Non simple blow ups for the Nirenberg problem on half spheres

Abstract

<p style='text-indent:20px;'>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) $\end{document}</tex-math></inline-formula> consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ 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 $\end{document}</tex-math></inline-formula>, which exhibit multiple blow up of <i>cluster-type</i> at the same boundary point. These solutions may have zero or non zero weak limit and may develop clusters at different boundary points. Such a blow up phenomena on half spheres drastically contrast with the case of the Nirenberg problem on spheres, where non simple blow up for finite energy solutions cannot occur and unveils an unexpected connection with vortex type problems arising in Euler equations in fluid dynamic and mean fields type equations in mathematical physics. We construct also, under suitable conditions on the restriction of <inline-formula><tex-math id="M6">\begin{document}$ 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.</p>

Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon

“…Estimating parameters within the nontrivial kernel of the linearized operator becomes particularly arduous in such scenarios. It is worth noting that circumventing the reliance on pointwise estimates and Pohozaev identities can be advantageous, especially in the exploration of non-compact variational problems where non-simple blow-ups may arise, such as in the singular mean-field equation with quantized singularities [9,24,34,35,19], and in the Nirenberg problem on half-spheres [2]. The presence of non-simple blow-up points significantly complicates the task of establishing pointwise C 0 -estimates, making our method particularly valuable in such contexts.…”

“…In this appendix we collect various estimates needed through the paper. Lemma 6.1 [2] Let a ∈ S n and λ > 0 be large. (i) Assume that τ ln λ is small enough, then it holds…”