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Shaoxiong Chen

Publications and source records attributed to Shaoxiong Chen.

9 recordsLinked to original sources
arXiv2026

A positive ground state for a planar Choquard equation with mixed diffusion and critical exponential growth

We study a two-dimensional Choquard equation driven by the mixed local and nonlocal operator $L:=-Δ+(-Δ)^s$, where the nonlinearity has critical exponential growth of Trudinger--Moser type. Under a coercive assumption on the potential and suitable one-sided assumptions on the nonlinearity, we prove the existence of a least energy positive solution. The proof combines Nehari manifold minimization, compactness below the critical Trudinger--Moser threshold, local regularity, and a strong maximum principle.

math.AP↗
arXiv2026

Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators

We study the Choquard equation involving mixed local and nonlocal operators \[ -Δu + (-Δ)^{s}u + V(x)u = \left(\frac{1}{|x|^μ} * F(u)\right) f(u) \quad \text{in } \mathbb{R}^{2}, \] where $s\in(0,1)$, $μ\in(0,2)$, $F(t)=\int_{0}^{t} f(τ)\,dτ$, and $f$ has subcritical exponential growth of Trudinger--Moser type. Under suitable assumptions on the potential $V$ and the nonlinearity $f$, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions.

math.AP↗
arXiv2026

Normalized solutions for a class of fractional Choquard equations with the HLS lower critical term and a nonlocal perturbation

In this paper, we study the mass-constrained fractional Choquard equation \( (-Δ)^s u = λu + α(I_μ* |u|^{\frac{2N-μ}{N}})|u|^{\frac{2N-μ}{N}-2}u + (I_μ* |u|^p)|u|^{p-2}u \) in \( \mathbb{R}^N \), under the constraint \( \int_{\mathbb{R}^N} |u|^2 \, dx = c^2 > 0 \), where \( N > 2s \), \( s \in (0,1) \), \( μ\in (0,N) \), \( α> 0 \), and \( 2 + \frac{2s-μ}{N} \le p < \frac{2N-μ}{N-2s} \). We first establish a nonexistence result in the \( L^2 \)-critical case \( p = 2 + \frac{2s-μ}{N} \). Then, in the \( L^2 \)-supercritical range, we prove the existence of normalized ground states in two complementary regimes determined by the quantity \( \mathcal{M}_1(c) \). Our approach is based on constrained variational methods, a min-max construction, and refined estimates for the associated fiber maps.

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*} Δ^2 u-βk^2\, Δu+αk^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) $α<0$, $β\in\mathbb{R}$; \qquad (b) $α>0$, $β<-2\sqrtα$; \qquad (c) $α=0$, $β<0$, \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↗
arXiv2026

Existence and concentration of ground state solutions for an exponentially critical Choquard equation involving mixed local-nonlocal operators

We study the Choquard equation involving mixed local and nonlocal operators \[-\varepsilon^{2}Δu+\varepsilon^{2s}(-Δ)^{s}u+V(x)u=\varepsilon^{μ-2}\left(\frac{1}{|x|^μ}*F(u)\right)f(u)\quad \text{in }\R^{2},\] where \(\varepsilon>0\), \(s\in(0,1)\), \(0<μ<2\), \(f\) has Trudinger--Moser critical exponential growth, and \(F(t)=\int_{0}^{t}f(τ)\,dτ\). By variational methods, combined with the Trudinger--Moser inequality and compactness arguments adapted to the critical growth and the nonlocal interaction term, we prove the existence of ground state solutions and describe their concentration behavior as \(\varepsilon\to0^{+}\).

math.AP↗
arXiv2025

Normalized solutions for a class of fractional Choquard equations with mixed nonlinearities

In this paper we study the following fractional Choquard equation with mixed nonlinearities: \[ \left\{ \begin{array}{l} (-Δ)^s u = λu + α\left( I_μ* |u|^q \right) |u|^{q-2} u + \left( I_μ* |u|^p \right) |u|^{p-2} u, \quad x \in \mathbb{R}^N, \\[4pt] \displaystyle \int_{\mathbb{R}^N} |u|^2 \,\mathrm{d}x = c^2 > 0. \end{array} \right. \] Here $N > 2s$, $s \in (0,1)$, $μ\in (0, N)$, and the exponents satisfy \[ \frac{2N - μ}{N} < q < p < \frac{2N - μ}{N - 2s}, \] while $α> 0$ is a sufficiently small parameter, $λ\in \mathbb{R}$ is the Lagrange multiplier associated with the mass constraint, and $I_μ$ denotes the Riesz potential. We establish existence and multiplicity results for normalized solutions and, in addition, prove the existence of ground state normalized solutions for $α$ in a suitable range.

math.AP↗
arXiv2023

High-content stimulated Raman histology of human breast cancer

Histological examination is crucial for cancer diagnosis, including hematoxylin and eosin (H&E) staining for mapping morphology and immunohistochemistry (IHC) staining for revealing chemical information. Recently developed two-color stimulated Raman histology could bypass the complex tissue processing to mimic H&E-like morphology. Yet, the underlying chemical features are not revealed, compromising the effectiveness of prognostic stratification. Here, we present a high-content stimulated Raman histology (HC-SRH) platform that provides both morphological and chemical information for cancer diagnosis based on un-stained breast tissues. Through spectral unmixing in the C-H vibration window, HC-SRH can map unsaturated lipids, cellular protein, extracellular matrix, saturated lipid, and water in breast tissue. In this way, HC-SRH provides excellent contrast for various tissue components. Considering rapidness is important in clinical trials, we implemented spectral selective sampling to boost the speed of HC-SRH by one order. We also successfully demonstrated the HC-SRH in a clinical-compatible fiber laser-based SRS microscopy. With the widely rapid tuning capability of the advanced fiber laser, a clear chemical contrast of nucleic acid and solid-state ester is shown in the fingerprint result.

q-bio.TO↗
arXiv2019

Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent

In present paper, we study the fractional Choquard equation $$\varepsilon^{2s}(-Δ)^s u+V(x)u=\varepsilon^{μ-N}(\frac{1}{|x|^μ}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$ $2^*_s=\frac{2N}{N-2s}$ and $0<μ<\min\{2s,N-2s\}$. Under suitable assumption on $V$ and $f$, we prove this problem has a nontrivial nonnegative ground state solution. Moreover, we relate the number of nontrivial nonnegative solutions with the topology of the set where the potential attains its minimum values and their's concentration behavior.

math.FA↗