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Liejun Shen

Publications and source records attributed to Liejun Shen.

10 recordsLinked to original sources
arXiv2024

Generalized Chern-Simons-Schrodinger system with critical exponential growth: the zero mass case

We consider the existence of ground state solutions for a class of zero-mass Chern-Simons-Schr\"{o}dinger systems \[ \left\{ \begin{array}{ll} \displaystyle -\Delta u +A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u)-a(x)|u|^{p-2}u, \newline \displaystyle \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,~\partial_1A_1+\partial_2A_2=0, \newline \displaystyle \partial_1A_0=A_2|u|^2,~ \partial_2A_0=-A_1|u|^2, \end{array} \right. \] where $a:\mathbb R^2\to\mathbb R^+$ is an external potential, $p\in(1,2)$ and $f\in \mathcal{C}(\mathbb R)$ denotes a nonlinearity that fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. By introducing an improvement of the version of Trudinger-Moser inequality, we are able to investigate the existence of positive ground state solutions for the given system using variational method.

math.AP↗
arXiv2024

Existence and concentration of normalized solutions for $p$-Laplacian equations with logarithmic nonlinearity

We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^p\Delta_p u+V(x)|u|^{p-2}u=\lambda |u|^{p-2}u+|u|^{p-2}u\log|u|^p ~\text{in}~\mathbb R^N,\newline \displaystyle \int_{\mathbb R^N}|u|^pdx=a^p\varepsilon^N, \end{array} \right. \] where $a,\varepsilon> 0$, $\lambda\in\mathbb R$ is known as the Lagrange multiplier, $\Delta_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla \cdot)$ denotes the usual $p$-Laplacian operator with $2\leq p < N$ and $V \in \mathcal{C}^0(\mathbb R^N)$ is the potential which satisfies some suitable assumptions. We prove that the number of positive solutions depends on the profile of $V$ and each solution concentrates around its corresponding global minimum point of $V$ in the semiclassical limit when $\varepsilon\to0^+$ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic $p$-Laplacian equations involving mass-supercritical nonlinearities.

math.AP↗
arXiv2024

Infinitely many solutions for a class of fractional Schrodinger equations coupled with neutral scalar field

We study the fractional Schr\"{o}dinger equations coupled with a neutral scalar field $$ (-\Delta)^s u+V(x)u=K(x)\phi u +g(x)|u|^{q-2}u, \quad x\in \mathbb{R}^3,\qquad (I-\Delta)^t \phi=K(x)u^2, \quad x\in \mathbb{R}^3, $$ where $(-\Delta)^s$ and $(I-\Delta)^t$ denote the fractional Laplacian and Bessel operators with $\frac{3}{4} <s<1$ and $0<t<1$, respectively. Under some suitable assumptions for the external potentials $V$, $K$ and $g$, given $q\in(1,2)\cup(2,2_s^*)$ with $2_s^*:= \frac{6}{3-2s}$, with the help of an improved Fountain theorem dealing with a class of strongly indefinite variational problems approached by Gu-Zhou [Adv. Nonlinear Stud., {\bf 17} (2017), 727--738], we show that the system admits infinitely many nontrivial solutions.

math.AP↗
arXiv2024

Planar Schr\"odinger-Poisson system with steep potential well: supercritical exponential case

We study a class of planar Schr\"{o}dinger-Poisson systems $$ -\Delta u+\lambda V(x)u+\phi u=f(u) , \quad x\in{\mathbb R}^2,\qquad \Delta \phi=u^2, \quad x\in{\mathbb R}^2, $$ where $\lambda>0$ is a parameter, $V\in C({\mathbb R}^2,{\mathbb R}^+)$ has a potential well $\Omega \triangleq\text{int}\, V^{-1}(0)$ and the nonlinearity $f$ fulfills the supercritical exponential growth at infinity in the Trudinger-Moser sense. By exploiting the mountain-pass theorem and elliptic regular theory, we establish the existence and concentrating behavior of ground state solutions for sufficiently large $\lambda$.

math.AP↗
arXiv2024

Normalized solutions to the Chern-Simons-Schr\"{o}dinger system: the supercritical case

We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schr\"{o}dinger type problems with supercritical exponential growth $$ -\Delta u +\lambda u+A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u),\quad \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,\quad \partial_1A_1+\partial_2A_2=0,\quad \partial_1A_0=A_2|u|^2,\quad \partial_2A_0=-A_1|u|^2,\quad \int_{\mathbb{R}^2}|u|^2dx=a^2, $$ where $a\neq0$, $\lambda\in \mathbb{R}$ is known as the Lagrange multiplier and $f\in C^1(\mathbb{R})$ denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger-Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regularity theory, we obtain that the problem has at least a ground state solution.

math.AP↗
arXiv2017

The existence and concentration of positive ground state solutions for a class of fractional Schrödinger-Poisson systems with steep potential wells

The present study is concerned with the following fractional Schrödinger-Poisson system with steep potential well: $$ \left\{% \begin{array}{ll} (-Δ)^s u+ \la V(x)u+K(x)ϕu= f(u), & x\in\R^3, (-Δ)^t ϕ=K(x)u^2, & x\in\R^3, \end{array}% \right. $$ where $s,t\in(0,1)$ with $4s+2t>3$, and $\la>0$ is a parameter. Under certain assumptions on $V(x)$, $K(x)$ and $f(u)$ behaving like $|u|^{q-2}u$ with $2<q<2_s^*=\frac{6}{3-2s}$, the existence of positive ground state solutions and concentration results are obtained via some new analytical skills and Nehair-Pohožaev identity. In particular, the monotonicity assumption on the nonlinearity is not necessary.

math.AP↗
arXiv2017

The existence of positive least energy solutions for a class of Schrodinger-Poisson systems involving critical nonlocal term with general nonlinearity

The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term with general nonlinearity: $$ \left\{ \begin{array}{ll} -Δu+V(x)u- ϕ|u|^3u= f(u), & x\in\mathbb{R}^3, -Δϕ= |u|^5, & x\in\mathbb{R}^3,\\ \end{array} \right. $$ Under certain assumptions on non-constant $V(x)$, the existence of a positive least energy solution is obtained by using some new analytical skills and Pohožaev type manifold. In particular, the Ambrosetti-Rabinowitz type condition or monotonicity assumption on the nonlinearity is not necessary.

math.AP↗
arXiv2017

Multiple positive solutions for Schrodinger-Poisson systems involving critical nonlocal term

The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term $$ \left\{ \begin{array}{ll} -Δu+u-K(x)ϕ|u|^3u=λf(x)|u|^{q-2}u, & x\in\mathbb{R}^3, -Δϕ=K(x)|u|^5, & x\in\mathbb{R}^3,\\ \end{array} \right. $$ where $1 0$ is a parameter. Under suitable assumptions on $K(x)$ and $f(x)$, there exists $λ_0=λ_0(q,S,f,K)>0$ such that for any $λ\in(0,λ_0)$, the above Schrödinger-Poisson system possesses at least two positive solutions by standard variational method, where a positive least energy solution will also be obtained.

math.AP↗
arXiv2016

Multiple positive solutions for a class of Kirchhoff type problems involving general critical growth

In this paper, we study the following nonlinear Kirchhoff problem involving critical growth: $$ \left\{% \begin{array}{ll} -(a+b\int_Ω|\nabla u|^2dx)Δu=|u|^4u+λ|u|^{q-2}u, u=0\ \ \text{on}\ \ \partialΩ, \end{array}% \right. $$ where $1 0$ are parameters and $Ω$ is a bounded domain in $\R^3$. We prove that there exists $λ_1=λ_1(q,Ω)>0$ such that for any $λ\in(0,λ_1)$ and $a,\ b>0$, the above Kirchhoff problem possesses at least two positive solutions and one of them is a positive ground state solution. We also establish the convergence property of the ground state solution as the parameter $b\searrow 0$. More generally, we obtain the same results about the following Kirchhoff problem: $$ \left\{% \begin{array}{ll} -(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx)Δu+u=Q(x)|u|^4u+λf(x)|u|^{q-2}u, u\in H^1(\mathbb{R}^3), \end{array}% \right. $$ for any $a,\ b>0$ and $λ\in \big(0,λ_0(q,Q,f)\big)$ under certain conditions of $f(x)$ and $Q(x)$. Finally, we investigate the depending relationship between $λ_0$ and $b$ to show that for any (large) $λ>0$, there exists a $b_0(λ)>0$ such that the above results hold when $b>b_0(λ)$ and $a>0$.

math.AP↗