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1. Normalized bound state solutions for mass supcritical fractional Schr\"{o}dinger equation with potential | |||
BAO Xin,OU Zeng-Qi,LV Ying | |||
Mathematics 06 March 2024 | |||
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Abstract:In this paper, we study the following fractional Schr\"{o}dinger equation with prescribed mass\begin{equation*}\left\{\begin{aligned}&(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u,\quad\text{in $\mathbb{R}^{N}$},\\&\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\quad u\in H^{s}(\mathbb{R}^{N}),\end{aligned}\right.\end{equation*}where $0<s<1$, $N>2s$, $2+\frac{4s}{N}<p<2_{s}^{*}:=\frac{2N}{N-2s}$, $c>0$, $\lambda\in \mathbb{R}$ and $a(x)\in C^{1}(\mathbb{R}^{N},\mathbb{R}^{+})$ is a potential function. By using a minimax principle, we prove the existence of bounded state solution for the above problem under various conditions on $a(x)$. | |||
TO cite this article:BAO Xin,OU Zeng-Qi,LV Ying. Normalized bound state solutions for mass supcritical fractional Schr\"{o}dinger equation with potential[OL].[ 6 March 2024] http://en.paper.edu.cn/en_releasepaper/content/4762069 |
2. Multiple positive solutions for Kirchhoff type equation involving concave-convex nonlinearities | |||
Qian-Li Chen,Zeng-Qi Ou | |||
Mathematics 05 February 2024 | |||
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Abstract:In this paper, we study the existence of multiple positive solution for the following equation: \begin{equation} \label{eq0} \left\{ \begin{aligned} &-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\right)\Delta u+V_{\lambda}(x)u=f(x)|u|^{q-2}u+g(x)|u|^{p-2}u ,&x\in \mathbb{R}^{3},\\ &u>0,u\in H^{1}(\mathbb{R}^{3}), \end{aligned} \right. \end{equation} where $a,b>0$ are constants, $1<q<2$, $4<p<6$ and $V_{\lambda}(x)=\lambda V^{+} (x)-V^{-}(x)$, $V^{\pm}=\max\{\pm V,0\}\ne 0$, $\lambda>0$. Assume that the functions $V, f, g$ satisfy the following conditions: $(V_{1})$\ $V^{+}\in C(\mathbb{R}^{3})$, $V^{-}\in L^{\frac{3}{2}}(\mathbb{R}^{3})$ with $\|V^{-} \|_{L^{\frac{3}{2}}}<aS$, where $S$ is the best Sobolev imbedding constant of $D^{1,2}(\mathbb{R}^{3})$ into $L^{6}(\mathbb{R}^{3})$; $(V_{2})$\ there exists $k>0$ such that $\{V^{+}<k\}=\{x\in \mathbb{R}^{3}: V^{+}(x)<k\}$ is nonempty and has finite measure; $(V_{3})$\ $\Omega:=$int$((V^{+} )^{-1}(0)$ is nonempty and has a smooth boundary with $\bar{\Omega} :=(V^{+})^{-1}(0)$, where $(V^{+})^{-1}(0):=\{x\in \mathbb{R}^{3}: V^{+}(x)=0\}$; $(f)$\ $f\in L^{\frac{6}{6-q}}(\mathbb{R}^{3})$ with the set $\{x\in \mathbb{R}^{3}: f(x)>0\}$ of positive measure; $(g_{1})$\ $g\in L^{\frac{6}{6-p}}(\mathbb{R}^3)$ with the set $\{x\in\mathbb{R}^{3}: g(x)>0\}$ of positive measure; $(g_{2})$ there exists a nonempty open set $\Omega_{g}\subset \Omega$ such that $g>0$ a.e. on $\Omega_{g}$. $(fg)$\ there is $\mu _{0}>1$ such that $$ \|f^{+}\|_{L^{\frac{6}{6-q}}}^{p-2}\|g^{+}\|_{L^{\frac{6}{6-p}}}^{2-q}< \left(\frac{\mu _{0}-1}{2d_6^{2}\mu_{0}(p-q)}\right)^{p-q}(2-q)^{2-q}(p-2)^{p-2}, $$ where $d_6$ is a embedding constant given in $\eqref{eq5}$ and $\|\cdot\|_{L^{s}}$ denotes the norm of Lebesgue space $L^{s}({\mathbb{R}^{3}})$. | |||
TO cite this article:Qian-Li Chen,Zeng-Qi Ou. Multiple positive solutions for Kirchhoff type equation involving concave-convex nonlinearities[OL].[ 5 February 2024] http://en.paper.edu.cn/en_releasepaper/content/4762075 |
3. Nehari manifold for fractional s($\cdot$)-Laplacian system involving concave-convex nonlinearities with magnetic field | |||
Feng Dong-Xue,Chen Wen-Jing | |||
Mathematics 07 June 2023 | |||
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Abstract:This paper is concerned with the nonlocal elliptic system driven by the variable-order fractional magnetic Laplace operator involving concave-convex nonlinearities\begin{equation*}\left\{\begin{array}{rl}(-\Delta)_{A}^{s(\cdot)} u&=\lambda~ a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x)+\beta(x)}c(x)|u|^{\alpha(x)-2}u| v| ^{\beta(x)}, \hspace{2mm}{\rm in}\ \Omega, \\(-\Delta)_{A}^{s(\cdot)} v&=\mu~ b(x)| v|^{q(x)-2}v+\frac{\beta(x)}{\alpha(x)+\beta(x)}c(x)| u|^{\alpha(x)}| v| ^{\beta(x)-2}v, \hspace{2.5mm}{\rm in}\ \Omega, \\u=v&=0 , \hspace{1cm} {\rm in}\ \mathbb{R}^N\backslash\Omega,\end{array}\right.\end{equation*}where $\Omega\subset\mathbb R^N, ~N\geq2$ is a smooth bounded domain, $\lambda, \mu>0$ are the parameters,$s\in C(\mathbb R^N\times \mathbb R^N, (0, 1))$ and $q, \alpha, \beta\in C(\overline{\Omega}, (1, \infty))$ are the variable exponents and$a, b, c\in C(\overline{\Omega}, [0, \infty))$ are the non-negative weight functions. $(-\Delta)_{A}^{s(\cdot)}$ is the variable-order fractional magnetic Laplace operator, the magnetic field is $A\in C^{0, \alpha}(\mathbb R^N, \mathbb R^N)$ with $\alpha\in(0, 1]$ and $u:\mathbb R^N\to\mathbb C$. Use Nehari manifold to prove that there exists $\Lambda>0$ such that $\forall\lambda+\mu<\Lambda$, this system obtains at least two non-negative solutions of theabove problem under some assumptions on $q, \alpha, \beta$. | |||
TO cite this article:Feng Dong-Xue,Chen Wen-Jing. Nehari manifold for fractional s($\cdot$)-Laplacian system involving concave-convex nonlinearities with magnetic field[OL].[ 7 June 2023] http://en.paper.edu.cn/en_releasepaper/content/4760785 |
4. Existence of the multi-peak solutions for a nonlinear fractional Kirchhoff equation withmagnetic fields | |||
SUN Qian,CHEN Wenjing | |||
Mathematics 29 March 2023 | |||
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Abstract:The aim of this paper focuses on the following class of fractional magnetic Kirchhofftype equation$$\left(a\varepsilon^{2s}+b\varepsilon^{4s-N}[u]_{A/\varepsilon}^2\right) (-\Delta)_{A/\varepsilon}^su+V(x)u=|u|^{p-1}u,\quad \mbox{in } \mathbb{R}^N,$$where $\varepsilon>0$ is a small parameter, $a,b>0$, $(-\Delta)_{A}^{s}$ is the fractional magneticLaplacian operator, $s\in (0,1)$, $p\in(1,\frac{N+2s}{N-2s})$ and $N\in(2s,4s)$, $A(x): \mathbb{R}^N\rightarrow \mathbb{R}^N$ is the bounded magnetic potential, $V(x): \mathbb{R}^N \rightarrow\mathbb{R}$ is a continuous potential function. Our approach is based on some decaying estimateand nondegenerate, we prove that the equation has multi-peak solutions concentrating at localminimum points of $V(x)$ by applying Lyapunov-Schmidt reduction method. | |||
TO cite this article:SUN Qian,CHEN Wenjing. Existence of the multi-peak solutions for a nonlinear fractional Kirchhoff equation withmagnetic fields[OL].[29 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759898 |
5. Existence of ground state solutions for coupled Choquard system with lower critical exponents | |||
WANG Fen-Fen,DENG Sheng-Bing | |||
Mathematics 29 March 2023 | |||
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Abstract:In this paper, we study the following coupled Choquard type system with Hardy--Littlewood--Sobolev lower critical exponents and a local nonlinear perturbation:\begin{equation*}\left\{ \arraycolsep=1.5pt \begin{array}{ll}-\Delta u+V(x)u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+ \lambda(I_\alpha*|v|^{p})|u|^{p-2}u, &\ \text{ in } \mathbb{R}^N,\\-\Delta v+V(x)v=\big(I_\alpha*|v|^{\frac{\alpha}{N}+1}\big)|v|^{\frac{\alpha}{N}-1}v+ \lambda(I_\alpha*|u|^{p})|v|^{p-2}v, &\ \text{ in } \mathbb{R}^N,\\ \end{array} \right.\end{equation*}where $N\geq 3$, $ \alpha \in (0,N)$, $I_{\alpha}:\mathbb{R}^N\backslash{\{0\}}\to\mathbb{R}$ is a Riesz potential,$V\in C(\mathbb{R}^N,[0,\infty))$ and satisfies some suitable conditions. In the case when $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N+1}$, $p=\frac{N+\alpha+2}{N+1}$, and $\frac{N+\alpha+2}{N+1}<p<\frac{N+\alpha}{N-2}$, respectively, we investigate the existence of positive ground states of this system if $\lambda>\lambda_{*}$ by variational approaches. | |||
TO cite this article:WANG Fen-Fen,DENG Sheng-Bing. Existence of ground state solutions for coupled Choquard system with lower critical exponents[OL].[29 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759877 |
6. Signed and sign-changing solutions for nonlinear Schr\"{o}dinger-Poisson system with two nonlocal terms in $\mathbb{R}^{3}$ | |||
Yi Wen, Zeng-Qi Ou | |||
Mathematics 03 March 2023 | |||
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Abstract:In this paper, we consider the following Schr\"{o}dinger-Poisson system \begin{equation} \left\{\begin{matrix} -\Delta u+V(x)u+\phi u=\left(\int_{\mathbb{R}^3}\frac{1}{p}|u|^pdx\right)^{\frac{2}{p}}|u|^{p-2}u+g(x)|u|^{q-2}u,& \mbox{in}\ \mathbb{R}^3,\\ -\Delta\phi=u^2,& \mbox{in}\ \mathbb{R}^3,\hfill\label{0.1} \end{matrix}\right. \end{equation} where $1<q<2<p<6$ and the functions $V(x), g(x)$ satisfy the certain conditions. Using variational methods and invariant sets of descending flow, we prove that system (\ref{0.1}) possesses three nontrivial solutions of mountain pass type (one positive, one negative and one sign-changing) and infinitely many high-energy sign-changing solutions. | |||
TO cite this article:Yi Wen, Zeng-Qi Ou. Signed and sign-changing solutions for nonlinear Schr\"{o}dinger-Poisson system with two nonlocal terms in $\mathbb{R}^{3}$[OL].[ 3 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759230 |
7. Infinitely many high energy solutions for Schr\"{o}dinger-Poisson system | |||
XIONG Biao,TANG Chun-lei | |||
Mathematics 15 February 2023 | |||
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Abstract:In this arcitle, we investigate the following Schr\"{o}dinger-Poisson system\begin{equation*} \begin{cases} -\Delta u+V(x)u+\phi u=f(u), & \text{ in }\R,\\ -\Delta \phi= u^2, & \text{ in }\R, \end{cases}\end{equation*}where $V(x)$ is coercive, $f$ satisfies that $\frac{1}{3}f(t)t\geq F(t)>0$ for every $t\in\RRR\setminus\{0\}$. Under certain assumptions about the above terms, we obtain infinitely many high energy solutions for the system by Symmetric mountain pass theorem. | |||
TO cite this article:XIONG Biao,TANG Chun-lei. Infinitely many high energy solutions for Schr\"{o}dinger-Poisson system[OL].[15 February 2023] http://en.paper.edu.cn/en_releasepaper/content/4759083 |
8. Ground state sign-changing solutions for a quasilinear Schr\"{o}dinger equation | |||
ZHAO Yan-Ping,WU Xing-Ping,TANG Chun-Lei | |||
Mathematics 02 February 2023 | |||
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Abstract:\ In this paper, we consider the existence of ground state solution and ground state sign-changing solution for the quasilinear Schr\"{o}dinger equation\begin{eqnarray}\label{1.1}-\triangle u-a(x)\triangle (u^2)u+V(x)u=f(x,u) &x\in \mathbb{R}^N\nonumber\end{eqnarray}where $N\geq3$, $V$ is coercive potential, $a(x)$ is a bounded function and $f\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$. The proof is based on variational methods, by using sign-changing Nehari manifold and deformation arguments, we can get a least energy sign-changing solution. | |||
TO cite this article:ZHAO Yan-Ping,WU Xing-Ping,TANG Chun-Lei. Ground state sign-changing solutions for a quasilinear Schr\"{o}dinger equation[OL].[ 2 February 2023] http://en.paper.edu.cn/en_releasepaper/content/4758883 |
9. Existence and asymptotic behavior of sign-changing solutions for the Schr\"{o}dinger-Bopp-Podolsky system with concave-convex nonlinearities | |||
Yi-Xin Hu, Xing-Ping Wu, Chun-Lei Tang | |||
Mathematics 28 March 2022 | |||
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Abstract:In this paper, we study the Schr\"{o}dinger-Bopp-Podolsky system with con-cave-convex nonlinearities. If $0< \lambda < \lambda^{*}$, the system has a sign-changing solution by variational methods. Besides, we argument the asymptotic behavior of the solution as $a\rightarrow 0$. | |||
TO cite this article:Yi-Xin Hu, Xing-Ping Wu, Chun-Lei Tang. Existence and asymptotic behavior of sign-changing solutions for the Schr\"{o}dinger-Bopp-Podolsky system with concave-convex nonlinearities[OL].[28 March 2022] http://en.paper.edu.cn/en_releasepaper/content/4757256 |
10. Existence of a positive solution to Kirchhoff type problemswithout compactness conditions | |||
DENG Landan, SHANG Yanying | |||
Mathematics 29 December 2021 | |||
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Abstract:In this paper, we consider a class of Kirchhoff problem with steep potential well,the nonlinear term does not require usual compactness conditions, under appropriate assumptions, we establish the existence of positive solutions by utilizing the truncation technique to overcome the lack of compactness. | |||
TO cite this article:DENG Landan, SHANG Yanying. Existence of a positive solution to Kirchhoff type problemswithout compactness conditions[OL].[29 December 2021] http://en.paper.edu.cn/en_releasepaper/content/4755924 |
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