Multiplicity of solutions to the Dirichlet problem for generalized Hénon equation

Kolonitskii, Sergey; Nazarov, Alexander I.

doi:10.1007/s10958-007-0299-7

Cited by 7 publications

(5 citation statements)

References 19 publications

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“…We will say that functions in

{scriptD}_{Õ false(k false)}^{s}

are

k

‐ radially symmetric (in Nazarov, 6 they are called

false(k, 0 false)

‐radial; see also Kolonitskii and Nazarov 4 and Nazarov 7 ). Notice that a

k

‐radially symmetric function depends only on

false| x_{1} false|

, …,

false| x_{m} false|

and is invariant with respect to permutations of its variables.…”

Section: Variational Toolsmentioning

confidence: 99%

“…are k-radially symmetric (in [23] they are called (k, 0)-radial, see also [17,24]). Notice that a k-radially symmetric function depends only on |x 1 |, .…”

Section: Variational Toolsmentioning

confidence: 99%

“…is crucially used to tackle certain problems driven by the Laplace operator −∆. We cite also [23,27,17,18,5,24], where the p-Laplacian or more general second order, possibly degenerate operators in divergence form are considered, and [1,4], that deal with fourthorder variational equations.…”

Section: Introductionmentioning

confidence: 99%

“…We cite for instance Gazzini and Musina 1 and Smets et al, 2 where the trivial equality

\int_{{double-struckR}^{n}} false| \nabla false(u φ_{μ} false) {false|}^{2} d x - \int_{{double-struckR}^{n}} false| \nabla u {false|}^{2} d x = μ \int_{{double-struckR}^{n}} false| x {false|}^{- 2} false| u {false|}^{2} d x

is crucially used to tackle certain problems driven by the Laplace operator

- normalΔ

. We cite also Caldiroli and Musina, 3 Kolonitskii and Nazarov, 4 Musina, 5 Nazarov, 6,7 and Scheglova, 8 where the

p

‐Laplacian or more general second order, possibly degenerate operators in divergence form are considered, and Bhakta and Musina 9 and Caldiroli and Cora 10 that deal with fourth‐order variational equations.…”

Section: Introductionmentioning

confidence: 99%

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A tool for symmetry breaking and multiplicity in some nonlocal problems

Musina

Nazarov

2020

Math Methods in App Sciences

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We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function u on R n and of its perturbation uϕ µ , where ϕ µ is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere S n−1 , thus providing a technical but rather powerful tool to detect symmetry breaking and multiplicity phenomena in variational equations driven by the fractional Laplace operator. A concrete application to a problem related to the fractional Caffarelli-Kohn-Nirenberg inequality is given.

show abstract

“…We will say that functions in

{scriptD}_{Õ false(k false)}^{s}

are

k

‐ radially symmetric (in Nazarov, 6 they are called

false(k, 0 false)

‐radial; see also Kolonitskii and Nazarov 4 and Nazarov 7 ). Notice that a

k

‐radially symmetric function depends only on

false| x_{1} false|

, …,

false| x_{m} false|

and is invariant with respect to permutations of its variables.…”

Section: Variational Toolsmentioning

confidence: 99%

“…are k-radially symmetric (in [23] they are called (k, 0)-radial, see also [17,24]). Notice that a k-radially symmetric function depends only on |x 1 |, .…”

Section: Variational Toolsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We cite for instance Gazzini and Musina 1 and Smets et al, 2 where the trivial equality

\int_{{double-struckR}^{n}} false| \nabla false(u φ_{μ} false) {false|}^{2} d x - \int_{{double-struckR}^{n}} false| \nabla u {false|}^{2} d x = μ \int_{{double-struckR}^{n}} false| x {false|}^{- 2} false| u {false|}^{2} d x

is crucially used to tackle certain problems driven by the Laplace operator

- normalΔ

. We cite also Caldiroli and Musina, 3 Kolonitskii and Nazarov, 4 Musina, 5 Nazarov, 6,7 and Scheglova, 8 where the

p

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A tool for symmetry breaking and multiplicity in some nonlocal problems

Musina

Nazarov

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…After that, much work has been made to study these non radial solutions: multiplicity, shape, asymptotic behavior. To have just an idea of the research on this topic, one can see for example [11], [4] and [9] for results about the critical and supercritical cases, [5] and [7] for the study of the asymptotic behavior of the maximum point and the existence of multi-peak solutions, [3] for the uniqueness of the radial solution for 2 < p < 2n+2α n−2 , [2] and [14] for results about above the p−Laplacian, [13] for the study of Hénon type system. See also the references in the quoted papers.…”

Section: Introductionmentioning

confidence: 99%