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Fukun Zhao

Publications and source records attributed to Fukun Zhao.

5 recordsLinked to original sources
arXiv2026

Fully sign-changing Nehari constraint vs sign-changing solutions of a competitive Schr\"{o}dinger system

We study a competitive nonlinear Schr\"odinger system in $\mathbb{R}^N$ whose nonlinear potential is localized in small regions that shrink to isolated points. Within a variational framework based on a fully sign-changing Nehari constraint and Krasnosel'skii genus, we construct, for all $\varepsilon>0$, a sequence of sign-changing solutions with increasing and unbounded energies, and after suitable translations they converge to a sequence of sign-changing solutions of the associated limiting system as $\varepsilon\to 0$ in $H^1$-norm. Moreover, these sign-changing solutions concentrate around the prescribed attraction points both in $H^1$-norm and $L^q$-norm for $q\in [1,\infty]$.

math.AP↗
arXiv2026

Multiplicity and concentration of dual solutions for a Helmholtz system

In this paper, we are concerned with the nonlinear Helmholtz system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -\Delta u-k^2 u=P(x)|v|^{p-2}v,\quad \text{in}\ \mathbb{R}^N, \\ -\Delta v-k^2v=Q(x)|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $N\geq3$, $P,Q: \mathbb{R}^N\rightarrow \mathbb{R}$ are two positive continuous functions, the exponents $p,q>2$ satisfy $\frac{1}{p}+\frac{1}{q}>\frac{N-2}{N}$. First, we obtained the existence of a ground state solution via a dual variational method. Moreover, the concentration behavior of such dual ground state solutions is established as $k\rightarrow\infty$, where a rescaling technique and the generalized Birman-Schwinger operator are involved. In addition, we also investigated the relation between the number of solutions and the topology of the set of the global maxima of the functions $P$ and $Q$.

math.AP↗
arXiv2026

Multiple standing waves of Helmholtz equation with mixed dispersion concentrating in the high frequency limit

In this paper, we study the nonlinear Helmholtz equation with mixed dispersion \begin{equation*} \Delta^2 u-\beta k^2\, \Delta u+\alpha k^4 u=W(x)\, |u|^{p-2}u~\text{in}~\mathbb{R}^N, \end{equation*} where the weight function $W(x)$ is continuous, nonnegative, and satisfies \[ \limsup_{|x|\to\infty} W(x) \;<\; \sup_{x\in\mathbb{R}^N} W(x). \] Within each of the following parameter ranges, \begin{center} (a) $\alpha<0$, $\beta\in\mathbb{R}$; \qquad (b) $\alpha>0$, $\beta<-2\sqrt{\alpha}$; \qquad (c) $\alpha=0$, $\beta<0$, \end{center} After a suitable rescaling, we obtain the existence of dual ground state solutions, which concentrate along the global maximizers of $W$ as $k\to\infty$. In addition, we establish the existence of multiple solutions associated with the set of global maximum points of $W$, and we further characterize the precise concentration behavior of these solutions.

math.AP↗
arXiv2025

Existence and multiplicity of $L^2$-Normalized solutions for the periodic Schr\"{o}dinger system of Hamiltonian type

In this paper, we study the following nonlinear Schr\"{o}dinger system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -\Delta u+V(x)u=\partial_v H(x,u,v)+\omega v, \ x \in \mathbb{R}^N, \\ -\Delta v+V(x)v=\partial_u H(x,u,v)+\omega u,\ x \in \mathbb{R}^N, \\ \displaystyle\int_{\mathbb{R}^N}|z|^2dx=a^2, \end{array}\right. \end{equation*} where the potential function $V(x)$ is periodic, $z:=(u,v):\mathbb{R}^N\rightarrow \mathbb{R}\times\mathbb{R}$, $\omega\in \mathbb{R}$ appears as a Lagrange multiplier, $a>0$ is a prescribed constant. The existence and multiplicity of $L^2$-normalized solutions for the above Schr\"{o}dinger system are obtained, and the combination of the Lyapunov-Schmidt reduction, a perturbation argument and the multiplicity theorem of Ljusternik-Schnirelmann is involved in the proof. In addition, a bifurcation result is also given.

math.AP↗