Related papers: Normalized ground states for nonlinear Schr\"{o}di…
We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho,…
This paper is devoted to studying the following nonlinear biharmonic Schr\"odinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} \Delta^{2}u-\lambda u=\mu|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\…
We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,} \] having…
In this paper, we look for solutions to the following coupled Schr\"{o}dinger system \begin{equation*} \begin{cases} -\Delta u+\lambda_{1}u=\alpha_{1}|u|^{p-2}u+\mu_{1}u^{3}+\rho v^{2}u & \text{in} \ \ \mathbb{R}^{N}, -\Delta…
This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schr\"odinger equation \begin{equation*} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=|u|^{p-2}u \quad\mathrm{in}\ \mathbb{R}^{N},…
In this paper we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\…
In this paper, we consider the following nonlinear Schr\"{o}dinger equations with mixed nonlinearities: \begin{eqnarray*} \left\{\aligned &-\Delta u=\lambda u+\mu |u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ &u\in…
In this paper, we study the existence of normalized solutions for the following quasilinear Schr\"odinger equation with Sobolev critical exponent: \begin{eqnarray*} -\Delta u-u\Delta (u^2)+\lambda…
We study the existence and nonexistence of normalized solutions $(u_a, \lambda_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear Schr\"{o}dinger equation with mixed nonlocal nonlinearities. This study can be viewed as a…
We propose a simple minimization method to show the existence of least energy solutions to the normalized problem \begin{cases} -\Delta u + \lambda u = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\ u \in H^1(\mathbb{R}^N), \\…
In this article, we study the existence of normalized ground state solutions for the following biharmonic nonlinear Schr\"{o}dinger equation with combined nonlinearities \begin{equation*} \Delta^2u=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u,\quad…
We study the existence of ground state normalized solution of the following Schr\"{o}dinger equation: \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda u=f(x,u), & x\in\mathbb{Z}^d \\ \Vert u\Vert_2^2=a \end{cases} \end{equation*}…
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, we aim to study the existence of ground state normalized solutions for the following quasilinear Schr\"{o}dinger equation $-\Delta u-\Delta(u^2)u=h(u)+\lambda u,\,\, x\in\R^N$, under the mass constraint…
We investigate the existence of normalized ground states to the system of coupled Schr\"odinger equations: \begin{equation}\label{eq:0.1} \begin{cases} -\Delta u_1 + \lambda_1 u_1 = \mu_1 |u_1|^{p_1-2}u_1 + \beta…
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 the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -\Delta u+\lambda_1u=\mu_1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -\Delta…
We study the existence of normalized solutions to the following logarithmic Schr\"{o}dinger equation \begin{equation*}\label{eqs01} -\Delta u+\lambda u=\alpha u\log u^2+\mu|u|^{p-2}u, \ \ x\in\R^N, \end{equation*} under the mass constraint…
We study the Sobolev critical Schr\"odinger-Bopp-Podolsky system \begin{gather*} -\Delta u+\phi u=\lambda u+\mu|u|^{p-2}u+|u|^4u\quad \text{in }\mathbb{R}^3, -\Delta\phi+\Delta^2\phi=4\pi u^2\quad \text{in } \mathbb{R}^3, \end{gather*}…
In present paper, we study the following nonlinear Schr\"{o}dinger equation with combined power nonlinearities \begin{align*} - \Delta u+V(x)u+\lambda u=|u|^{2^*-2}u+\mu |u|^{q-2}u \quad \quad \text{in} \ \mathbb{ R}^N, \ N\geq 3…