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-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author, a similar quantization phenomenon does not hold for this fourth order problem. Assuming that the $u_k$'s are radially symmetrical, we push further the previous analysis. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way.
We consider the following class of focusing $L^2$-supercritical fourth-order nonlinear Schrödinger equations \[ i\partial_t u - Δ^2 u + μΔu = - |u|^αu, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N, \] where $N\geq 2$, $μ\geq 0$, and $\frac{8}{N}<α<α^*$ with $α^*:=\frac{8}{N-4}$ if $N\geq 5$ and $α^*=\infty$ if $N\leq 4$. By using the localized Morawetz estimates and radial Sobolev embedding, we establish the energy scattering below the ground state threshold for the equation with radially symmetric initial data. We also address the existence of finite time blow-up radial solutions to the equ…
We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded domains; moreover, near the origin, the gradient graph of such a solution is continuous if and only if the graph is special Lagrangian.
The necessary and sufficient conditions for a regular positive entire solution $u$ of the biharmonic equation: \begin{equation} \label{0.1} -Δ^2 u=u^{-p} \;\; \mbox{in $\R^N \; (N \geq 3)$}, \;\; p>1 \end{equation} to be a radially symmetric solution are obtained via the moving plane method (MPM) of a system of equations. It is well-known that for any $a>0$, \eqref{0.1} admits a unique minimal positive entire radial solution ${\underline u}_a (r)$ and a family of non-minimal positive entire radial solutions $u_a (r)$ such that $u_a (0)={\underline u}_a (0)=a$ and $u_a (r) \geq {\underline u}_a…
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 radially symmetric, energy critical wave maps from (1 + 2)-dimensional Minkowski space into the unit sphere $\mathbb{S}^m$, $m \geq 1$, and prove global regularity and scattering for classical smooth data of finite energy. In addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. This extends and complements the beautiful classical work of Christodoulou-Tahvildar-Zadeh [3, 4] and Struwe [31, 33] as well as of Nahas [22] on radial w…
We are concerned on the fourth-order elliptic equation \begin{equation}\tag{$P_λ$} \left\{ \begin{array}[c]{ll} Δ^2 u- Δu + V(x)u -λΔ[ρ(u^2)]ρ'(u^2)u= f(u)\, \, \mbox{in} \, \, \mathbb{R}^N, & u\in W^{2,2}(\mathbb{R}^N), \end{array} \right. \end{equation} where $Δ^2 = Δ(Δ)$ is the biharmonic operator, $3\leq N\leq 6$, the radially symmetric potential $V$ may change sign and $\inf_{\mathbb{R}^N}V(x)=-\infty$ is allowed. If $f$ satisfies a type of nonquadracity and monotonicity conditions and $ρ$ is a suitable smooth function, we prove, via variational approach, the existence of a radially symme…
We consider radial solutions of the slightly subcritical problem $-Δu_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as $\varepsilon\to 0$) of all local minima and maxima of $u_\varepsilon$ as well as its derivative at roots. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. A…
This note studies the asymptotic behavior of global solutions to the fourth-order generalized Hartree equation $$i\dot u+Δ^2 u\pm(I_α*|u|^p)|u|^{p-2}u=0.$$ Indeed, for both attractive and repulsive sign, the scattering is obtained in the mass super-critical and energy sub-critical regimes, with radial setting.
In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schrödinger equation. We firstly establish the profile decomposition of bounded sequences in $\dot{H}^{γ_{\text{c}}} \cap \dot{H}^2$. We also prove a compactness lemma and a variational characterization of ground states related to the equation. As a result, we obtain the $\dot{H}^{γ_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{γ_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{…
We investigate the next Trudinger-Moser critical equations, \[ \begin{cases} -Δu=λue^{u^2+α|u|^β}&\text{ in }B,\\ u=0&\text{ on }\partial B, \end{cases} \] where $α>0$, $(λ,β)\in(0,\infty)\times(0,2)$ and $B\subset \mathbb{R}^2$ is the unit ball centered at the origin. We classify the asymptotic behavior of energy bounded sequences of radial solutions. Via the blow--up analysis and a scaling technique, we deduce the limit profile, energy, and several asymptotic formulas of concentrating solutions together with precise information of the weak limit. In particular, we obtain a new necessary cond…
This manuscript proves the energy scattering of global solutions to a repulsive fourth-order generalized Hartree equation with non-radial data in the inter-critical regime. This work uses a new approach due to Dodson-Murphy [4] and extends the previous work [14] by removing the spherically symmetric assumption on the data.
The chemotaxis system \begin{align*} u_t &= Δu - \nabla \cdot (u\nabla v), \\ v_t &= Δv - uv, \end{align*} is considered under the boundary conditions $\frac{\partial u}{\partialν}- u\frac{\partial v}{\partialν}=0$ and $v=v_\star$ on $\partialΩ$, where $Ω\subset\mathbb{R}^n$ is a ball and $v_\star$ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case $n=2$, while global weak solutions are constructed when $n\in \{3,4,5\}$. This is achieved by analyzing an energy-t…
This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation $$i\dot u+Δ^2 u+F(x,u)=0 .$$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to Dodson-Murphy \cite{dm}.
We verify the existence of radial positive solutions for the semi-linear equation $$ -\,Δu=u^{p}\,-\,V(y)\,u^{q},\,\quad\quad u>0,\quad\quad\mbox{ in }\mathbb{R}^N$$ where $N\geq 3$, $p$ is close to $p^*:=(N+2)/(N-2)$, and $V$ is a radial smooth potential. If $q$ is super-critical, namely $q>p^*$, we prove that this Problem has a radial solution behaving like a super-position of bubbles blowing-up at the origin with different rates of concentration, provided $V(0)<0$. On the other hand, if $N/(N-2)<q<p^*$, we prove that this Problem has a radial solution behaving like a super-position of {\it …
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 are concerned with solutions to the linear wave equation. We give an asymptotic formula for large time, valid in the energy space, via an operator related to the Radon transform. This allows us to show that the energy is concentrated near the light cone. This allows to derive further expressions the exterior energy (outside a shifted light cone). We in particular generalize the formulas of [CKS14] obtained in the radial setting. In odd dimension, we study the discrepancy of the exterior energy regarding initial energy, and prove in the general case the results of [KLLS15] (which were restri…
We study the ground states of the following generalization of the Kirchhoff-Love functional, $$J_σ(u)=\int_Ω\dfrac{(Δu)^2}{2} - (1-σ)\int_Ωdet(\nabla^2u)-\int_ΩF(x,u),$$ where $Ω$ is a bounded convex domain in $\mathbb{R}^2$ with $C^{1,1}$ boundary and the nonlinearities involved are of sublinear type or superlinear with power growth. These critical points correspond to least-energy weak solutions to a fourth-order semilinear boundary value problem with Steklov boundary conditions depending on $σ$. Positivity of ground states is proved with different techniques according to the range of the pa…
By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*} \int_{\mathbb{R}^N}|x|^α|Δu|^2 dx \geq S^{rad}(N,α)\left(\int_{\mathbb{R}^N}|x|^{-α}|u|^{p^*_α} dx\right)^{\frac{2}{p^*_α}}, \quad u\in C^\infty_c(\mathbb{R}^N), \end{equation*} where $N\geq 3$, $4-N<α<2$, $p^*_α=\frac{2(N-α)}{N-4+α}$. Then we obtain the explicit form of the unique (up to scaling) radial positive solution $U_{λ,α}$ to the weighted fourth-order Hardy (for $α>0$) or…
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 consider the drift-diffusion equation $u_t-εΔu + \nabla \cdot(u\nabla K^*u)=0$ in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case $K(x)=-|x|$. We quantify the mass concentration phenomenon, a genuinely nonlinear effect, for radially symmetric solutions of this equation for small diffusivity $ε$ studied in our previous paper [3], obtaining optimal sharp upper and lower bounds for Sobolev norms.
In this note, we consider the following problem, \begin{equation*} \begin{cases} -Δu=(1+g(x))u^{\frac{N+2}{N-2}},\ u>0\text{ in }B,\\ u=0\text{ on }\partial B, \end{cases} \end{equation*} where $N\ge3$ and $B\subset \mathbb{R}^N$ is a unit ball centered at the origin and $g(x)$ is a radial Hölder continuous function such that $g(0)=0$. We prove the existence and nonexistence of radial solutions by the variational method with the concentration compactness analysis and the Pohozaev identity.
In this paper, we consider the weighted fourth order equation $$Δ(|x|^{-α}Δu)+λ\text{div}(|x|^{-α-2}\nabla u)+μ|x|^{-α-4}u=|x|^βu^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq 5$, $-n<α<n-4$, $p>1$ and $(p,α,β,n)$ belongs to the critical hyperbola $$\frac{n+α}{2}+\frac{n+β}{p+1}=n-2.$$ We prove the existence of radial solutions to the equation for some $λ$ and $μ$. On the other hand, let $v(t):=|x|^{\frac{n-4-α}{2}}u(|x|)$, $t=-\ln |x|$, then for the radial solution $u$ with non-removable singularity at origin, $v(t)$ is a periodic function if $α\in (-2,n-4)$ and $λ$, $μ…
Typically, aggregation-diffusion is modeled by parabolic equations that combine linear or nonlinear diffusion with a Fokker-Planck convection term. Under very general suitable assumptions, we prove that radial solutions of the evolution process converge asymptotically in time towards a stationary state representing the balance between the two effects. Our parabolic system is the gradient flow of an energy functional, and in fact we show that the stationary states are minimizers of a relaxed energy. Here, we study radial solutions of an aggregation-diffusion model that combines nonlinear fast d…
We show that the solutions of the three-dimensional critical defocusing nonlinear wave equation with Neumann boundary conditions outside a ball and radial initial data scatter. This is to our knowledge the first result of scattering for a nonlinear wave equation with Neumann boundary conditions. Our proof uses the scheme of concentration-compactness/rigidity introduced by Kenig and Merle, extending it to our setup, together with the so-called channels of energy method to rule out compact-flow solutions. We also obtain, for the focusing equation, the same exact scattering/blow-up dichotomy belo…
We study qualitative properties for nonnegative solutions to a conformally invariant coupled system of fourth order equations involving critical exponents. For solutions defined in the punctured space, there exist essentially two cases to analyze. If the origin is a removable singularity, we prove that non-singular solutions are rotationally invariant and weakly positive. More precisely, they are the product of a fourth order spherical solution by a unit vector with nonnegative coordinates. If the origin is a non-removable singularity, we show that the solutions are radially symmetric and stro…
