 1. This result has been generalized by Dancer, Hillman and Pistoia ( [9]) to the case of a not contractible domain. On the other hand, Dancer and Yan ( [11]) showed that if Ω is a strictly convex domain and k ≥ 2, then problem (1.1) does not admit a k-peak positive solutions (see also [24] for the proof when k = 2).…”

Section: Introductionmentioning

confidence: 71%

Solutions with multiple alternate sign peaks along a boundary geodesic to a semilinear Dirichlet problem

D’Aprile¹,

Pistoia²

2012

Preprint

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We study the existence of sign-changing multiple interior spike solutions for the following Dirichlet problemwhere Ω is a smooth and bounded domain of R N , ε is a small positive parameter, f is a superlinear, subcritical and odd nonlinearity. In particular we prove that if Ω has a plane of symmetry and its intersection with the plane is a two-dimensional strictly convex domain, then, provided that k is even and sufficiently large, a k-peak solution exists with alternate sign peaks aligned along a closed curve near a geodesic of ∂Ω.

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Section: Introductionmentioning

confidence: 71%

Solutions with multiple alternate sign peaks along a boundary geodesic to a semilinear Dirichlet problem

D’Aprile¹,

Pistoia²

2012

Preprint

Self Cite

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“…By using the Lyapunov-Schmidt reduction, the authors proved the existence of positive multi-peak solutions under the homogeneous Dirichlet boundary condition in a general domain Ω with nontrivial topology. For the further related results, we refer to [6,14,19,21,23] and the reference therein.…”

Section: Introductionmentioning

confidence: 99%

A new type of nodal solutions to singularly perturbed elliptic equations with supercritical growth

Liu¹,

Wei²,

Zhang³

2022

Preprint

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In this paper, we aim to investigate the following class of singularly perturbed elliptic problemand f is a nonlinearity of C 1 class with supercritical growth. By a reduction argument, we show that there exists a nodal solution uε with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimension N −1 as ε → 0. In particular, we establish different concentration phenomena of four peaks when the parameter η > 2, η = 2 and η < 2. Contents 1. Introduction 1 2. Preliminary results 4 3. The energy estimates 9 4. Proof of Theorem 1.2 24 4.1. Case η < 2 24 4.2. Case η > 2 27 5. Proof of Theorem 1.3 28 References 35

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A new type of nodal solutions to singularly perturbed elliptic equations with supercritical growth

Liu

Wei

Zhang

2022

Journal of Differential Equations

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Deformation retracts to the fat diagonal and applications to the existence of peak solutions of nonlinear elliptic equations

Abstract: We consider the equation −ε 2 u = u p − u q in a bounded, smooth domain ⊂ ‫ޒ‬ N with homogeneous Dirichlet boundary conditions when eitherWe prove the existence of multiple positive solutions in the case of small diffusion provided the domain is not contractible.

Cited by 5 publications

References 28 publications

Solutions with multiple alternate sign peaks along a boundary geodesic to a semilinear Dirichlet problem

Solutions with multiple alternate sign peaks along a boundary geodesic to a semilinear Dirichlet problem

A new type of nodal solutions to singularly perturbed elliptic equations with supercritical growth

A new type of nodal solutions to singularly perturbed elliptic equations with supercritical growth

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