相关论文: Normalized solutions for Schr\"{o}dinger-Bopp-Podo…
In this paper, we consider the existence, multiplicity and nonexistence of solutions for the following equation \begin{equation*} \begin{cases} \begin{aligned} &-\Delta u+\omega u=\mu u^{p-1}+u^{q-1},~ u>0 \quad &&\text { in } \Omega, \\…
In this article we study the quasi-linear equation \[ \left\{ \begin{aligned} \mathrm{div}\, \mathcal A(x,u,\nabla u)&=\mathcal B(x,u,\nabla u)&&\text{in }\Omega,\\ u\in H^{1,p}_{loc}&(\Omega;wdx) \end{aligned} \right. \] where $\mathcal A$…
Given a smooth bounded domain $\Omega\subset \mathbb R^3$, we consider the following nonlinear Schr\"odinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+ \phi u -\abs{u}^{p-2}u = \omega u & \quad \text{in }…
In this paper,we show that spherical bounded energy solution of the defocusing 3D energy critical Schr\"odinger equation with harmonic potential, $(i\partial_t + \frac {\Delta}2+\frac {|x|^2}2)u=|u|^4u$, exits globally and scatters to free…
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…
This paper focuses on the normalized solutions for the planar Schr\"{o}dinger-Poisson system with a two-electron interaction, which models the effect between electrons and the electrostatic potential they generate. As the parameters vary,…
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…
In this paper we establish a new existence result for the quasilinear elliptic problem \[ -{\rm div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x,u)|\nabla u|^p + V(x)|u|^{p-2} u = g(x,u)\quad\mbox{ in } \mathbb{R}^N, \] with $N\ge 2$,…
In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schr\"odinger system \[ i\partial_t u_{j}+\partial_{xx}u_{j}+ \left(\sum_{k=1}^{3} a_{kj}…
In this paper, we consider the existence of normalized solutions for the following biharmonic nonlinear Schr\"{o}dinger system \begin{equation*} \begin{cases} \Delta^2u+\alpha_{1}\Delta u+\lambda u=\beta r_{1}|u|^{r_{1}-2}|v|^{r_{2}} u &…
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 minimizing problem with two constraints: \[ \inf \left\{ E(u) | u=(u_1,u_2), \ \| u_1 \|_{L^2}^2 = \alpha_1, \ \| u_2 \|_{L^2}^2 = \alpha_2 \right\}, \] where $\alpha_1,\alpha_2 > 0$ and $E(u)$ is…
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…
It is shown that if $p \ge 3$ and $u \in W^{1,p}(\Omega,\mathbb{R}^N)$ solves the inhomogenous $p$-Laplace system \[ \operatorname{div} (|\nabla u|^{p-2} \nabla u) = f, \qquad f \in W^{1,p'}(\Omega,\mathbb{R}^N), \] then locally the…
We are devoted to the study of the following nonlinear $p$-Laplacian Schr\"odinger equation with $L^{p}$-norm constraint \begin{align*} \begin{cases} &-\Delta_{p} u=\lambda |u|^{p-2}u +|u|^{r-2}u\quad\mbox{in}\quad\Omega,\\…
In this article we study the quasi-linear equation \[\mathrm{div}\, \mathcal A(x,u,\nabla u)=\mathcal B(x,u,\nabla u)\quad \text{in }\Omega,\qquad u\in H^{1,p}_{loc}(\Omega;w_1dx)\] where $\mathcal A$ and $\mathcal B$ are functions…
In this paper, we study the following fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=g(u) & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-\Delta)^t\phi=u^2,\,\,…
In this article we study the existence of solutions to the system \begin{equation*}\left\{ \begin{array}{ll} -\left(a+b\int_{\Omega}|\nabla u|^{2}\right)\Delta u +\phi u= f(x, u) &\text{in }\Omega \hbox{} -\Delta \phi= u^{2} &\text{in…
In this paper we study the existence and regularity results of normalized solutions to the following quasilinear elliptic Choquard equation with critical Sobolev exponent and mixed diffusion type operators: \begin{equation*}…
This paper concerns the normalized ground states for the nonlinear Schr\"{o}dinger equation in the Bopp-Podolsky electrodynamics. This equation has a nonlocal nonlinearity and a mass supercritical power nonlinearity, both of which have deep…