arXiv subjects
Publications and source records attributed to Juncheng Wei.
Starting from a bound state (positive or sign-changing) solution to $$ -Δω_m =|ω_m|^{p-1} ω_m -ω_m \ \ \mbox{in}\ \R^n, \ ω_m \in H^2 (\R^n)$$ and solutions to the Helmholtz equation $$ Δu_0 + λu_0=0 \ \ \mbox{in} \ \R^n, \ λ>0, $$ we build new Dancer's type entire solutions to the nonlinear scalar equation $$ -Δu =|u|^{p-1} u-u \ \ \mbox{in} \ \R^{m+n}. $$
Using the method of sub-super-solution, we construct a solution of $(-Δ)^su-cu_z-f(u)=0$ on $\R^3$ of pyramidal shape. Here $(-Δ)^s$ is the fractional Laplacian of sub-critical order $1/2<s<1$ and $f$ is a bistable nonlinearity. Hence, the existence of a traveling wave solution for the parabolic fractional Allen-Cahn equation with pyramidal front is asserted. The maximum of planar traveling wave solutions in various directions gives a sub-solution. A super-solution is roughly defined as the one-dimensional profile composed with the signed distance to a rescaled mollified pyramid. In the main estimate we use an expansion of the fractional Laplacian in the Fermi coordinates.
Let $X$ be an asymptotically hyperbolic manifold and $M$ its conformal infinity. This paper is devoted to deduce several existence results of the fractional Yamabe problem on $M$ under various geometric assumptions on $X$ and $M$: Firstly, we handle when the boundary $M$ has a point at which the mean curvature is negative. Secondly, we re-encounter the case when $M$ has zero mean curvature and is either non-umbilic or umbilic but non-locally conformally flat. As a result, we replace the geometric restrictions given by González-Qing (Analysis and PDE, 2013) and González-Wang (arXiv:1503.02862) with simpler ones. Also, inspired by Marques (Comm. Anal. Geom., 2007) and Almaraz (Pacific J. Math., 2010), we study lower-dimensional manifolds. Finally, the situation when $X$ is Poincaré-Einstein, $M$ is either locally conformally flat or 2-dimensional is covered under the validity of the positive mass theorem for the fractional conformal Laplacians.
We consider the following class of equations with exponential nonlinearities on a compact surface $M$: $$ - Δu = ρ_1 \left( \frac{h_1 \,e^{u}}{\int_M h_1 \,e^{u} } - \frac{1}{|M|} \right) - ρ_2 \left( \frac{h_2 \,e^{-u}}{\int_M h_2 \,e^{-u} } - \frac{1}{|M|} \right), $$ which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h_1, h_2$ are smooth positive functions and $ρ_1, ρ_2$ are two positive parameters. We start by proving a concentration phenomena for the above equation, which leads to a-priori bound for the solutions of this problem provided $ρ_i\notin 8π\mathbb{N}, \, i=1,2$. Then we study the blow up behavior when $ρ_1$ crosses $8π$ and $ρ_2 \notin 8π\mathbb{N}$. By performing a suitable decomposition of the above equation and using the shadow system that was introduced for the $SU(3)$ Toda system, we can compute the Leray-Schauder topological degree for $ρ_1 \in (0,8π) \cup (8π,16π)$ and $ρ_2 \notin 8π\mathbb{N}$. As a byproduct our argument, we give new existence results when the underlying manifold is a sphere and a new proof for some known existence result.
The aim of this paper is to use a selection process and a careful study of the interaction of bubbling solutions to show a classification result for the blow-up values of the elliptic sinh-Gordon equation $$Δu+h_1e^u-h_2e^{-u}=0 \quad \mathrm{in}~B_1\subset\mathbb{R}^2.$$ In particular we get that the blow-up values are multiple of $8π.$ It generalizes the result of Jost, Wang, Ye and Zhou \cite{jwyz} where the extra assumption $h_1 = h_2$ is crucially used.
We consider the problem of finding positive solutions of the problem $Δu - λu +u^5 = 0$ in a bounded, smooth domain $Ω$ in $\mathbb{R}^3$, under zero Neumann boundary conditions. Here $λ$ is a positive number. We analyze the role of Green's function of $-Δ+λ$ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when $λ$ is regarded as a parameter. As a special case of our results, we find and characterize a positive value $λ_*$ such that if $λ-λ^*>0$ is sufficiently small, then this problem is solvable by a solution $u_λ$ which blows-up by bubbling at a certain interior point of $Ω$ as $λ\downarrow λ_*$.
We derive monotonicity formulae for solutions of the fractional Hénon-Lane-Emden equation \begin{equation*} (-Δ)^{s} u=|x|^a |u|^{p-1} u \ \ \ \text{in } \ \ \mathbb{R}^n, \end{equation*} when $0 0$ and $p>1$. Then, we apply these formulae to classify stable solutions of the above equation.
We classify finite Morse index solutions of the fractional Lane-Emden equation $(-Δ)^{s} u=|u|^{p-1} u \ \ \ \mathbb{R}^n $ for $1<s<2$. For the local case, $s=1$ and $s=2$ this classification was done by Farina in [10] and Davila, Dupaigne, Wang and Wei in [8], respectively. Moreover, for the nonlocal case, $0<s<1$, finite Morse index solutions are classified by Davila, Dupaigne and Wei in [7].
We give negative answers to Lin-Ni's conjecture for any four and six dimensional domains. No condition on the symmetry, geometry nor topology of the domain is needed.
We study sign-changing radial solutions for the following semi-linear elliptic equation \begin{align*} Δu-u+|u|^{p-1}u=0\quad{\rm{in}}\ \mathbb{R}^N,\quad u\in H^1(\mathbb{R}^N), \end{align*} where $1<p<\frac{N+2}{N-2}$, $N\geq3$. It is well-known that this equation has a unique positive radial solution and sign-changing radial solutions with exactly $k$ nodes. In this paper, we show that such sign-changing radial solution is also unique when $p$ is close to $\frac{N+2}{N-2}$. Moreover, those solutions are non-degenerate, i.e., the kernel of the linearized operator is exactly $N$-dimensional.
Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [\hat{h}])$. The fractional Yamabe problem addresses to solve \[P^γ[g^+,\hat{h}] (u) = cu^{n+2γ\over n-2γ}, \quad u > 0 \quad \text{on } M\] where $c \in \mathbb{R}$ and $P^γ[g^+,\hat{h}]$ is the fractional conformal Laplacian whose principal symbol is $(-Δ)^γ$. In this paper, we construct a metric on the half space $X = \mathbb{R}^{n+1}_+$, which is conformally equivalent to the unit ball, for which the solution set of the fractional Yamabe equation is non-compact provided that $n \ge 24$ for $γ\in (0, γ^*)$ and $n \ge 25$ for $γ\in [γ^*,1)$ where $γ^* \in (0, 1)$ is a certain transition exponent. The value of $γ^*$ turns out to be approximately 0.940197.
We prove that the following pointwise inequality holds \begin{equation*} -Δu \ge \sqrt\frac{2}{(p+1)-c_n} |x|^{\frac{a}{2}} u^{\frac{p+1}{2}} + \frac{2}{n-4} \frac{|\nabla u|^2}{u} \ \ \text{in}\ \ \mathbb{R}^n \end{equation*} where $c_n:=\frac{8}{n(n-4)}$, for positive bounded solutions of the fourth order Hénon equation that is \begin{equation*} Δ^2 u = |x|^a u^p \ \ \ \ \text {in }\ \ \mathbb{R}^n \end{equation*} for some $a\ge0$ and $p>1$. Motivated by the Moser's proof of the Harnack's inequality as well as Moser iteration type arguments in the regularity theory, we develop an iteration argument to prove the above pointwise inequality. As far as we know this is the first time that such an argument is applied towards constructing pointwise inequalities for partial differential equations. An interesting point is that the coefficient $\frac{2}{n-4}$ also appears in the fourth order $Q$-curvature and the Paneitz operator. This in particular implies that the scalar curvature of the conformal metric with conformal factor $u^\frac{4}{n-4}$ is positive.
We consider critical exponent semi-linear elliptic equation with coefficient K(x) periodic in its first k variables, with 2k smaller than n-2. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in Rk, including infinite lattices. We also show that for 2k greater than or equal to n-2, no such solutions exist.
We consider the following singularly perturbed Neumann problem \begin{eqnarray*} \ve^2 Δu -u +u^p = 0 \, \quad u>0 \quad {\mbox {in}} \quad Ω, \quad {\partial u \over \partial ν}=0 \quad {\mbox {on}} \quad \partial Ω, \end{eqnarray*} where $p>2$ and $Ω$ is a smooth and bounded domain in $\R^2$. We construct a new class of solutions which consist of large number of spikes concentrating on a {\bf segment} of the boundary which contains a local minimum point of the mean curvature function and has the same mean curvature at the end points. We find a continuum limit of ODE systems governing the interactions of spikes and show that the mean curvature function acts as {\em friction force}.
We prove that the non-radial sign-changing solutions to the nonlinear Schrödinger equation \begin{equation*} Δu-u+|u|^{p-1}u=0 \mbox{ in }\R^N, \quad u \in H^1 (\R^N ) \end{equation*} constructed by Musso, Pacard and Wei is non-degenerate. This provides the first example of non-degenerate sign-changing solution with finite energy to the above nonlinear Schrödinger equation.
In this article we prove that for locally defined singular SU(n+1) Toda systems in R^2, the profile of fully bubbling solutions near the singular source can be accurately approximated by global solutions. The main ingredients of our new approach are the classification theorem of Lin-Wei-Ye and the non-degeneracy of the linearized Toda system, which make us overcome the difficulties that come from the lack of symmetry and the singular source.
On any closed Riemannian manifold of dimension greater than $7$, we construct examples of background physical coefficients for which the Einstein-Lichnerowicz equation possesses a non-compact set of positive solutions. This yields in particular the existence of an infinite number of positive solutions in such cases.
In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation $α^2 Δu + u(1-u^2)=0, \quad \hbox{in }Ω\subset \R^N $ where $N=3$, $Ω$ is a smooth bounded domain and $\A>0$ is a small parameter. We provide asymptotic behavior which shows that, as $α\to 0$, the level sets of the solutions collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature that intersects orthogonally $\partial Ω$ of the domain and that is non-degenerate respect to $Ω$. We provide explicit examples of surfaces to which our result applies.