Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, part I

Antonio Ambrosetti; Andrea Malchiodi; Wei Ming Ni

doi:10.1007/s00220-003-0811-y

Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, part I

Antonio Ambrosetti
, Andrea Malchiodi
, Wei Ming Ni

School of Mathematics

Research output: Contribution to journal › Article › peer-review

178 Scopus citations

Abstract

We deal with the existence of positive radial solutions concentrating on spheres to a class of singularly perturbed elliptic problems like -ε² Δu + V(|x|)u = u^p, u ∈ H¹ (ℝⁿ). 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.

Original language	English (US)
Pages (from-to)	427-466
Number of pages	40
Journal	Communications in Mathematical Physics
Volume	235
Issue number	3
DOIs	https://doi.org/10.1007/s00220-003-0811-y
State	Published - Apr 2003

Publisher link

10.1007/s00220-003-0811-y

Find It

Find It at U of M Libraries

Cite this

@article{394cf4a959ce40179229d8ac8d453473,

title = "Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, part I",

abstract = "We deal with the existence of positive radial solutions concentrating on spheres to a class of singularly perturbed elliptic problems like -ε2 Δu + V(|x|)u = up, u ∈ H1 (ℝn). Under suitable assumptions on the auxiliary potential M(r) = rn-1 Vθ(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.",

author = "Antonio Ambrosetti and Andrea Malchiodi and Ni, \{Wei Ming\}",

year = "2003",

month = apr,

doi = "10.1007/s00220-003-0811-y",

language = "English (US)",

volume = "235",

pages = "427--466",

journal = "Communications in Mathematical Physics",

issn = "0010-3616",

publisher = "Springer New York",

number = "3",

}

TY - JOUR

T1 - Singularly perturbed elliptic equations with symmetry

T2 - Existence of solutions concentrating on spheres, part I

AU - Ambrosetti, Antonio

AU - Malchiodi, Andrea

AU - Ni, Wei Ming

PY - 2003/4

Y1 - 2003/4

N2 - We deal with the existence of positive radial solutions concentrating on spheres to a class of singularly perturbed elliptic problems like -ε2 Δu + V(|x|)u = up, u ∈ H1 (ℝn). Under suitable assumptions on the auxiliary potential M(r) = rn-1 Vθ(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.

AB - We deal with the existence of positive radial solutions concentrating on spheres to a class of singularly perturbed elliptic problems like -ε2 Δu + V(|x|)u = up, u ∈ H1 (ℝn). Under suitable assumptions on the auxiliary potential M(r) = rn-1 Vθ(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.

UR - https://www.scopus.com/pages/publications/0038609648

UR - https://www.scopus.com/pages/publications/0038609648#tab=citedBy

U2 - 10.1007/s00220-003-0811-y

DO - 10.1007/s00220-003-0811-y

M3 - Article

AN - SCOPUS:0038609648

SN - 0010-3616

VL - 235

SP - 427

EP - 466

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -

Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, part I

Abstract

Publisher link

Find It

Other files and links

Fingerprint

Cite this