arXiv subjects

Chengxiang Zhang

Publications and source records attributed to Chengxiang Zhang.

13 recordsLinked to original sources

arXiv2026

Segregated Bubbling Solutions for a Critical Schr\"odinger System of Brezis--Nirenberg Type with Sublinear Competitive Coupling

We construct segregated bubbling solutions for a two-component critical Schr\"odinger system of Brezis--Nirenberg type in a smooth bounded domain $\Omega\subset\mathbb R^N$, $N\geq5$, with any fixed competitive coupling $\beta<0$. Suppose that the Robin function has two distinct prescribed critical points, each satisfying a local degree condition. For every sufficiently small $\epsilon>0$, the system admits a nonnegative weak solution with both components nontrivial. Each component has a single-bubble profile and concentrates at one of the prescribed points. Each component also vanishes identically in a ball centered at the other concentration point; after rescaling by the natural bubble length, the radius of this ball tends to infinity. The main obstruction is that $p=N/(N-2)\in(1,2)$, so the gradient of the interaction potential $(s,t)\mapsto |s|^p|t|^p$ is not differentiable when one component vanishes and the other is nonzero. Hence the usual global Lyapunov--Schmidt reduction cannot be applied directly. We first solve a nonlinear exterior problem variationally. The resulting dead cores remove the cross-component coupling from the inner localization regions, where a projected critical reduction can then be carried out. The remaining scale and center equations are solved by Brouwer degree theory.

math.AP↗

arXiv2026

Uniqueness of bound states for sublinear elliptic equations

We investigate the uniqueness of radial bound state solutions to the sublinear elliptic equation \[ \begin{cases} -\Delta u - u + |u|^{q-2}u = 0 & \text{in } \mathbb{R}^n,\cr u(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q\in(1,2)$ and $n\geq 2$. A distinctive feature of this problem is the non-Lipschitz singularity of the nonlinearity at the origin, which gives rise to compactly supported ground states and bound states. Using a shooting argument together with a detailed analysis of the linearized variation with respect to the initial value, we prove that for every prescribed integer $k\geq 1$, the equation admits exactly one radial bound state solution with $k$ simple zeros, up to sign reflection and spatial translation. In addition, our analysis yields a classification of radial solutions according to the initial value and describes their behavior near the finite support boundary.

math.AP↗

arXiv2026

Segregated Solutions to Critical Elliptic Systems in High Dimensions ($N \geq 5$)

We study the existence of multiple segregated solutions to the critical coupled Schrödinger system \[ \begin{cases} -Δu_{1} = K_1(| y|) | u_{1}|^{2^*-2}u_{1}+β| u_{2}|^{\frac{2^{*}}{2}}| u_{1}|^{\frac{2^{*}}{2}-2}u_{1}, & y\in \mathbb R^N,\\ -Δu_{2} = K_2(| y|) | u_{2}|^{2^*-2}u_{2}+β| u_{1}|^{\frac{2^{*}}{2}}| u_{2}|^{\frac{2^{*}}{2}-2}u_{2}, & y\in\mathbb R^N,\\ u_{1},u_{2}\geq0, u_{1},u_{2}\in C_0(\mathbb R^{N})\cap D^{1,2}(\mathbb R^N), \end{cases} \] with $N \geq 5$, $2^* = \frac{2N}{N-2}$, radial potentials $K_1, K_2 > 0$,and repulsive coupling $β< 0$.Under the assumption that $K_1$ and $K_2$ attain local maxima at distinct radii $r_0 \ne ρ_0$ with precise asymptotic expansions near these points, we prove the existence of infinitely many non-radial segregated solutions $(u_{1,k}, u_{2,k})$ for all sufficiently large integers $k$. These solutions exhibit multiple bumps concentrating on two separate circles of radius $r_0$ and $ρ_0$ respectively. Moreover, each component develops a "dead core'' near the concentration points of the other. The proof overcomes the sublinear and non-smooth nature of the coupling term ($2^*/2 -1 < 1$) by constructing a tailored complete metric space and combining a finite-dimensional reduction with a novel tail minimization argument.

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arXiv2025

Multi-bump solutions for sublinear elliptic equations with nonsymmetric coefficients

We investigate the existence of nonnegative bump solutions to the sublinear elliptic equation \[ \begin{cases} -Δv - K(x)v + |v|^{q-2}v = 0 & \text{in } \mathbb{R}^N, \\ v(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q \in (1,2)$, $ N \geq 2$, and the potential $K \in L^p_{\mathrm{loc}}(\mathbb{R}^N)$ with $p > N/2$ is a function without any symmetry assumptions. Under the condition that $\|K - 1\|_{L^p_{\mathrm{loc}}}$ is sufficiently small, we construct infinitely many solutions with arbitrarily many bumps. The construction is challenged by the sensitive interaction between bumps, whose limiting profiles have compact support. The key to ensuring their effective separation lies in obtaining sharp estimates of the support sets. Our method, based on a truncated functional space, provides precisely such control. We derive qualitative local stability estimates in region-wise maximum norms that govern the size of each bump's essential support, confining its core to a designated region and minimizing overlap. Crucially, these estimates are uniform in the number of bumps, which is the pivotal step in establishing the existence of solutions with infinitely many bumps.

math.AP↗

arXiv2025

Segregated solutions for nonlinear Schrödinger systems with sublinear coupling terms

We establish the existence of infinitely many nonnegative, segregated solutions for the sublinearly coupled Schrödinger system \begin{equation*} \left\{\begin{aligned}-Δu+K_1(x)u&=μu^{p-1}+ (σ_1+1)βu^{σ_1}v^{σ_2+1}, &x\in\mathbb{R}^N&, -Δv+K_2(x)v&=νv^{p-1}+(σ_2+1)βu^{σ_1+1}v^{σ_2}, &x\in\mathbb{R}^N&,\end{aligned}\right. \end{equation*}where $N \geq 2$, $p \in (2,2^*)$, $2^* = 2N/(N-2)$ ($2^* = \infty$ if $N=2$), $K_j$ are radial potentials, $μ, ν> 0$, $β\in \mathbb{R}$, and critically $σ_j \in (0,1)$. The sublinear coupling exponents $σ_j$ introduce fundamental challenges due to nonsmooth nonlinearities and singularities in standard reduction methods. To overcome this, we develop an enhanced Lyapunov-Schmidt reduction framework. By recasting the problem within a specially constructed metric space of local minimizers for an outer boundary value problem, we derive sharp a priori estimates enabling contraction mapping arguments. This approach circumvents the limitations of classical methods for sublinear couplings. We further uncover a novel "dead core" phenomenon: solutions $(u_\ell, v_\ell)$ exhibit non-strict positivity with topological segregation. Specially, for $N=2$ and large integers $\ell$, there exist radii $0 < R_1 < R_2$ such that $\text{supp } u_\ell \subset B_{R_2}(0)$, $\text{supp } v_\ell \subset \mathbb{R}^N \setminus B_{R_1}(0)$, and $u_\ell + v_\ell \to 0$ uniformly in $B_{R_2}(0) \setminus B_{R_1}(0)$ as $\ell \to \infty$. Our methodology provides a versatile framework for handling nonsmooth nonlinearities in reduction techniques.

math.AP↗

arXiv2024

Uniqueness and Nondegeneracy of ground states of $ -Δu + (-Δ)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$}$ when $s$ is close to $0$ and $1$

We are concerned with the mixed local/nonlocal Schrödinger equation \begin{equation} - Δu + (-Δ)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$,} \end{equation} for arbitrary space dimension $n\geqslant1$, $s\in(0,1)$, and $p\in(0,2^*-2)$ with $2^*$ the critical Sobolev exponent. We provide the existence and several fundamental properties of nonnegative solutions for the above equation. And then, we prove that, if $s$ is close to $0$ and $1$, respectively, such equation then possesses a unique (up to translations) ground state, which is nondegenerate.

math.AP↗

arXiv2024

A New Approach to Solving Singularly Perturbed NLS at Local Potential Maxima

This paper presents a new approach for addressing the singularly perturbed nonlinear Schrödinger (NLS) equation: \begin{equation} -\varepsilon^2Δv + V(x) v =f(v),\ v>0,\ \lim_{|x|\to \infty} v(x)=0, \end{equation} where $V$ possesses a local maximum point and $f$ satisfies the Berestycki-Lions conditions.The key to our approach is the derivation of a refined lower bound on the gradient norm.

math.AP↗

arXiv2023

Normalized clustering peak solutions for Schrödinger equations with general nonlinearities

We are concerned with the normalized $\ell$-peak solutions to the nonlinear Schrödinger equation \[ -\varepsilon^2Δv+V(x)v=f(v)+λv,\quad \int_{\mathbb{R}^N}v^2 =α\varepsilon^N. \] Here $λ\in \mathbb{R}$ will arise as a Lagrange multiplier, $V$ has a local maximum point, and $f$ is a general $L^2$-subcritical nonlinearity satisfying a nonlipschitzian property that $\lim_{s\to0} f(s)/s=-\infty$. The peaks of solutions that we construct cluster near a local maximum of $V$ as $\varepsilon\to0$. Since there is no information about the uniqueness or nondegeneracy for the limiting system, a delicate lower gradient estimate should be established when the local centers of mass of functions are away from the local maximum of $V$. We introduce a new method to obtain this estimate, which is significantly different from the ideas in del Pino and Felmer (Math. Ann. 2002), where a special gradient flow with high regularity is used, and in Byeon and Tanaka (J. Eur. Math. Soc. 2013 \& Mem. Amer. Math. Soc. 2014), where an extra translation flow is introduced. We also give the existence of ground state solutions for the autonomous problem, i.e., the case $V\equiv0$. The ground state energy is not always negative and the strict subadditive property of ground state energy here is achieved by strict concavity.

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arXiv2021

Qualitative analysis on logarithmic Schrödinger equation with general potential

In this paper, we study the existence, uniqueness, nondegeneracy and some qualitative properties of positive solutions for the logarithmic Schrödinger equations: \[ -Δu+ V(|x|) u=u\log u^2, u\in H^1(\mathbb R^N). \] Here $N\geq 2$ and $V\in C^2((0,+\infty))$ is allowed to be singular at $0$ and repulsive at infinity (i.e., $V(r)\to-\infty$ as ${r\to\infty}$). Under some general assumptions, we show the existence, uniqueness and nondegeneracy of this equation in the radial setting.Specifically, these results apply to singular potentials such as $V(r)=α_{1}\log r+α_2 r^{α_3}+α_4$ with $α_1>1-N$, $α_2, α_3\geq 0$ and $α_4\in\mathbb R$, which is repulsive for $α_1<0$ and $α_2=0$. We also investigate the connection between some power-law nonlinear Schrödinger equation with a critical frequency potential and the logarithmic-law Schrödinger equation with $V(r)=α\log r$, $α>1-N$, proving convergence of the unique positive radial solution from the power type problem to the logarithmic type problem. Under a further assumption, we also derive the uniqueness and nondegeneracy results in $H^1(\mathbb R^N)$ by showing the radial symmetry of solutions.

math.AP↗

arXiv2019

Bound states for logarithmic Schrodinger equations with potentials unbounded below

We study the existence and concentration behavior of the bound states for the following logarithmic Schrödinger equation \begin{equation*} \begin{cases} -\varepsilon^2Δv+V(x)v=v\log v^2 \ \ &\text {in}\ \ \mathbb R^N,\\ v(x)\to 0 \ \ &\text {as}\ \ |x|\to\infty, \end{cases} \end{equation*} where $N\geq 1$, $\varepsilon>0$ is a small parameter, and $V$ may be unbounded below at infinity with a speed of at most quadratic strength. We show that around various types of local topological critical points of the potential function, positive bound state solutions exist and concentrate as $\varepsilon\to0$.

math.AP↗

arXiv2017

Remarks on the Clark theorem

The Clark theorem is important in critical point theory. For a class of even functionals it ensures the existence of infinitely many negative critical values converging to $0$ and it has important applications to sublinear elliptic problems. We study the convergence of the corresponding critical points and we give a characterization of accumulation points of critical points together with examples, in which critical points with negative critical values converges to non-zero critical point. Our results improve the abstract results in Kajikiya [Ka1] and Liu-Wang [LW].

math.AP↗

arXiv2016

Multi-bump solutions for logarithmic Schrödinger equations

We study spatially periodic logarithmic Schrödinger equations: \begin{equation}\tag{LS} -Δu + V(x)u=Q(x)u\log u^2, \quad u>0\quad \text{in}\ \mathbb{R}^N, \end{equation} where $N\geq 1$ and $V(x)$, $Q(x)$ are spatially $1$-periodic functions of class $C^1$. We take an approach using spatially $2L$-periodic problems ($L\gg 1$) and we show the existence of infinitely many multi-bump solutions of $(LS)$ which are distinct under $\mathbb{Z}^N$-action.

math.AP↗

arXiv2015

The Brezis-Nirenberg problem for nonlocal systems

By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

math.AP↗