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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 .
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 .
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 .
<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>
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