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In the present paper, we consider the Chern-Simons-Schr\"{o}dinger system \begin{equation} \left\{ \begin{aligned} &-\varepsilon^{2}\Delta u+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\,\,\,\,x\in \mathbb{R}^2,\\ &\partial_1 A_0 = A_2…

We are concerned with the following nonlinear Schr\"odinger equation $$-\varepsilon^2\Delta u+ V(x)u=|u|^{p-2}u,~u\in H^1(\R^N),$$ where $N\geq 3$, $2<p<\frac{2N}{N-2}$. For $\varepsilon$ small enough and a class of $V(x)$, we show the…

This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic system involving the Trudinger-Moser exponential growth nonlinearities. We first study the existence of solutions for the following…

This paper is concerned with the bifurcation from infinity of the nonlinear Schr\"odinger equation $$-\Delta u+V(x)u=\lambda u+f(x,u),\hspace{0.4cm} x\in \mathbb{R}^N.$$ We treat this problem in the framework of dynamical systems by…

We prove that the non-radial sign-changing solutions to the nonlinear Schr\"odinger equation \begin{equation*} \Delta u-u+|u|^{p-1}u=0 \mbox{ in }\R^N, \quad u \in H^1 (\R^N ) \end{equation*} constructed by Musso, Pacard and Wei is…

In this paper, we consider the following Schr\"odinger-Poisson system \begin{equation*} \begin{cases} - \Delta u+\lambda V(x)u+ \mu\phi u=|u|^{p-2}u &\text{in $\mathbb{R}^3$},\cr -\Delta \phi=u^{2} &\text{in $\mathbb{R}^3$}, \end{cases}…

The paper considers the following nonhomogeneous Schr\"odinger-Maxwell system -\Delta u + u+\lambda\phi (x) u =|u|^{p-1}u+g(x),\ x\in \mathbb{R}^3, -\Delta\phi = u^2, \ x\in \mathbb{R}^3, . \leqno{(SM)} where $\lambda>0$, $p\in(1,5)$ and…

For the first time, the general nonlinear Schr\"odinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class…

In this paper, we are concerned with qualitative properties of multi-peak solutions of the following nonlinear Schr\"{o}dinger equations \begin{equation*} -\Delta u+V(x)u= u^{p-\varepsilon},\,\,\,u>0,\,\,\,\text{in}\,\,\,\mathbb{R}^N,…

We obtain sufficient conditions for the existence and uniqueness of solutions with non-negative components to general quasilinear parabolic problems \begin{equation*} \partial_t u^k = \sum_{i,j=1}^n a_{ij} (t,x,u)\partial^2_{x_i x_j}\!u^k +…

We consider the semilinear electromagnetic Schr\"{o}dinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(\Omega,\mathbb{C}), where $\Omega=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq…

We consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb R^{3}$ $$\left\{ \begin{array}{c} -\varepsilon^{2} \Delta u + V(x)u + \phi u = f(u)\\ -\varepsilon^{2} \Delta \phi + \varepsilon^{4} \Delta^{2}\phi = 4\pi\varepsilon…

In this paper we consider the model semilinear Neumann system $$\left\{ \begin{array}{lll} -\Delta u+a(x)u=\lambda c(x) F_u(u,v)& {\rm in} & \Omega,\\ -\Delta v+b(x)v=\lambda c(x) F_v(u,v)& {\rm in} & \Omega,\\ \frac{\partial u}{\partial…

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…

We study long time dynamics of non-radial solutions to the focusing inhomogeneous nonlinear Schr\"odinger equation. By using the concentration/compactness and rigidity method, we establish a scattering criterion for non-radial solutions to…

This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation \begin{equation} \tag{$P$} \label{equ1} -\Delta u+ V(x) u= u\log u^2, \quad u\in H^1(\mathbb{R}^N), \end{equation} and its…

In this paper, we study the following $k$-coupled nonlinear Schr\"odinger system with Sobolev critical exponent: \begin{equation*} \left\{ \begin{aligned} -\Delta u_i & +\lambda_iu_i =\mu_i u_i^{2^*-1}+\sum_{j=1,j\ne i}^{k} \beta_{ij}…

In this paper we consider the problem \begin{equation*} \left \{ \begin{array}{l} -\Delta u \pm \phi u + W'(x,u) = 0\hbox{ in } \mathbb{R}^2,\newline \Delta \phi = u^2 \hbox{ in } \mathbb{R}^2, \end{array} \right. \end{equation*} where $W$…

In this paper we are focusing to the following Schr\"odinger-Maxwell system $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$: \[ \begin{cases} -\Delta_{g}u+\beta(x)u+eu\phi=\Psi(\lambda,x)f(u) & \mathrm{in}\ M -\Delta_{g}\phi+\phi=qu^{2} &…

We study the existence of positive solutions for the system of fractional elliptic equations of the type, \begin{equation*} \begin{array}{rl} (-\Delta)^{\frac{1}{2}} u &=\frac{p}{p+q}\lambda f(x)|u|^{p-2}u|v|^q + h_1(u,v)…