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In this paper, we study the nonlinear Schr\"{o}dinger equation $ -\Delta u+V(x)u=f(x,u) $on the lattice graph $ \mathbb{Z}^{N}$. Using the Nehari method, we prove that when $f$ satisfies some growth conditions and the potential function $V$…

In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schr\"{o}dinger operators…

We study the Schr\"{o}dinger equation: \begin{equation} - \Delta u+V(x)u=f(x,u) ,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{equation} where $V$ is periodic and $f$ is periodic in the $x$-variables, 0 is in a gap of the spectrum of the…

We study the nonlinear Helmholtz equation $(\Delta - \lambda^2)u = \pm |u|^{p-1}u$ on $\mathbb{R}^n$, $\lambda > 0$, $p \in \mathbb{N}$ odd, and more generally $(\Delta_g + V - \lambda^2)u = N[u]$, where $\Delta_g$ is the (positive)…

We study the problem \begin{eqnarray*} (-\Delta)^s u &=& u^p - u^q \quad\text{in }\quad \mathbb{R}^N, u &\in& \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), u&>0& \quad\text{in}\quad\mathbb{R}^N, \end{eqnarray*} where $s\in(0,1)$ is a…

We consider a semiclassical $2\times 2$ matrix Schr\"odinger operator of the form $P=-h^2\Delta {\bf I}_2 + {\rm diag}(x_n-\mu, \tau V_2(x)) +hR(x,hD_x)$, where $\mu$ and $\tau$ are two small positive constants, $V_2$ is real-analytic and…

In this paper, we consider the singularly perturbed fractional Schr\"{o}dinger equation \begin{equation*} \epsilon^{2\alpha}(-\Delta)^\alpha u+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $\epsilon>0$ is a small parameter,…

We study the existence of solutions of the following nonlinear Schr\"odinger equation \begin{equation*} -\Delta u + \Big(V(x)-\frac{\mu}{|x|^2}\Big) u = f(x,u) \hbox{ for } x\in\mathbb{R}^N\setminus\{0\}, \end{equation*} where…

We look for solutions to a fractional Schr\"odinger equation of the following form $$ (-\Delta)^{\alpha / 2} u + \left( V(x) - \frac{\mu}{|x|^{\alpha}} \right) u = f(x,u)-K(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N \setminus \{0\}, $$ where $V$…

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[(-\Delta+m^2)^\sigma u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $0<\sigma<1$, $V$ is a…

We prove the existence of $L^2$ solutions to the Helmholtz equation $(-\Delta - 1)u = f$ in ${\mathbb R}^n$ assuming the given data $f$ belongs to $L^{(2n+2)/(n+5)}({\mathbb R}^n)$ and satisfies the "Fredholm condition" that $\hat{f}$…

We study the existence and non-existence of nontrivial weak solution of $$ {\Delta^2u-\mu\frac{u}{|x|^{4}} = \frac{|u|^{q_{\beta}-2}u}{|x|^{\beta}}+|u|^{q-2}u\quad\textrm{in ${\mathbb R}^N$,}} $$ where $N\geq 5$,…

We study the Choquard equation involving mixed local and nonlocal operators $$-\Delta u+(-\Delta)^{s}u+V(x)u=(\frac{1}{|x|^{\mu}}* F(u))f(u)\quad\text{in }\R^{2},$$ where $s\in(0,1)$, $\mu\in(0,2)$, $F(t)=\int_{0}^{t} f(\tau)\,d\tau$, and…

We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \ast |u|^p)|u|^{p - 2} u= |u|^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_\alpha$ is the Riesz potential of…

The present study is concerned with the following fractional Schr\"{o}dinger-Poisson system with steep potential well: $$ \left\{% \begin{array}{ll} (-\Delta)^s u+ \la V(x)u+K(x)\phi u= f(u), & x\in\R^3, (-\Delta)^t \phi=K(x)u^2, &…

We look for a solutions to a nonlinear, fractional Schr\"odinger equation $$(-\Delta)^{\alpha / 2}u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where potential $V$ is coercive or $V=V_{per} + V_{loc}$ is a sum of periodic…

We study the following Choquard type equation in the whole plane $(C) -\Delta u+V(x)u=(I_2\ast F(x,u))f(x,u),x\in\mathbb{R}^2$ where $I_2$ is the Newton logarithmic kernel, $V$ is a bounded Schr\"odinger potential and the nonlinearity…

In this paper, we consider the following quasilinear Schr\"{o}dinger equation \begin{align*} -\Delta u-u\Delta(u^{2})=k(x)\left\vert u\right\vert ^{q-2}u-h(x)\left\vert u\right\vert ^{s-2}u\text{, }u\in D^{1,2}(\mathbb{R}^{N})\text{,}…

We establish several existence results for traveling-wave solutions of the nonlocal derivative nonlinear Schr\"odinger equation with general coefficients by variational methods. We study associated minimization problems in the subcritical…

Let $\sigma\in(0,1)$ with $\sigma\neq\frac{1}{2}$. We investigate the fractional nonlinear Schr\"odinger equation in $\mathbb R^d$: $$i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-\Delta)^\sigma$ is the…