相关论文: Concentration behavior of normalized ground states…

We are concerned with $L^2$-constraint minimizers for the Kirchhoff functional $$ E_b(u)=\int_{\Omega}|\nabla u|^2\mathrm{d}x+\frac{b}{2}\left(\int_\Omega|\nabla u|^2\mathrm{d}x\right)^2+\int_\Omega…

This paper considers ground states of mass subcritical rotational nonlinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u+i\Omega(x^\perp\cdot\nabla u)=\mu u+\rho^{p-1}|u|^{p-1}u \,\ \text{in} \,\ \mathbb{R}^2, \end{equation*}…

We consider the following time-independent nonlinear $L^2$-critical Schr\"{o}dinger equation \[ -\Delta u(x)+V(x)u(x)-a|x|^{-b}|u|^{1+\frac{4-2b}{N}}=\mu u(x)\,\ \hbox{in}\,\ \mathbb{R}^N, \] where $\mu\in\mathbb{R}$, $a>0$, $N\geq 1$,…

We investigate normalized solutions for a class of nonlinear Schr\"{o}dinger (NLS) equations with potential $V$ and inhomogeneous nonlinearity $g(|u|)u=|u|^{q-2}u+\beta |u|^{p-2}u$ on a bounded domain $\Omega$. Firstly, when…

In this paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_{\Omega}\vert\nabla u\vert^2dx)\Delta u+\lambda u=\vert u\vert^{p-2}u & \text{…

We study the existence and blow-up behavior of minimizers for $E(b)=\inf\Big\{\mathcal{E}_b(u) \,|\, u\in H^1(R^2), \|u\|_{L^2}=1\Big\},$ here $\mathcal{E}_b(u)$ is the Kirchhoff energy functional defined by $\mathcal{E}_b(u)= \int_{R^2}…

This paper is devoted to the study of normalized solutions to the Kirchhoff type equation with a logarithmic perturbation\[-\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2 \,\mathrm{d}x \right) \Delta u=\lambda u+|u|^{p-2}u+u\log u^2,\quad x…

Let $a>0,b>0$ and $V(x)\geq0$ be a coercive function in $\mathbb R^2$. We study the following constrained minimization problem on a suitable weighted Sobolev space $\mathcal{H}$: \begin{equation*}…

In this paper, we study the existence and the concentration behavior of minimizers for $i_V(c)=\inf\limits_{u\in S_c}I_V(u)$, here $S_c=\{u\in H^1(\R^N)|~\int_{\R^N}V(x)|u|^2<+\infty,~|u|_2=c>0\}$ and…

This paper is concerned with the existence of solutions to the problem $$-\left(a+ b\int_{\mathbb{R}^{N}}|\nabla u|^{2} dx \right)\Delta u +V(x)u+\lambda u = |u|^{p-2}u,\ \ x \in \mathbb{R}^{N},\ \ \lambda \in \mathbb{R}^{+} $$ where $a,…

In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)\Delta u+V(x)u+\lambda u=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying…

This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities \begin{equation*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u(x)|^{2}\right) \Delta u =\lambda…

In this paper, we consider the following singularly perturbed Kirchhoff equation \begin{equation*} -(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+V(x)u=P(x)|u|^{p-2}u+Q(x)|u|^4u,\quad x\in\mathbb{R}^3,…

We are concerned with the following Kirchhoff type equation $$-\varepsilon^2 M \left(\varepsilon^{2-N} \int_{\mathbb{R}^N} | \nabla u|^2\, \mathrm{d} x\right) \Delta u+V(x)u = f(u),\ x \in \mathbb{R}^N,\ \ N\ge2, $$ where $M \in…

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schr\"odinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-\delta u+\lambda_1…

In this paper, by using variational methods we study the existence of positive solutions for the following Kirchhoff type problem: $$ \left\{ \begin{array}{ll} -\left(a+b\mathlarger{\int}_{\Omega}|\nabla u|^{2}dx\right)\Delta u+V(x)u=u^{5},…

We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth \[\left\{\begin{gathered} - \Bigl({\varepsilon ^2}a + \varepsilon b\int_{{\R^3}} {{{\left| {\nabla u} \right|}^2}}…

We study the following nonlinear Hartree-type equation \begin{equation*} -\Delta u+V(x)u-a(\frac{1}{|x|^\gamma}\ast |u|^2)u=\lambda u,~\text{in}~\mathbb{R}^N, \end{equation*} where $a>0$, $N\geq3$, $\gamma\in(0,2)$ and $V(x)$ is an external…

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $\lambda\to 0$ and…

We consider ground states of $L^2$-subcritical nonlinear Schr\"{o}dinger equation (1.1), which can be described equivalently by minimizers of the following constraint minimization problem $$ e(\rho):=\inf\{E_{\rho}(u):u\in…