Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

Ayed, Mohamed Ben; Ghoudi, Rabeh; Bouh, Kamal Ould

doi:10.1007/s00030-011-0145-y

Cited by 11 publications

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References 23 publications

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“…This problem has been studied on half spheres of dimensions n = 2, 3, 4. See the papers [13][14][15]17,25,29,30] and the references therein. Very much like the case of spheres, to recover compactness one considers here the following subcritical approximation…”

Section: (P)mentioning

confidence: 99%

“…Moreover in dimensions n = 2, 3 counting index criteria have been established, see [14,17,25,29]. Furthermore under additional condition on K 1 it has been proved in [15] that all blow up points are isolated simple, but already in dimension n = 4 counting index formulae, under the above non degeneracy conditions fail. More surprisingly and in contrast with the case of closed spheres, the Nirenberg problem on half spheres may have non simple blow up points, even for finite energy bubbling solutions of (P ε ) see [1,2].…”

Section: (P)mentioning

confidence: 99%

“…2) Let i, j ∈ I b := {k : a k ∈ ∂S n + } and let μ i and μ j be defined by (15). Assume that μ j ≤ c μ i for some constant c , then: (i) either there exists a constant c such that…”

Section: Lemma 52 1)mentioning

confidence: 99%

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The Nirenberg problem on high dimensional half spheres: the effect of pinching conditions

Ahmedou

Ayed

2021

Calc. Var.

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In this paper we study the Nirenberg problem on standard half spheres $$(\mathbb {S}^n_+,g), \, n \ge 5$$ ( S + n , g ) , n ≥ 5 , which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: $$\begin{aligned} (\mathcal {P}) \quad {\left\{ \begin{array}{ll} -\Delta _{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 &{}\quad \text{ in } \mathbb {S}^n_+, \\ \frac{\partial u}{\partial \nu }\, =\, 0 &{}\quad \text{ on } \partial \mathbb {S}^n_+. \end{array}\right. } \end{aligned}$$ ( P ) - Δ g u + n ( n - 2 ) 4 u = K u n + 2 n - 2 , u > 0 in S + n , ∂ u ∂ ν = 0 on ∂ S + n . where $$K \in C^3(\mathbb {S}^n_+)$$ K ∈ C 3 ( S + n ) is a positive function. This problem has a variational structure but the related Euler–Lagrange functional $$J_K$$ J K lacks compactness. Indeed it admits critical points at infinity, which are limits of non compact orbits of the (negative) gradient flow. Through the construction of an appropriate pseudogradient in the neighborhood at infinity, we characterize these critical points at infinity, associate to them an index, perform a Morse type reduction of the functional $$J_K$$ J K in their neighborhood and compute their contribution to the difference of topology between the level sets of $$J_K$$ J K , hence extending the full Morse theoretical approach to this non compact variational problem. Such an approach is used to prove, under various pinching conditions, some existence results for $$(\mathcal {P})$$ ( P ) on half spheres of dimension $$n \ge 5$$ n ≥ 5 .

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Section: (P)mentioning

confidence: 99%

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The Nirenberg problem on high dimensional half spheres: the effect of pinching conditions

Ahmedou

Ayed

2021

Calc. Var.

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“…It is easy to see that a necessary condition for solving the problem is that K has to be positive somewhere. Note that some related problems of type (P ε ), in case of bounded domains, were studied in [4,5,9,[12][13][14] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Construction of sign-changing solutions for a harmonic equation with subcritical exponent

Bouh

2015

Monatsh Math

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In this paper, we study the harmonic equation involving subcritical exponentunit ball in R n , n ≥ 5 with Euclidean metric g 0 , ∂B n = S n−1 is its boundary, K is a function on S n−1 and ε is a small positive parameter. We construct solutions of the subcritical equation (P ε ) which blow up at two different critical points of K . Furthermore, we construct solutions of (P ε ) which have two bubbles and blow up at the same critical point of K .

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“…This problem has been studied on half spheres of dimensions n = 3, 4. See the papers [18,14,8,9,10] and the references therein.…”

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Non simple blow ups for the Nirenberg problem on half spheres

Ahmedou¹,

Ayed²

2022

DCDS

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In this paper we study a Nirenberg type problem on standard half spheres <inline-formula><tex-math id="M1">\begin{document}$ (\mathbb{S}^n_+,g_0) $\end{document}</tex-math></inline-formula> consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} (\mathcal{P}) \quad \begin{cases} - \Delta_{g_0} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 \quad \mbox{in } \mathbb{S}^n_+, \\ \frac{\partial u}{\partial \nu }\, = \, 0 \quad \mbox{on } \partial \mathbb{S}^n_+, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M2">\begin{document}$ K \in C^3(\mathbb{S}^n_+) $\end{document}</tex-math></inline-formula> is a positive function. We construct, under generic conditions on the function <inline-formula><tex-math id="M3">\begin{document}$ K $\end{document}</tex-math></inline-formula>, finite energy solutions of a subcritical approximation of <inline-formula><tex-math id="M4">\begin{document}$ (\mathcal{P}) $\end{document}</tex-math></inline-formula> on half spheres of dimension <inline-formula><tex-math id="M5">\begin{document}$ n \geq 5 $\end{document}</tex-math></inline-formula>, which exhibit multiple blow up of cluster-type at the same boundary point. These solutions may have zero or non zero weak limit and may develop clusters at different boundary points. Such a blow up phenomena on half spheres drastically contrast with the case of the Nirenberg problem on spheres, where non simple blow up for finite energy solutions cannot occur and unveils an unexpected connection with vortex type problems arising in Euler equations in fluid dynamic and mean fields type equations in mathematical physics. We construct also, under suitable conditions on the restriction of <inline-formula><tex-math id="M6">\begin{document}$ K $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M7">\begin{document}$ \partial \mathbb{S}^n_+ $\end{document}</tex-math></inline-formula>, approximate solutions of arbitrarily large energy and Morse index.

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Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

Cited by 11 publications

References 23 publications

The Nirenberg problem on high dimensional half spheres: the effect of pinching conditions

The Nirenberg problem on high dimensional half spheres: the effect of pinching conditions

Construction of sign-changing solutions for a harmonic equation with subcritical exponent

Non simple blow ups for the Nirenberg problem on half spheres

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