Related papers: Ground state solutions for the nonlinear Schroding…

We prove existence and multiplicity of bound and ground state solutions, under appropriate conditions on the parameters, for a bi-harmonic stationary system of coupled nonlinear Schr\"odinger--Korteweg-de Vries equations.

We prove the existence results for the Schr\"odinger equation of the form $$ -\Delta u + V(x) u = g(x,u), \quad x \in \mathbb{R}^N, $$ where $g$ is superlinear and subcritical in some periodic set $K$ and linear in $\mathbb{R}^N \setminus…

In this paper we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schr\"{o}dinger equations with square root of the Laplacian $$ \left\{ \begin{array}{lr}…

We consider the focusing nonlinear Schr\"odinger equation with the critical inverse square potential. We give the first proof of the uniqueness of the ground state solution. Consequently, we obtain a sharp Hardy-Gagliardo-Nirenberg…

This paper is devoted to studying the following nonlinear biharmonic Schr\"odinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} \Delta^{2}u-\lambda u=\mu|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\…

We study the following class of linearly coupled Schr\"{o}dinger elliptic systems $$\left\{ \begin{array}{lr} -\Delta u+V_{1}(x)u=\mu|u|^{p-2}u+\lambda(x)v, & \quad x\in\mathbb{R}^{N}, \\ -\Delta v+V_{2}(x)v=|v|^{q-2}v+\lambda(x)u, &…

We consider in the whole plane the Hamiltonian coupling of semilinear Schroedinger equations which have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground state solutions is compact up to translations.…

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 study the existence of ground state standing waves, of prescribed mass, for the nonlinear Schr\"{o}dinger equation with mixed power nonlinearities \begin{equation*} i \partial_t v + \Delta v + \mu v |v|^{q-2} + v |v|^{2^* - 2} = 0, \quad…

This paper explores the existence and properties of ground states, including both energy and action ground states, for nonlinear Dirac equations with power-type potentials. \begin{equation*} -i c\sum\limits_{k=1}^3\alpha_k\partial_k u +mc^2…

Mathematical proofs are presented concerning the existence of solutions of the Maxwell equations with suitable boundary conditions. In particular it is stated that the well known "delayed potentials" provide effective solutions of the…

We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- \Delta u + u = \bigl(I_\alpha \ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_\alpha) is a Riesz potential, under almost…

We consider the nonlinear Schr\"odinger equation on a unit ball in one and two dimensions with Dirichlet boundary conditions, which have stabilizing effect on solutions behavior. In particular, we confirm that the ground state solutions are…

In this paper, we study the following class of nonlinear equations: $$ -\Delta u+V(x) u = \left[|x|^{-\mu}*(Q(x)F(u))\right]Q(x)f(u),\quad x\in\mathbb{R}^2, $$ where $V$ and $Q$ are continuous potentials, which can be unbounded or vanishing…

The goal is a construction of stationary solutions close to a non-trivial combination of two plane waves at high energies for a periodic non-linear Schroedinger equation in dimension two. The corresponding isoenergetic surfaces are…

In this paper, we study the existence of a ground state solution, that is, a non trivial solution with least energy, of a noncooperative semilinear elliptic system on a bounded domain. By using the method of the generalized Nehari manifold…

It is established ground states and multiplicity of solutions for a nonlocal Schr\"{o}dinger equation $(-\Delta )^s u + V(x) u = \lambda a(x) |u|^{q-2}u + b(x)f(u)$ in $\mathbb{R}^N,$ $u \in H^s(\mathbb{R}^N),$ where $0<s<\min\{1,N/2\},$…

We consider the stationary semilinear Schr\"odinger equation $-\Delta u + a(x) u = f(x,u)$, $u\in H^1(\R^N)$, where $a$ and $f$ are continuous functions converging to some limits $a_\infty>0$ and $f_\infty=f_\infty(u)$ as $|x|\to\infty$. In…

We consider the stationary magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u=\bigg(\frac{1}{|x|^{\alpha}}*F(|u|)\bigg)\frac{f(|u|)}{|u|}{u},\] where $A: \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a vector potential, $V$ is…

We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equation $$ -\Delta u+u=(I_\alpha*F(u))F'(u)\qquad\text{in }\mathbb{R}^2, $$ where $I_\alpha$ is the Riesz potential of order $\alpha$ on the…