Given an isoparametric function $f$ on the $n$-dimensional round sphere, we consider functions of the form $u=w\circ f$ to reduce the semilinear elliptic problem \[ -Δ_{g_0}u+λu=λ | u\ | ^{p-1}u\qquad\text{ on }\mathbb{S}^n \] with $λ>0$ and $1<p$, into a singular ODE in $[0,π]$ of the form $w'' + \frac{h(r)}{\sin r} w' + \fracλ{\ell^2}\ (| w|^{p-1}w - w\ )=0$, where $h$ is an strictly decreasing function having exactly one zero in this interval and $\ell$ is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a sequence of sign-changing solutions to the first problem which are constant on the isoparametric hypersurfaces associated to $f$ and blowing-up at one or two of the focal submanifolds generating the isoparametric family. Our methods apply also when $p>\frac{n+2}{n-2}$, i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets.
We provide the first example of a sequence of {\em nondegenerate}, in the sense of Duyckaerts-Kenig-Merle \cite{DKM}, nodal nonradial solutions to the critical Yamabe problem $$ -ΔQ= |Q|^{\frac{2}{n-2}} Q, \ \ Q \in {\mathcal D}^{1,2} (\R^n). $$
For each $n\geq 3$ we establish the existence of a nodal solution $u$ to the Yamabe problem on the round sphere $(\mathbb{S}^n,g)$ which satisfies $$\int_{\mathbb{S}^n}|u|^{2^*}dV_g < 2m_n\mathrm{vol}(\mathbb{S}^n),$$ where $m_3=9,$ $m_4= 7,$ $m_5=m_6=6$, and $m_n= 5\ \text{if }n\geq 7.$
We study an optimal M-partition problem for the Yamabe equation on the round sphere, in the presence of some particular symmetries. We show that there is a correspondence between solutions to this problem and least-energy sign-changing symmetric solutions to the Yamabe equation on the sphere with precisely M nodal domains. The existence of an optimal partition is established through the study of the limit profiles of least-energy solutions to a weakly coupled competitive elliptic system on the sphere.
We construct a new family of entire solutions to the Yamabe equation $$-Δu=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u \mbox{ in }\mathcal{D}^{1,2}(\mathbb{R}^n).$$ If $n=3$, our solutions have maximal rank, being the first example in odd dimension. Our construction has analogies with the doubling of the equatorial spheres in the construction of minimal surfaces in $S^3(1)$.
Given a compact Riemannian manifold $(M,g)$ without boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$-div_{g}(a\nabla u)+bu=c|u|^{2^{\ast}-2}u\quad on\ M$$ where $a,b,c\in C^{\infty}(M)$, $a$ and $c$ are positive, $-div_{g}(a\nabla)+b$ is coercive, and $2^{\ast}=\frac{2m}{m-2}$ is the critical Sobolev exponent. In particular, if $R_{g}$ denotes the scalar curvature of $(M,g)$, we give conditions which guarantee that the Yamabe problem $$Δ_{g}u+\frac{m-2}{4(m-1} R_{g}u=κu^{2^{\a…
We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with umbilic boundary provided the dimension of the manifold is n>7 and that the Weyl tensor W(x) is not vanishing on the boundary of M.
We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.
We prove that the fractional Yamabe equation $\mathcal L_γu=|u|^\frac{4γ}{Q-2γ}u$ on the Heisenberg group $\mathbb H^n$ has $[\frac{n+1}{2}]$ sequences of nodal (sign-changing) weak solutions whose elements have mutually different nodal properties, where $\mathcal L_γ$ denotes the CR fractional sub-Laplacian operator on $\mathbb H^n$, $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$, and $γ\in \bigcup_{k=1}^n[k,\frac{kQ}{Q-1})$. Our argument is variational, based on a Ding-type conformal pulling-back transformation of the original problem into a problem on the CR sphere $S^{2n+1}$ combin…
We consider a closed cohomogeneity one Riemannian manifold $(M^n,g) $ of dimension $n\geq 3$. If the Ricci curvature of $M$ is positive, we prove the existence of infinite nodal solutions for equations of the form $-Δ_g u + λu = λu^q$ with $λ>0$, $q>1$. In particular for a positive Einstein manifold which is of cohomogeneity one or fibers over a cohomogeniety one Einstein manifold we prove the existence of infinite nodal solutions for the Yamabe equation, with a prescribed number of connected components of its nodal domain.
We give conditions for the existence of regular optimal partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$. To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if $\dim M\geq 10$, $(M,g)$ is not locally conformally flat and satisfies an additional geometric assumption whenever $\dim M=10$. Moreover, we show that the limit profiles of the components of the solution separat…
Given an isoparametric function $f$ on the $n$-dimensional sphere, we consider the space of functions $w\circ f$ to reduce the Yamabe equation on the round sphere into a singular ODE on $w$ in the interval $[0,π]$, of the form $w" + (h(r)/\sin r)w'+λ(\vert w\vert^{4/n-2}w - w)=0$, where $h$ is a monotone function with exactly one zero on $[0,π]$ and $λ>0$ is a constant. The natural boundary conditions in order to obtain smooth solutions are $w'(0)=0$ and $w'(π)=0$. We show that for any positive integer $k$ there exists a solution with exactly $k$-zeroes yielding solutions to the Yamabe equatio…
In this paper, we aim to investigate the following class of singularly perturbed elliptic problem $$ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\triangle {u}+|x|^ηu =|x|^ηf(u)& \mbox{in}\,\, A, u=0 & \mbox{on}\,\, \partial A, \end{array} \right. $$ where $\varepsilon>0$, $η\in\mathbb{R}$, $A=\{x\in\R^{2N}:\,\,0<a<|x|<b\}$, $N\ge2$ and $f$ is a nonlinearity of $C^1$ class with supercritical growth. By a reduction argument, we show that there exists a nodal solution $u_\e$ with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimens…
Through desingularization of Clifford torus, we prove the existence of a sequence of nondegenerate (in the sense of Duyckaerts-Kenig-Merle nodal nonradial solutions to the critical Yamabe problem $$-Δu=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u,\qquad u\in {\mathcal{D}}^{1,2}(\mathcal{R}^n). $$ The case $n=4$ is the first example in the literature of a solution with {\em maximal rank} ${\mathcal N}=2n+1+\frac{n(n-1)}{2}$.
We prove the existence of regular optimal $G$-invariant partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$ when $G$ is a compact group of isometries of $M$ with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy $G$-invariant solution with nontrivial components and we show that the limit profiles of the its components separate spatially as the competition parameter goes to $-\infty$, givin…
We study the problem% \[ -Δv+λv=| v| ^{p-2}v\text{ in }Ω,\text{\qquad}v=0\text{ on $\partialΩ$},\text{ }% \] for $λ\in\mathbb{R}$ and supercritical exponents $p,$ in domains of the form% \[ Ω:=\{(y,z)\in\mathbb{R}^{N-m-1}\times\mathbb{R}^{m+1}:(y,| z| )\inΘ\}, \] where $m\geq1,$ $N-m\geq3,$ and $Θ$ is a bounded domain in $\mathbb{R}% ^{N-m}$ whose closure is contained in $\mathbb{R}^{N-m-1}\times(0,\infty)$. Under some symmetry assumptions on $Θ$, we show that this problem has infinitely many solutions for every $λ$ in an interval which contains $[0,\infty)$ and $p>2$ up to some number which i…
Let $Ω$ be a bounded domain in $\R^n$, $n\ge 3$ with smooth boundary $\partialΩ$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$ -Δu=|u|^{{n+2\over n-2} +\ve -1 } u \, \, \mbox{ in } Ω, \quad \quad u=0 \mbox{ on } \partial Ω, $$ where $\ve$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem $$ -Δw = |w|^{\frac{4}{n-2}} w, \ \ w \in {\mathcal D}^{1,2} (\R^n) $$ which has large number ($3n$) of kernels.
Sign-changing solutions to the Yamabe problem on manifolds with boundary
Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）；Dipartimento di Matematica e Fisica, Università della Campania “Luigi Vanvitelli”（坎帕尼亚路易吉范维泰利大学数学与物理系）；Dipartimento SBAI, Sapienza Università di Roma（罗马智慧大学SBAI系）
We consider the semi-Riemannian Yamabe type equations of the form \[ -\square u + λu = μ\vert u\vert^{p-1}u\quad\text{ on }M \] where $M$ is either the semi-Euclidean space or the pseudosphere of dimension $m\geq 3$, $\square$ is the semi-Riemannian Laplacian in $M$, $λ\geq0$, $μ\in\mathbb{R}\smallsetminus\{0\}$ and $p>1$. Using semi-Riemannian isoparametric functions on $M$, we reduce the PDE into a generalized Emden-Fowler ODE of the form \[ w''+q(r)w'+λw = μ\vert w\vert^{p-1}w\quad\text{ on } I, \] where $I\subset\mathbb{R}$ is $[0,\infty)$ or $[0,π]$, $q(r)$ blows-up at $0$ and $w$ is subj…
Multiple sign-changing and semi-nodal normalized solutions for a Gross-Pitaevskii type system on bounded domains: the $L^2$-supercritical case
Beijing University of Technology（北京工业大学）；Tsinghua University（清华大学）；Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623)（玛丽和路易斯·巴斯德大学，法国国家科学研究中心，LmB实验室）
We prove the existence of positive solutions for the supercritical nonlinear fractional Schrödinger equation $(-Δ)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1), \ n>2s$. We show that if $V(x)=o(|x|^{-2s})$ as $|x|\to +\infty$, then for $p>\frac{n+2s-1}{n-2s-1}$, this problem admits a continuum of solutions. More generally, for $p>\frac{n+2s}{n-2s}$, conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: th…
In this paper we prove an existence result to the problem $$\left\{\begin{array}{ll} -Δu = |u|^{p-1} u \qquad & \text{in} Ω, \\ u= 0 & \text{on} \partialΩ, \end{array} \right. $$ where $Ω$ is a bounded domain in ${\mathbb R}^{N}$ which is a perturbation of the annulus. Then there exists a sequence $p_1<p_2<..$ with $\lim\limits_{k\rightarrow+\infty}p_k=+\infty$ such that for any real number $p>1$ and $p\ne p_k$ there exist at least one solution with $m$ nodal zones. In doing so, we also investigate the radial nodal solution in an annulus: we provide an estimate of its Morse index and analyze t…
