Entire Solutions for a Class of Fourth-Order Semilinear Elliptic Equations with Weights

Caldiroli, Paolo; Cora, Gabriele

doi:10.1007/s00009-015-0519-1

Cited by 7 publications

(8 citation statements)

References 20 publications

(41 reference statements)

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“…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%

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.

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“…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%

A tool for symmetry breaking and multiplicity in some nonlocal problems

Musina

Nazarov

2020

Math Methods in App Sciences

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“…It is known from (1.1) that α ∈ (4 − N, N ). Equation (P) and problem (Q) have been studied by many authors recently, in particular, in the case of the pure biharmonic operator, see, for example, [2,3,5,6,8,9,10,11,13,16,18,19,20,21,23,25] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation

Zhang

Wan

Wang

2019

Commun. Contemp. Math.

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“…Some fundamental results about this equation have been established, such as the weighted Sobolev inequalities, existence of solutions, Caffarelli-Kohn-Nirenberg type inequalities and so on. We refer the readers to [2,3,4,7,8,10,14] and the references therein.…”

mentioning

confidence: 99%

“…. Notice that if λ, µ satisfy any of these three conditions, then K 0 ≥ 0, K 2 ≥ 0, K 2 2 − 4K 0 ≥ 0 and (1.10)-(1.11) hold.…”

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confidence: 99%

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Classification of positive radial solutions to a weighted biharmonic equation

Yan¹

2021

CPAA

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In this paper, we consider the weighted fourth order equation<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u = |x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\}, $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ n\geq 5 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ -n<\alpha<n-4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ p>1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ (p,\alpha,\beta,n) $\end{document}</tex-math></inline-formula> belongs to the critical hyperbola<disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \frac{n+\alpha}{2}+\frac{n+\beta}{p+1} = n-2. $\end{document} </tex-math></disp-formula>We prove the existence of radial solutions to the equation for some <inline-formula><tex-math id="M5">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. On the other hand, let <inline-formula><tex-math id="M7">\begin{document}$ v(t): = |x|^{\frac{n-4-\alpha}{2}}u(|x|) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ t = -\ln |x| $\end{document}</tex-math></inline-formula>, then for the radial solution <inline-formula><tex-math id="M9">\begin{document}$ u $\end{document}</tex-math></inline-formula> with non-removable singularity at origin, <inline-formula><tex-math id="M10">\begin{document}$ v(t) $\end{document}</tex-math></inline-formula> is a periodic function if <inline-formula><tex-math id="M11">\begin{document}$ \alpha \in (-2,n-4) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> satisfy some conditions; while for <inline-formula><tex-math id="M14">\begin{document}$ \alpha \in (-n,-2] $\end{document}</tex-math></inline-formula>, there exists a radial solution with non-removable singularity and the corresponding function <inline-formula><tex-math id="M15">\begin{document}$ v(t) $\end{document}</tex-math></inline-formula> is not periodic. We also get some results about the best constant and symmetry breaking, which is closely related to the Caffarelli-Kohn-Nirenberg type inequality.

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Entire Solutions for a Class of Fourth-Order Semilinear Elliptic Equations with Weights

Cited by 7 publications

References 20 publications

A tool for symmetry breaking and multiplicity in some nonlocal problems

A tool for symmetry breaking and multiplicity in some nonlocal problems

Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation

Classification of positive radial solutions to a weighted biharmonic equation

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