We study blow-up and quantization phenomena for a sequence of solutions $(u_k)$ to the prescribed $Q$-curvature problem $$ (-Δ)^nu_k= Q_ke^{2nu_k}\quad \text{in }Ω\subset\mathbb{R}^{2n},\quad \int_Ωe^{2nu_k}dx\leq C,$$ under natural assumptions on $Q_k$. It is well-known that, up to a subsequence, either $(u_k)$ is bounded in a suitable norm, or there exists $β_k\to\infty$ such that $ u_k=β_k(φ+o(1))$ in $Ω\setminus (S_1\cup S_φ)$ for some non-trivial non-positive $n$-harmonic function $φ$ and for a finite set $S_1$, where $S_φ$ is the zero set of $φ$. We prove quantization of the total curvature $\int_{\tildeΩ}Q_ke^{2nu_k}dx$ on the region $\tildeΩ\Subset(Ω\setminus S_φ)$. We also consider a non-local case in dimension three.
We consider generic 2 x 2 singular Liouville systems on a smooth bounded domain in the plane having some symmetry with respect to the origin. We construct a family of solutions to which blow-up at the origin and whose local mass at the origin is a given quantity depending on the parameters of the system. We can get either finitely many possible blow-up values of the local mass or infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials.
We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-Δu= \la a(x) e^u-4πN δ_0\;\hbox{ in } Ω,\quad u=0 \;\hbox{ on }\partial Ω,$$ where $Ω$ is a smooth and bounded domain in $\R^2$ such that $0\inΩ$, $a(x)$ is a positive smooth function, $N$ is a positive integer and $\la>0$ is a small parameter. Here $δ_0$ defines the Dirac measure with pole at $0$. We find conditions on the function $a$ and on the domain $Ω$ under which there exists a solution $u_\la$ blowing up at $0$ and satisfying $\la\into a(x)e^{u_\la} \to 8π(N+1)$ as $\la\to 0^+$.
We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in $R^{2m}$, then that of a closed manifold and, finally, the particular case of the sphere $S^{2m}$. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in $R^{2m}$, concentration phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have…
We consider the drift-diffusion equation $$ u_t-\varepsilon Δu+\nabla\cdot(u\nabla K\star u)=0 $$ in the whole space with global-in-time bounded solutions. Mass concentration phenomena for radially symmetric solutions of this equation with small diffusivity are studied.
We consider a reaction-diffusion-advection equation arising from a biological model of migrating species. The qualitative properties of the globally attracting solution are studied and in some cases the limiting profile is determined. In particular, a conjecture of Cantrell, Cosner and Lou on concentration phenomena is resolved under mild conditions. Applications to a related parabolic competition system is also discussed.
Small-energy regularity for stationary weak solutions of the Liouville equation
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Mathematics, Yunnan Normal University（云南师范大学数学学院）
We study the existence of steady states to the Keller-Segel system with linear chemotactical sensitivity function on a smooth bounded domain in $\mathbb R^N,$ $N\ge3,$ having rotational symmetry. We find three different types of chemoattractant concentration which concentrate along suitable $(N-2)-$dimensional minimal submanifolds of the boundary. The corresponding density of the cellular slime molds exhibit in the limit one or more Dirac measures supported on those boundary submanifolds.
We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.
We consider here solutions of the nonlinear fractional Schrödinger equation $$ε^{2s}(-Δ)^s u+V(x)u=u^p.$$ We show that concentration points must be critical points for $V$. We also prove that, if the potential $V$ is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point as $ε$ tends to zero. In addition, if the potential $V$ is radial, then the minimizer is unique provided $ε$ is small.
We study the multiplicity and concentration of complex valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik-Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.
We perform a semiclassical analysis for the planar Schrödinger-Poisson system \[ \cases{ -\varepsilon^{2} Δψ+V(x)ψ= E(x) ψ\quad \text{in $\mathbb{R}^2$},\cr -ΔE= |ψ|^{2} \quad \text{in $\mathbb{R}^2$}, \cr } \tag{$SP_\varepsilon$} \] where $\varepsilon$ is a positive parameter corresponding to the Planck constant and $V$ is a bounded external potential. We detect solution pairs $(u_\varepsilon, E_\varepsilon)$ of the system $(SP_\varepsilon)$ as~$\ge \rightarrow 0$.
In this paper, we show that any solution of the nonlinear Schr{ö}dinger equation $iu\_t+Δu\pm|u|^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on the Bourgain's one~\cite{MR99f:35184}, which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega~\cite{MR1671214}. We also generalize to higher dimensions the results in Keraani~\cite{MR2216444} and Merle and Vega~\cite{…
In this paper, we deal with the following singular perturbed fractional elliptic problem $ ε^{} (-Δ)^{1/2}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}, $ where $ (-Δ)^{1/2}u$ is the square root of the Laplacian and $f(s)$ has exponential critical growth. Under suitable conditions on $f(s)$, we construct a localized bound state solution concentrating at an isolated component of the positive local minimum points of the potential of $V$ as $ε$ goes to $0.$
We study the existence of solutions with multiple concentration to the following boundary value problem $$-Δu=\e^2 e^u-4π\sum_{p\in Z}α_p δ_{p}\;\hbox{in} Ω,\quad u=0 \;\hbox{on}\partial Ω,$$ where $Ω$ is a smooth and bounded domain in $\R^2$, $α_{p}$'s are positive numbers, $Z\subset Ω$ is a finite set, $δ_p$ defines the Dirac mass at $p$, and $\e>0$ is a small parameter. In particular we extend the result of Del-Pino-Kowalczyk-Musso (\cite{delkomu}) to the case of several singular sources. More precisely we prove that, under suitable restrictions on the weights $α_p$, a solution exists with …
We consider the following nonlinear fractional Choquard equation $$ \varepsilon^{2s}(-Δ)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{μ-N}\left(\frac{1}{|x|^μ}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $0<μ<2s$, $N\geq 3$, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a smooth magnetic potential, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive potential with a local minimum and $f$ is a continuous nonlinearity with subcritical growth. By using variational methods…
We consider a spatially extended mesoscopic FitzHugh-Nagumo model with strong local interactions and prove that its asymptotic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. As the local interactions strongly dominate, the weak solution to the mesoscopic equation under consideration converges to the local equilibrium, which has the form of Dirac distribution concentrated to an averaged membrane potential. Our approach is based on techniques widely developed in kinetic theory (Wasserstein distance, relative entropy method), where macroscopic quantities…
We consider the following Liouville-type equation with exponential Neumann boundary condition: $$ -Δ\tilde u = \varepsilon^2 K(x) e^{2\tilde u}, \quad x\in D, \qquad \frac{\partial \tilde u}{\partial n} + 1 = \varepsilon κ(x) e^{\tilde u}, \quad x\in\partial D, $$ where $D\subset \mathbb R^2$ is the unit disc, $\varepsilon^2 K(x)$ and $\varepsilon κ(x)$ stand for the prescribed Gaussian curvature and the prescribed geodesic curvature of the boundary, respectively. We prove the existence of concentration solutions if $κ(x) + \sqrt{K(x)+κ(x)^2}$ ($x\in\partial D$) has a strictly local extremum p…
Relaxation of variational problems in the space of functions with bounded $\mathcal{B}$-variation: interaction with measures and lack of concentration phenomena
In this paper we deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger-Kirchhoff type equation \begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}} \iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy + \frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2} dx\right)[\varepsilon^{2s} (-Δ)^{s}u+ V(x)u]= f(u) \, \mbox{in} \mathbb{R}^{3} \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $(-Δ)^{s}$ is the fractional Laplacian, $M$ is a Kirchhoff function, $V$ is a continuous positive potential …
In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems $$ -Δu+V(x)u=A(εx)f(u), \quad x \in \R^{N}, \eqno{(P)_ε} $$ where $N \geq 2$, $ε>0$, $A:\R^{N}\rightarrow\R$ is a continuous function that satisfies $$ 0<\inf_{x\in\R^{N}}A(x)\leq\lim_{|x|\rightarrow+\infty}A(x)<\sup_{x\in\R^{N}}A(x)=A(0),\eqno{(A)} $$ $f:\R\rightarrow\R$ is a continuous function having critical growth, $V:\R^{N}\rightarrow\R$ is a continuous and $\Z^{N}$--periodic function with $0\notinσ(Δ+V)$. By using variational methods, we prove the existen…
We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order $3$ in $\R^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of $\R^3$ as the restriction of a metric on $\R^4_+$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $Φ$ in $\R^4_+$), in analogy with a $4$-dimensional result of Adimurthi-Robert-Struwe, and construct examples of su…
We study nonnegative, measure-valued solutions to nonlinear drift type equations modelling concentration phenomena related to Bose-Einstein particles. In one spatial dimension, we prove existence and uniqueness for measure solutions. Moreover, we prove that all solutions blow up in finite time leading to a concentration of mass only at the origin, and the concentrated mass absorbs increasingly the mass converging to the total mass as time goes to infinity. Our analysis makes a substantial use of independent variable scalings and pseudo-inverse functions techniques.
The seminal work \cite{bm} by Brezis and Merle has been pioneering in studying the bubbling phenomena of the mean field equation with singular sources. When the vortex points are not collapsing, the mean field equation possesses the property of the so-called "bubbling implies mass concentration". Recently, Lin and Tarantello in \cite{lt} pointed out that the "bubbling implies mass concentration" phenomena might not hold in general if the collapse of singularities occurs. In this paper, we shall construct the first concrete example of non-concentrated bubbling solution of the mean field equatio…
In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-Δ)^{s}u + V(x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $N\geq 2$, $(-Δ)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive continuous function, $M: [0, \infty)\rightarrow \mathbb{R}$ is a Kirchhoff function satisfying suitable co…
In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-Δ)^{s}u+V(x) u=Q_{u}(u, v)+\frac{1}{2^{*}_{s}}K_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\varepsilon^{2s} (-Δ)^{s}u+W(x) v=Q_{v}(u, v)+\frac{1}{2^{*}_{s}}K_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} u, v>0 &\mbox{ in } \R^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-Δ)^{s}$ is the fractional Laplacian operator, $V:\mathbb{R}^{N}\rightar…
For a smooth, bounded domain $Ω$, $s\in(0,1)$, $p\in \left(1,\frac{n+2s}{n-2s}\right)$ we consider the nonlocal equation $$ ε^{2s} (-Δ)^s u+u=u^p \quad {\mbox{in}}Ω$$ with zero Dirichlet datum and a small parameter $ε>0$. We construct a family of solutions that concentrate as $ε\to 0$ at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case $s=1$, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due …
In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first author in [15] but many questions were left open. Similar questions were also explicitly asked in subsequent papers, see Del Pino-Musso-Ruf [12], Malchiodi-Martinazzi [30] or Martinazzi [34]. We answer all of them, proving in particular that blow up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will …
This paper studies the concentration phenomena to nonlinear Schrodinger equations with magnetic potentials and constant electric potentials. We find that the magnetic field plays an important role in the location of concentrations if the electric potential is constant. This is a completely new result compared with the case of non-constant electric potentials.
We let $Ω$ be a smooth bounded domain of $\mathbb{R}^4$ and a sequence of fonctions $(V_k)_{k\in\mathbb{N}}\in C^0(Ω)$ such that $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(Ω)$. We consider a sequence of functions $(u_k)_{k\in\mathbb{N}}\in C^4(Ω)$ such that $$Δ^2 u_k=V_k e^{4u_k}$$ in $Ω$ for all $k\in\mathbb{N}$. We address in this paper the question of the asymptotic behaviour of the $(u_k)'s$ when $k\to +\infty$. The corresponding problem in dimension 2 was considered by Brézis-Merle and Li-Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-…
In this paper we study the concentration profile of various kind of symmetric solutions of some semilinear elliptic problems arising in astrophysics and in diffusion phenomena. Using a reduction method we prove that doubly symmetric positive solutions in a $2m$-dimensional ball must concentrate and blow up on $(m-1)$-spheres as the concentration parameter tends to infinity. We also consider axially symmetric positive solutions in a ball in $\mathbb{R}^N$, $N \geq 3$, and show that concentration and blow up occur on two antipodal points, as the concentration parameter tends to infinity.
We study the asymptotic profile, as $\hbar\rightarrow 0$, of positive solutions to $$-\hbar^2Δu+V(x)u-\hbar^{2+γ}uΔu^2=K(x)|u|^{p-2}u,\ \ x\in \mathbb{R}^N $$ where $γ\geq 0$ is a parameter with relevant physical interpretations, $V$ and $K$ are given potentials and $N\geq 5$. We investigate the concentrating behavior of solutions when $γ>0$ and, differently form the case $γ=0$ where the leading potential is $V$, the concentration is here localized by the source potential $K$. Moreover, surprisingly for $γ>0$ we find a different concentration behavior of solutions in the case $p=\frac{2N}{N-2}…
In this paper, we study the limiting behavior of Riemann solutions to the Euler equations of one-dimensional compressible fluid flow as $γ$ tends to one. We show that the limit solution forms the delta wave to the pressureless Euler system of one-dimensional compressible fluid flow in the distribution sense. Some numerical results exhibiting the phenomena of concentration are also presented.
We study the asymptotic behavior of the principal eigenvalue of a weakly coupled, cooperative linear elliptic system in a stationary ergodic heterogeneous medium. The system arises as the so-called multigroup diffusion model for neutron flux in nuclear reactor cores, the principal eigenvalue determining the criticality of the reactor in a stationary state. Such systems have been well-studied in recent years in the periodic setting, and the purpose of this work is to obtain results in random media. Our approach connects the linear eigenvalue problem to a system of quasilinear viscous Hamilton-J…
This paper is concerned with the spatial propagation of nonlocal dispersal equations with bistable or multistable nonlinearity in exterior domains. We obtain the existence and uniqueness of an entire solution which behaves like a planar wave front as time goes to negative infinity. In particular, some disturbances on the profile of the entire solution happen as the entire solution comes to the interior domain. But the disturbances disappear as the entire solution is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile and cont…
We consider the possible concentration in phase space of a sequence of eigenfunctions (or, more generally, a quasimode) of an operator whose principal symbol has completely integrable Hamilton flow. The semiclassical wavefront set $WF_h$ of such a sequence is invariant under the Hamilton flow. In principle this may allow concentration of $WF_h$ along positive codimension sub-tori of a Liouville torus $\mathcal{L}$ if there exist rational relations among the frequencies of the flow on $\mathcal{L}.$ We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. Th…
The pioneering work of Brezis-Merle [7], Li-Shafrir [27], Li [26] and Bartolucci-Tarantello [4] showed that any sequence of blow up solutions for (singular) mean field equations of Liouville type must exhibit a "mass concentration" property. A typical situation of blow-up occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case Lin-Tarantello in [30] pointed out that the phenomenon: "bubbling implies mass concentration" might not occur and new scenarios open for investigation. In this paper, we present two explicit examp…
In their seminal paper "Oscillations and concentrations in weak solutions of the incompressible fluid equations", R. DiPerna and A. Majda introduced the notion of measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a …
In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation $$ \left\{ \begin{array}{l} ε^{2α}(-Δ)_ρ^α u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^α(\mathbb{R}^n) \end{array} \right. $$ where $ε$ is a positive parameter, $0< α< 1$, $1<p<\frac{n+2α}{n-2α}$, $n>2α$; $(-Δ)_ρ^α$ is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius $ρ(x)>0$, $ρ, Q, K$ are competing functions. We study the existence of ground state and we analyze the behavior of se…
Concentration Phenomena of Normalized Solutions of Critical Biharmonic Equations with Combined Nonlinearities in $\mathbb{R}^{N}$
College of Mathematics, Changchun Normal University（长春师范大学数学学院）；Faculty of Education, University of Ljubljana（卢布尔雅那大学教育学院）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Mathematics, Physics and Mechanics, University of Ljubljana（卢布尔雅那大学数学、物理与力学研究所）
In this paper we study the existence of solutions and their concentration phenomena of a singularly perturbed semilinear Schrodinger equation with the presence of the critical Sobolev exponent.
In this article we study some aspects of dispersive and concentration phenomena for the Schrödinger equation posed on hyperbolic space $\mathbb{H}^n$, in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties : we prove that the dispersion inequality is valid, in a stronger form than the one on $\mathbb{R}^n$. However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for …
On $(M,g)$ a compact riemannian $4-$manifold we consider the prescribed $Q-$curvature equation defined on $M$ with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on $\mathbb R^4$ and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an in…
