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In this paper, we study the existence of normalized solutions for the nonautonomous Schr\"{o}dinger-Poisson equations \begin{equation}\nonumber -\Delta u+\lambda u +\left(\vert x \vert ^{-1} * \vert u \vert ^{2} \right)…

This is the first of two articles dealing with the equation $(-\Delta)^{s} v= f(v)$ in $\mathbb{R}^{n}$, with $s\in (0,1)$, where $(-\Delta)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a L\'evy process. This…

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroedinger Equation $- \epsilon^2 \Delta \psi + V(x) \psi = |\psi|^{p-1} \psi$ on a manifold or in the Euclidean space. Here V represents the potential, p is…

We present a new generalization of the steepest descent method introduced by Deift and Zhou for matrix Riemann-Hilbert problems and use it to study the semiclassical limit of the focusing nonlinear Schroedinger equation with real analytic,…

In this paper, we study normalized solutions to the Chern-Simons-Schr\"odinger system, which is a gauge-covariant nonlinear Sch\"oridnger system with a long-range electromagnetic field, arising in nonrelativistic quantum mechanics theory.…

In this paper, we search for normalized solutions to a fractional, nonlinear, and possibly strongly sublinear Schr\"odinger equation $$(-\Delta)^s u + \mu u = g(u) \quad \hbox{in $\mathbb{R}^N$},$$ 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…

We are concerned with the following Schr\"odinger-Newton problem \begin{equation} -\varepsilon^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon^2} \big(\int_{\mathbb R^3}\frac{u^2(\xi)}{|x-\xi|}d\xi\big)u,~x\in \mathbb R^3. \end{equation} For…

Consider the equation \begin{equation*} -\Delta_p u =\lambda |u|^{p-2}u+\mu|u|^{q-2}u+|u|^{p^\ast-2}u\ \ {\rm in}\ \R^N \end{equation*} under the normalized constraint $$\int_{ \R^N}|u|^p=c^p,$$ where $-\Delta_pu={\rm div} (|\nabla…

In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb…

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 examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{eqnarray} \left\{ \begin{array}{lcl} \hfill \mathcal L u &=& \lambda F(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill \mathcal L v &=& \gamma G(u,v)…

We consider the problem $-\Delta u+\lambda u=u^{p-1}$, where $u\in H^1_0(\Omega)$ verifies $\|u\|_{L^2}=m>0$, and $\lambda\in [0,+\infty)$. Here, $\mathbb{R}^N\setminus\Omega$ is nonempty and compact. We prove the existence of a solution…

This paper constructs solutions to linear and nonlinear Schr\"odinger-type equations in two and three spatial dimensions that exhibit prescribed, extraordinary gradient amplification and localization. For any finite time interval $[0,T]$,…

We consider the following singularly perturbed nonlinear elliptic problem: $$-\e^2\Delta u+V(x)u=f(u),\ u\in H^1(\mathbb{R^N}),$$ where $N\ge 3$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a solution $u_\e$…

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate…

We investigate the existence and the singular limit of normalized ground states for focusing doubly nonlinear Schr\"odinger equations with both standard and concentrated nonlinearities on two-dimensional square grids. First, we provide…

We consider the nonlinear Schr\"{o}dinger equation $-\Delta u+(\lambda a(x)+1)u=|u|^{p-1}u$ on a locally finite graph $G=(V,E)$. We prove via the Nehari method that if $a(x)$ satisfies certain assumptions, for any $\lambda>1$, the equation…

This paper investigates a local central limit theorem for a normalized sequence of random variables belonging to a fixed order Wiener chaos and converging to the standard normal distribution. We prove, without imposing any additional…

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 $V$…