In this paper we study a Nirenberg type problem on standard half spheres $(\mathbb{S}^n_+,g_0)$ consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary $\partial \mathbb{S}^n_+$. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: \begin{equation*} (\mathcal{P}) \quad \begin{cases}
-\D_{g_0} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+,
\frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \end{equation*} where $K \in C^2(\mathbb{S}^n_+)$ is a positive function.
We construct, under generic conditions on the function $K$, finite energy solutions of a subcritical approximation of $(\mathcal{P})$ on half spheres of dimension $n \geq 5$, which exhibit multiple blow up of \emph{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 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 subsolutions 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 $K$ on $\partial \mathbb{S}^n_+$, approximate solutions of arbitrarily large energy and Morse index
In this paper we perform a refined blow up analysis of finite energy approximated solutions to a Nirenberg type problem on half spheres. The later consists of prescribing, under minimal boundary conditions, the scalar curvature to be a given function. In particular we give a precise location of blow up points and blow up rates. Such an analysis shows that the blow up picture of the Nirenberg problem on half spheres is far more complicated that in the case of closed spheres. Indeed besides the combination of interior and boundary blow ups, there are non simple blow up points for subcritical sol…
We show a convergence result of the fractional Laplacian for sequences of nonnegative functions without uniform boundedness near infinity. As an application, we construct a sequence of solutions to the fractional Nirenberg problem that blows up in the region where the prescribed functions are negative. This is a different phenomenon from the classical Nirenberg problem.
We establish theorems on the existence and compactness of solutions to the $σ_2$-Nirenberg problem on the standard sphere $\mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order der…
We consider the Brezis-Nirenberg problem: $$-Δu =λu + |u|^{p-1}u\qquad \mbox{in}\,\, Ω,\quad u=0\,\, \mbox{on}\,\,\ \partialΩ,$$ where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$, $p=\frac{N+2}{N-2}$ and $λ>0$. In this paper we prove that, if $Ω$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $λ\rightarrow λ_1$, where $λ_1=λ_1(Ω)$ denotes the first eigenvalue of $-Δ$ on $Ω$, with zero Dirichlet boundary condition.
The main objective of this paper is to establish the convergence for the fractional $p$-Laplacian of nonnegative sequence of functions with $p>2$. Further, we show the blow-up phenomena for solutions to the extended Nirenberg problem modeled by fractional $p$-Laplacian with the prescribed negative functions.
In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $n=3,$ $σ=1/2,$ when the prescribing $σ$-curvature function satisfies the $(n-2σ)$-flatness condition near its critical points. The compactness results are new and optimal. In addition, we obtain a degree-counting formula of all solutions. From our results, we can know where blow up occur. Moreover, for any finite distinct points, the sequence of solutions that blow up precisely at these points can be constructed. We extend the results of Li in \cite[CPAM…
We show that the classical Brezis-Nirenberg problem $$ -Δu=u|u| + λu\ \hbox{in}\ Ω, u=0\ \hbox{on}\ \partialΩ, $$ when $Ω$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $Ω$ as $λ$ approaches a suitable value $λ_0>0.$
Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations $P_σ(v)= Kv^{\frac{n+2σ}{n-2σ}}$ on the standard unit sphere $\mathbb{S}^n$ for all $σ\in (0,n/2)$, where $P_σ$ is the intertwining operator of order $2σ$. Finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures. When $σ=1$, it is the prescribing scalar curvature problem or the Nirenberg problem,…
This paper is a continuation of our earlier work "[T. Jin, Y.Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, to appear in J. Eur. Math. Soc.]", where compactness results were given on a fractional Nirenberg problem. We prove two existence results stated there. We also obtain a fractional Aubin inequality.
In this paper we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{\frac{N+2}{N-2}}+\varepsilon u, &{\text{in}~Ω},\\ u>0, &{\text{in}~Ω},\\ u=0, &{\text{on}~\partial Ω}. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small $\varepsilon>0$ was obtained by Musso and Pistoia. However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem fo…
Let $(M,g)$ be a closed locally conformally flat Riemannian manifold of dimension $n \ge 7$ and of positive Yamabe type. If $ξ_0$ denotes a non-degenerate critical point of the mass function we prove the existence, for any $ k \ge 1$ and $\varepsilon >0$, of a positive blowing-up solution $u_{\varepsilon}$ of $$\triangle_g u_{\varepsilon} +\big( c_n S_g +\varepsilon h\big) u_{\varepsilon} = u_{\varepsilon}^{2^*-1},$$ that blows up like the superposition of $k$ positive bubbles concentrating at different speeds at $ξ_0$. The method of proof combines a finite-dimensional reduction with the sharp…
This paper is devoted to study a class of integral type Brezis-Nirenbreg problem on the Heisenberg group. It is a class of new nonlinear integral equations on the bounded domains of Heisenberg group and related to the CR Yamabe problems on the CR manifold. Based on the sharp Hardy-Littlewood-Sobolev inequalities, the nonexistence and existence results are obtained by Pohozaev type identity, variational method and blow-up analysis, respectively.
Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg (GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained. Applying the GN inequality, we show that the finite-time blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality.
We prove that for the Navier Stokes equation with dissipation $(-Δ)^α$, where $1<α<{5/4}$, and smooth initial data, the Hausdorff dimension of the singular set at time of first blow up is at most $5-4α$. This unifies two directions from which one might approach the Clay prize problem.
In this article, we provide a simple method for constructing dispersive blow-up solutions to the nonlinear Schrödinger equation. Our construction mainly follows the approach in Bona, Ponce, Saut and Sparber [2]. However, we make use of the dispersive estimate to enjoy the smoothing effect of the Schrödinger propagator in the integral term appearing in Duhamel's formula. In this way, not only do we simplify the argument, but we also reduce the regularity requirement to construct dispersive blow-ups. In addition, we provide more examples of dispersive blow-ups by constructing solutions that blow…
We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation $i \partial_t u + Δu = - |x|^b |u|^{p-1} u$ in the radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$. We show the global existence and energy scattering in the inter-critical regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. …
In their work on a sharp compactness theorem for the Yamabe problem, Khuri, Marques and Schoen apply a refined blow-up analysis (what we call `second order blow-up argument' in this article) to obtain highly accurate approximate solutions for the Yamabe equation. As for the conformal scalar curvature equation on S^n with n > 3, we examine the second order blow-up argument and obtain refined estimate for a blow-up sequence near a simple blow-up point. The estimate involves local effect from the Taylor expansion of the scalar curvature function, global effect from other blow-up points, and the b…
We consider the supercritical inhomogeneous nonlinear Schrödinger equation (INLS) $$i\partial_t u+Δu+|x|^{-b}|u|^{2σ}u=0,$$ where $(2-b)/N<σ<(2-b)/(N-2)$ and $0<b<\min\{2,N\}$. We prove a Gagliardo-Nirenberg type estimate and use it to establish sufficient conditions for global existence and blow-up in $H^1(\mathbb{R}^N)$.
In this article we study a normalised double obstacle problem with polynomial obstacles $ p^1\leq p^2$ under the assumption that $ p^1(x)=p^2(x)$ iff $ x=0$. In dimension two we give a complete characterisation of blow-up solutions depending on the coefficients of the polynomials $p^1, p^2$. In particular, we see that there exists a new type of blow-ups, that we call double-cone solutions since the coincidence sets $\{u=p^1\}$ and $\{u=p^2\}$ are cones with a common vertex. We prove the uniqueness of blow-up limits, and analyse the regularity of the free boundary in dimension two. In particula…
For dimensions $N \geq 4$, we consider the Brézis-Nirenberg variational problem of finding \[ S(εV) := \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω|\nabla u|^2 \, dx +ε\int_ΩV\, |u|^2 \, dx}{\left(\int_Ω|u|^q \, dx \right)^{2/q}}, \] where $q=\frac{2N}{N-2}$ is the critical Sobolev exponent and $Ω\subset \mathbb{R}^N$ is a bounded open set. We compute the asymptotics of $S(0) - S(εV)$ to leading order as $ε\to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient in…
We consider in this paper some class of perturbation for the semilinear wave equation with critical (in the conformal transform sense) power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using a two-step argument based on interpolation and a critical Gagliardo-Nirenberg inequality, we show that the blow-up rate of any sigular solution is given by the solution of the non perturbed associated ODE, namely $u" = u^p$.
We study the Keller-Segel model of chemotaxis and develop a composite particle-grid numerical method with adaptive time stepping which allows us to accurately resolve singular solutions. The numerical findings (in two dimensions) are then compared with analytical predictions regarding formation and interaction of singularities obtained via analysis of the stochastic differential equations associated with the Keller-Segel model.
We consider the focusing nonlinear Schrödinger equation with the critical inverse square potential. We give the first proof of the uniqueness of the ground state solution. Consequently, we obtain a sharp Hardy-Gagliardo-Nirenberg interpolation inequality. Moreover, we provide a complete characterization for the minimal mass blow-up solutions to the time dependent problem.
In Kenig and Toro's two-phase free boundary problem, one studies how the regularity of the Radon-Nikodym derivative $h= dω^-/dω^+$ of harmonic measures on complementary NTA domains controls the geometry of their common boundary. It is now known that $\log h \in C^{0,α}(\partial Ω)$ implies that pointwise the boundary has a unique blow-up, which is the zero set of a homogeneous harmonic polynomial. In this note, we give examples of domains with $\log h \in C(\partial Ω)$ whose boundaries have points with non-unique blow-ups. Philosophically the examples arise from oscillating or rotating a blow…
The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of $\mathbb{R}^n \to \mathbb{R}^n$ associated with the $n$-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case ($n>1$) are described. Existence or nonexistence of blow-ups in different dimensions, boundness of certain linear combinations of blow-up derivatives and the first occurrence of the gradient catastrophe are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any di…
We characterize the dynamics of the finite time blow up solutions with minimal mass for the focusing mass critical Hartree equation with $H^1(\mathbb{R}^4)$ data and $L^2(\mathbb{R}^4)$ data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we also analyze the mass concentration phenomenon of such blow up solutions.
We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows to define global solutions, even in the presence of blow-ups of the freezing rate. We provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Holder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root…
