Odd symmetry of least energy nodal solutions for the Choquard equation

We consider the Choquard equation (also known as the stationary Hartree equation or Schrödinger–Newton equation)

$?\mathrm{\Delta }u+u=(I_{\alpha }?|u{|}^p)|u{|}^{p?2}u.$

Here $I_{\alpha }$ stands for the Riesz potential of order $\alpha \in (0,N)$, and $\frac{N?2}{N+\alpha }<\frac{1}{p}\le \frac{1}{2}$. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when α is either close to 0 or close to N.     

On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.     

Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications

In this paper, we study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Minkowski-curvature problem

$\{\begin{array}{c}?{(u^{\prime }/\sqrt{1?u^{\prime 2}})}^{\prime }=\lambda f(u),\text{ in }(?L,L),\hfill \\ u(?L)=u(L)=0,\hfill \end{array}$

where $\lambda ,L>0$, $f\in C[0,\infty )\cap C^2(0,\infty )$ and $f(u)>0$ for $u\ge 0$. Furthermore, we show that, for sufficiently large $L>0$, the bifurcation curve $S_L$ may have arbitrarily many turning points. Finally, we apply these results to obtain the global bifurcation diagrams for Ambrosetti–Brezis–Cerami problem, Liouville–Bratu–Gelfand problem and perturbed Gelfand problem with the Minkowski-curvature operator, respectively. Moreover, we will make two lists which show the different properties of bifurcation curves for Minkowski-curvature problems, corresponding semilinear problems and corresponding prescribed curvature problems.     

Global uniqueness in an inverse problem for time fractional diffusion equations

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d?2$, with boundary ?M, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T)\times M$, $T>0$, with time-fractional Caputo derivative of order $\alpha \in (0,1)\cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solutions on a subset of ?M at fixed time. In the “flat” case where M is a compact subset of ${\mathbb{R}}^d$, two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation $\rho {?}_t^{\alpha }u?\mathrm{div}\left(a\mathrm{?}u\right)+qu=0$ on $(0,T)\times M$ are recovered simultaneously.     

Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball

In this article we obtain positive singular solutions of

(1)$?\mathrm{\Delta }u=|\mathrm{?}u{|}^p\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a small $C^2$ perturbation of the unit ball in ${\mathbb{R}}^N$. For $\frac{N}{N?1}<p<2$ we prove that if Ω is a sufficiently small $C^2$ perturbation of the unit ball there exists a singular positive weak solution u of (1). In the case of $p>2$ we prove a similar result but now the positive weak solution u is contained in $C^{0,\frac{p?2}{p?1}}(\stackrel{￣}{\mathrm{\Omega }})$ and yet is not in $C^{0,\frac{p?2}{p?1}+\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ for any $\epsilon >0$.     

Inviscid limit for the derivative Ginzburg–Landau equation with small data in modulation and Sobolev spaces

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg–Landau equation $u_t=(\nu +\mathrm{i})\mathrm{\Delta }u+{\overrightarrow{\lambda }}_1?\mathrm{?}({|u|}^{2}u)+({\overrightarrow{\lambda }}_2?\mathrm{?}u){|u|}^2+\alpha {|u|}^{2\delta }u$, where $\delta \in \mathbb{N}$, ${\overrightarrow{\lambda }}_1,{\overrightarrow{\lambda }}_2$ are complex constant vectors, $\nu \in [0,1]$, $\alpha \in \mathbb{C}$. For $n\ge 3$, we show that it is uniformly global well posed for all $\nu \in [0,1]$ if initial data $u_0$ in modulation space $M_{2,1}^s$ and Sobolev spaces $H^{s+n/2}$ ($s>3$) and $6u_0{6}_{L^2}$ is small enough. Moreover, we show that its solution will converge to that of the derivative Schrödinger equation in $C(0,T;L^2)$ if $\nu \to 0$ and $u_0$ in $M_{2,1}^s$ or $H^{s+n/2}$ with $s>4$. For $n=2$, we obtain the local well-posedness results and inviscid limit with the Cauchy data in $M_{1,1}^s$ ($s>3$) and $6u_0{6}_{L^1}?1$.     

Regularity of the Eikonal equation with two vanishing entropies

Let $\mathrm{\Omega }?{\mathbb{R}}^2$ be a bounded simply-connected domain. The Eikonal equation $\left|\mathrm{?}u\right|=1$ for a function $u:\mathrm{\Omega }?{\mathbb{R}}^2\to \mathbb{R}$ has very little regularity, examples with singularities of the gradient existing on a set of positive $H^1$ measure are trivial to construct. With the mild additional condition of two vanishing entropies we show ?u is locally Lipschitz outside a locally finite set. Our condition is motivated by a well known problem in Calculus of Variations known as the Aviles–Giga problem. The two entropies we consider were introduced by Jin, Kohn [26], Ambrosio, DeLellis, Mantegazza [2] to study the Γ-limit of the Aviles–Giga functional. Formally if u satisfies the Eikonal equation and if

(1)$\mathrm{?}?({\stackrel{?}{\mathrm{\Sigma }}}_{e_1{e}_2}(\mathrm{?}{u}^{\perp }))=0\text{ and }\mathrm{?}?({\stackrel{?}{\mathrm{\Sigma }}}_{{?}_1{?}_2}(\mathrm{?}{u}^{\perp }))=0\text{ distributionally in }\mathrm{\Omega },$

where ${\stackrel{?}{\mathrm{\Sigma }}}_{e_1{e}_2}$ and ${\stackrel{?}{\mathrm{\Sigma }}}_{{?}_1{?}_2}$ are the entropies introduced by Jin, Kohn [26], and Ambrosio, DeLellis, Mantegazza [2], then ?u is locally Lipschitz continuous outside a locally finite set.Condition (1) is motivated by the zero energy states of the Aviles–Giga functional. The zero energy states of the Aviles–Giga functional have been characterized by Jabin, Otto, Perthame [25]. Among other results they showed that if ${\lim }_{n\to \infty }?I_{{?}_n}(u_n)=0$ for some sequence $u_n\in W_0^{2,2}(\mathrm{\Omega })$ and $u={\lim }_{n\to \infty }?u_n$ then ?u is Lipschitz continuous outside a finite set. This is essentially a corollary to their theorem that if u is a solution to the Eikonal equation $\left|\mathrm{?}u\right|=1$ a.e. and if for every “entropy” Φ (in the sense of [18], Definition 1) function u satisfies $\mathrm{?}?[\mathrm{\Phi }(\mathrm{?}{u}^{\perp })]=0$ distributionally in Ω then ?u is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result in that we require only two entropies to vanish.The method of proof is to transform any solution of the Eikonal equation satisfying (1) into a differential inclusion $DF\in K$ where $K?M^{2\times 2}$ is a connected compact set of matrices without Rank-1 connections. Equivalently this differential inclusion can be written as a constrained non-linear Beltrami equation. The set K is also non-elliptic in the sense of Sverak [32]. By use of this transformation and by utilizing ideas from the work on regularity of solutions of the Eikonal equation in fractional Sobolev space by Ignat [23], DeLellis, Ignat [15] as well as methods of Sverak [32], regularity is established.     

The uniqueness of an indefinite nonlinear diffusion problem in population genetics,part II

We study the following Neumann problem which models the “complete dominance” case of population genetics of two alleles.

$\{\begin{array}{c}{u}_t=d{u}^{″}+g(x)u^2(1?u)\text{in}(0,1)\times (0,\infty ),\hfill \\ 0\le u\le 1\text{in}(0,1)\times (0,\infty ),\hfill \\ u^{\prime }(0,t)=u^{\prime }(1,t)=0\text{in}(0,\infty ),\hfill \end{array}$

where g changes sign in $(0,1)$. It is known that this equation has a nontrivial steady state $u_d$ for d sufficiently small [5]. It has been conjectured by Nagylaki and Lou [2] that $u_d$ is a unique nontrivial steady state if ${\int }_{\mathrm{\Omega }}g(x)dx\ge 0$. This was proved in [6] if g changes sign only once. In this paper under additional condition on $g(x)$ we treat the case when g has multiple zeros.     

A Hardy type inequality for W02,1(Ω) functions

We consider functions $u\in W_0^{2,1}(\Omega )$, where $\Omega ?{\mathbb{R}}^N$ is a smooth bounded domain. We prove that $\frac{u(x)}{d(x)}\in W_0^{1,1}(\Omega )$ with$6\mathrm{?}\left(\frac{u(x)}{d(x)}\right)6_{L^1(\Omega )}?C6u{6}_{W^{2,1}(\Omega )},$ where d is a smooth positive function which coincides with $\mathrm{dist}(x,?\Omega )$ near ?Ω and C is a constant depending only on d and Ω.     

On a nonlinear Schrödinger equation with a localizing effect

We consider the nonlinear Schrödinger equation associated to a singular potential of the form $a{|u|}^{?(1?m)}u+bu$, for some $m\in (0,1)$, on a possible unbounded domain. We use some suitable energy methods to prove that if $\mathrm{Re}(a)+\mathrm{Im}(a)>0$ and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any $t>0$. This property contrasts with the behavior of solutions associated to regular potentials $(m?1)$. Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential $a{|u|}^{?(1?m)}u$. The existence of solutions is obtained by some compactness methods under additional conditions. To cite this article: P. Bégout, J.I. Díaz, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Existence and convexity of local solutions to degenerate Hessian equations

In this work, we prove the existence of convex solutions to the following k-Hessian equation

$S_k[u]=K(y)g(y,u,Du)$

in the neighborhood of a point $(y_0,u_0,p_0)\in {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n$, where $g\in C^{\infty },g(y_0,u_0,p_0)>0$, $K\in C^{\infty }$ is nonnegative near $y_0$, $K(y_0)=0$ and $\text{Rank}(D_y^{2}K)(y_0)\ge n?k+1$.     

Majoration of the dimension of the space of concatenated solutions to a specific pantograph equation

For each $\lambda \in {\mathbb{N}}^{?}$, we consider the integral equation:

$\underset{\lambda y}{\overset{\lambda x}{\int }}f(t)\mathrm{d}t=f(x)?f(y)\text{ for every }(x,y)\in {\mathbb{R}}_{+}^2\text{,}$

where f is the concatenation of two continuous functions $f_a,f_b:[0,\lambda ]\to \mathbb{R}$ along a word $u=u_0{u}_1?\in {\{a,b\}}^{\mathbb{N}}$ such that $u=\sigma (u)$, where σ is a λ-uniform substitution satisfying some combinatorial conditions.There exists some non-trivial solutions ([1]). We show in this work that the dimension of the set of solutions is at most two.     

Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well

In this paper, we study the following fractional Kirchhoff equations

$\{\begin{array}{c}(a+b{\int }_{{\mathbb{R}}^N}|{(?△)}^{\frac{\alpha }{2}}u{|}^2\mathrm{d}x){(?△)}^{\alpha }u+\lambda V(x)u=(|x{|}^{?\mu }?G(u))g(u),\hfill \\ u\in H^{\alpha }({\mathbb{R}}^N),N\ge 3,\hfill \end{array}$

where $a,b>0$ are constants, and ${(?△)}^{\alpha }$ is the fractional Laplacian operator with $\alpha \in (0,1),2<2_{\alpha ,\mu }^{?}=\frac{2N?\mu }{N?2\alpha }\le 2_{\alpha }^{?}=\frac{2N}{N?2\alpha }$, $0<\mu <2\alpha$, $\lambda >0$, is real parameter. $2_{\alpha }^{?}$ is the critical Sobolev exponent. g satisfies the Berestycki–Lions-type condition (see [2]). By using Poho?aev identity and concentration-compact theory, we show that the above problem has at least one nontrivial solution. Furthermore, the phenomenon of concentration of solutions is also explored. Our result supplements the results of Lü (see [8]) concerning the Hartree-type nonlinearity $g(u)=|u{|}^{p?1}u$ with $p\in (2,6?\alpha )$.