arXiv subjects

Qiaoran Wu

Publications and source records attributed to Qiaoran Wu.

3 recordsLinked to original sources

arXiv2026

Multiple sign-changing and semi-nodal normalized solutions for a Gross-Pitaevskii type system on bounded domains: the $L^2$-supercritical case

In this paper we investigate the existence of multiple sign-changing and semi-nodal normalized solutions for an $m$-coupled elliptic system of the Gross-Pitaevskii type: \begin{equation} \left\{ \begin{aligned} &-Δu_j + λ_j u_j = \sum_{k=1 }^mβ_{kj} u_k^2 u_j, \quad u_j \in H_0^1(Ω), &\int_Ωu_j^2dx = c_j, \quad j = 1,2,\cdots,m. \end{aligned} \right. \end{equation} Here, $Ω\subset \mathbb{R}^N$ ($N = 3,4$) is a bounded domain. The constants $β_{kj} \neq 0$ and $c_j > 0$ are prescribed constants, while $λ_1, \cdots, λ_m$ are unknown and appear as Lagrange multipliers. This is the first result in the literature on the existence and multiplicity of sign-changing and semi-nodal normalized solutions of couple Schrödinger system in all regimes of $β_{kj}$. The main tool which we use is a new skill of vector linking and this article attempts for the first time to use linking method to search for solutions of a coupled system. Particularly, to obtain semi-nodal normalized solutions, we introduce partial vector linking which is new up to our knowledge. Moreover, by investigating the limit process as $\vec{c}=(c_1,\ldots,c_m) \to \vec{0}$ we obtain some bifurcation results. Note that when $N=4$, the system is of Sobolev critical.

math.AP↗

arXiv2023

Normalized solutions for p-Laplacian equations with potential

In this paper, we consider the existence of normalized solutions for the following $p$-Laplacian equation \begin{equation*} \left\{\begin{array}{ll} -Δ_{p}u-V(x)\lvert u\rvert^{p-2}u+λ\lvert u\rvert^{p-2}u=\lvert u\rvert^{q-2}u&\mbox{in}\ \mathbb{R}^N, \int_{\mathbb{R}^N}\lvert u\rvert^pdx=a^p, \end{array}\right. \end{equation*} where $N\geqslant 1$, $p>1$, $p+\frac{p^2}{N} 0$ and $λ\in\mathbb{R}$ is a Lagrange multiplier which appears due to the mass constraint. Firstly, under some smallness assumptions on $V$, but no any assumptions on $a$, we obtain a mountain pass solution with positive energy, while no solution with negative energy. Secondly, assuming that the mass $a$ has an upper bound depending on $V$, we obtain two solutions, one is a local minimizer with negative energy, the other is a mountain pass solution with positive energy.

math.AP↗

arXiv2023

Normalized solutions for $p$-Laplacian equation with critical Sobolev exponent and mixed nonlinearities

In this paper, we consider the existence and multiplicity of normalized solutions for the following $p$-Laplacian critical equation \begin{align*} \left\{\begin{array}{ll} -Δ_{p}u=λ\lvert u\rvert^{p-2}u+μ\lvert u\rvert^{q-2}u+\lvert u\rvert^{p^*-1}u&\mbox{in}\ \mathbb{R}^N, \int_{\mathbb{R}^N}\lvert u\rvert^pdx=a^p, \end{array}\right. \end{align*} where $1 0$, $μ\in\mathbb{R}$ and $λ\in\mathbb{R}$ is a Lagrange multiplier. Using concentration compactness lemma, Schwarz rearrangement, Ekeland variational principle and mini-max theorems, we obtain several existence results under $μ>0$ and other assumptions. We also analyze the asymptotic behavior of there solutions as $μ\rightarrow 0$ and $μ$ goes to its upper bound. Moreover, we show the nonexistence result for $μ<0$ and get that the $p$-Laplacian equation has infinitely solutions by genus theory when $p<q<p+\frac{p^2}{N}$.

math.AP↗