Related papers: Concentration phenomena for the Schr\"odinger-Pois…

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 the paper we consider the following quasilinear Schr\"odinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases} - \varepsilon^2 \Delta u + (V + \phi) u = u |u|^{p - 1} \newline - \Delta \phi - \beta \Delta_4 \phi = u^2,…

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}…

In this paper, we study the following nonlinear Schr\"odinger-Poisson type equation \begin{equation*} \begin{cases} -\varepsilon^2\Delta u+V(x)u+K(x)\phi u=f(u)&\text{in}\ \mathbb{R}^3,\\ -\varepsilon^2\Delta \phi=K(x)u^2&\text{in}\…

We study the problem (-\epsilon\mathrm{i}\nabla+A(x)) ^{2}u+V(x)u=\epsilon ^{-2}(\frac{1}{|x|}\ast|u|^{2}) u, u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),\text{\ \ \ \}\epsilon\nabla u+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}), where…

We prove the existence of a ground state positive solution of Schr\"odinger-Poisson systems in the plane of the form $$ -\Delta u + V(x)u + \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \qquad\text{in}\…

We consider the equation $$-\Delta u+u=Q_\varepsilon(x)|u|^{p-2}u,\qquad u\in H^1(\mathbb{R}^N),$$ where $Q_\varepsilon$ takes the value $1$ on each ball $B_\varepsilon(y)$, $y\in\mathbb{Z}^N$, and the value $-1$ elsewhere. We establish the…

In this paper, we are concerned with the coupled nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} -\varepsilon^{2}\Delta u+a(x)u=\mu_{1}u^{3}+\beta v^{2}u \ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\ -\varepsilon^{2}\Delta…

This work examines a quasilinear Schr\"odinger-Poisson system involving a critical nonlinearity, expressed as \[ -\Delta u + \phi u + \lambda u = |u|^{q-2} u + |u|^4 u, \quad x \in \Omega_r, \] \[ -\Delta \phi - \varepsilon^4 \Delta_4 \phi…

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,\,\,…

Consider the following nonlinear Schr\"odinger-Bopp-Podolsky system in $\mathbb{R}^3$: $$ \begin{cases} -\varepsilon^2 \Delta u + (V + \phi) u = u |u|^{p-1}; \\ a^2 \Delta^2 \phi - \Delta \phi = 4 \pi u^2, \end{cases} $$ where $a,…

In this paper, we study the following semilinear Schr\"odinger equation $$ -\epsilon^2\triangle u+ u+ V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}), $$ where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function $V$ is bounded in…

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger 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 an…

In this article we study the existence and concentration behavior of bound states for a nonlinear Schr\"odinger-Poisson system with a parameter $\varepsilon>0$. Under some suitable conditions on the potential functions, we prove that for…

In this paper, we study the following Schr\"odinger-Poisson system: $$ \left\{\aligned&-\Delta u+V_\lambda(x)u+K(x)\phi u=f(x,u)&\quad\text{in }\bbr^3,\\ &-\Delta\phi=K(x)u^2&\quad\text{in }\bbr^3,\\…

We study the concentration phenomenon for solutions of the fractional nonlinear Schr\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation…

We study the weakly coupled nonlinear Schr\"odinger system \begin{equation*} \begin{cases} -\Delta u_1 = \mu_1 u_1^{p} +\beta u_1^{\frac{p-1}{2}} u_2^{\frac{p+1}{2}}\text{ in } \Omega,\\ -\Delta u_2 = \mu_2 u_2^{p} +\beta…

We study a class of planar Schr\"{o}dinger-Poisson systems $$ -\Delta u+\lambda V(x)u+\phi u=f(u) , \quad x\in{\mathbb R}^2,\qquad \Delta \phi=u^2, \quad x\in{\mathbb R}^2, $$ where $\lambda>0$ is a parameter, $V\in C({\mathbb R}^2,{\mathbb…

In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schr\"odinger system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)…

In this paper, we are concerned with the Schr\"{o}dinger-Poisson system \begin{equation} (0.1)\qquad -\Delta u + u +\phi u = |u|^{p-2}u \quad \text{in}\ \mathbb{R}^{d},\qquad \Delta \phi= u^{2} \quad \text{in}\ \mathbb{R}^{d}.…