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We consider singular quasilinear elliptic systems with homogeneous Dirichlet boundary condition. Using Leray-Schauder topological degree, combined with the sub-supersolutions method and suitable truncation arguments, we establish the…

In this paper, we study the following fractional Schr\"{o}dinger-Poisson system involving competing potential functions \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=K(x)f(u)+Q(x)|u|^{2_s^{\ast}-2}u,…

We consider the following anisotropic sinh-Poisson tpye equation with a Hardy or H\'{e}non term: $$-\mathrm{div} (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2\alpha}(e^u-e^{-u}) \quad\mathrm{in}\quad \Omega,$$ $$\frac{\partial u}{\partial…

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 consider the problem {\Delta}u+V(x)u = f'(u) in RN. Here the nonlinearity has a double power behavior and V is invariant under an orthogonal involution, with V ({\infty}) = 0. An existence theorem of one pair of solutions which change…

We study the existence and multiplicity of positive normalized solutions with prescribed $L^{2}$-norm for the Sobolev critical Schr\"odinger equation $-\Delta U + V(x) U = \lambda U + |U|^{2^*-2} U$ in $\mathbb{R}^N$, $\int_{\mathbb{R}^N}…

In this paper, we deal with the planar Schr\"{o}dinger-Poisson system \begin{equation*}\begin{cases} -\Delta u + V(x) u + \phi u = b|u|^{p-2} u \ &\text{in}\ \mathbb{R}^{2},\\\Delta \phi= u^{2} &\text{in}\ \mathbb{R}^{2},\end{cases}…

This paper is concerned with the following planar Schr\"{o}dinger-Poisson system \begin{equation*} \begin{cases} -\triangle{u}+V(x)u+\phi{(x)}|u|^{p-2}u=f(x,u),&\text{in $\mathbb{R}^{2}$}, \triangle{\phi}=|u|^{p},&\text{in…

Goal of this paper is to study positive semiclassical solutions of the nonlinear Schr\"odinger equation $$ \varepsilon^{2s}(- \Delta)^s u+ V(x) u= f(u), \quad x \in \mathbb{R}^N,$$ where $s \in (0,1)$, $N \geq 2$, $V \in…

In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schr\"{o}dinger equation given by \begin{align*} \left\{ \begin{aligned} &-\epsilon^2 \Delta u+V( x)u=\lambda u+u \log u^2,…

We study the existence of nonradial sign-changing solutions to the Schroedinger-Poisson system in dimension N>=3. We construct nonradial sign-changing multi-peak solutions whose peaks are displaced in suitable symmetric configurations and…

The aim of this work is the study of the existence of normalized solutions to the nonlinear Schr\"odinger equation with nonlocal nonlinearities: \begin{equation}\nonumber \left\{\begin{aligned} &-\Delta u =\lambda…

In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schr\"odinger equation $$ \left\{ \begin{array}{l} \epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u =…

We consider the Schr\"odinger-Poisson-Slater (SPS) system in $\R^3$ and a nonlocal SPS type equation in balls of $\mathbb R^3$ with Dirichlet boundary conditions. We show that for every $k\in\mathbb N$ each problem considered admits a nodal…

In this paper, a class of coupled systems of nonlinear Schrodinger equations with sign-changing potential, including the linearly coupled case, is considered. The existence of non-trivial bound state solutions via linking methods for cones…

We consider the standing-wave problem for a nonlinear Schr\"{o}dinger equation, corresponding to the semilinear elliptic problem \begin{equation*} -\Delta u+V(x)u=|u|^{p-1}u,\ u\in H^1(\mathbb{R}^2), \end{equation*} where $V(x)$ is a…

In this paper, we consider the following nonlinear Schr\"odinger system in $R^3$: \begin{align*} -\Delta u_j +P_j(x) u=\mu_j u_j^3+\sum\limits_{i=1,i\neq j}^N\beta_{ij}u_i^2u_j, \end{align*} where $N\geq3$, $P_j$ are nonnegative radial…

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,\,\,…

We look for ground state solutions to the following nonlinear Schr\"{o}dinger equation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic…

Consider the Schr\"odinger--Bopp--Podolsky system \[ \begin{cases} -\epsilon^2\Delta u+(V+K\phi)u=u|u|^{p-1};\newline \Delta^2\phi-\Delta\phi=4\pi K u^2 \end{cases} ~\text{in}~\mathbb{R}^3 \] for sufficiently small $\epsilon>0$, where…