Existence and concentration of ground state solutions for a critical nonlocal Schr\"odinger equation in R2\R^2

We study the following singularly perturbed nonlocal Schr\"{o}dinger equation -\vr^2\Delta u +V(x)u =\vr^{\mu-2}\Big[\frac{1}{|x|^{\mu}}\ast F(u)\Big]f(u) \quad \mbox{in} \quad \R^2, where $V(x)$ is a continuous real function on $\R^2$, $F(s)$ is the primitive of $f(s)$, $0<\mu<2$ and \vr is a positive parameter. Assuming that the nonlinearity $f(s)$ has critical exponential growth in the sense of Trudinger-Moser, we establish the existence and concentration of solutions by variational methods.Comment: 3

On Holographic Insulators and Supersolids

We obtain holographic realizations for systems that have strong similarities to Mott insulators and supersolids, after examining the ground states of Einstein-Maxwell-scalar systems. The real part of the AC conductivity has a hard gap and a discrete spectrum only. We add momentum dissipation to resolve the delta function in the conductivity due to translational invariance. We develop tools to directly calculate the Drude weight for a large class of solutions and to support our claims. Numerical RG flows are also constructed to verify that such saddle points are IR fixed points of asymptotically AdS_4 geometries.Comment: 52 pages, jheppub, 15 figures; v2: minor corrections, references adde

Ground state solutions for diffusion system with superlinear nonlinearity

In this paper, we study the following diffusion system \begin{equation*} \begin{cases} \partial_{t}u-\Delta_{x} u +b(t,x)\cdot \nabla_{x} u +V(x)u=g(t,x,v),\\ -\partial_{t}v-\Delta_{x} v -b(t,x)\cdot \nabla_{x} v +V(x)v=f(t,x,u) \end{cases} \end{equation*} where $z=(u,v)\colon\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$, $b\in C^{1}(\mathbb{R}\times\mathbb{R}^{N}, \mathbb{R}^{N})$ and $V(x)\in C(\mathbb{R}^{N},\mathbb{R})$. Under suitable assumptions on the nonlinearity, we establish the existence of ground state solutions by the generalized Nehari manifold method developed recently by Szulkin and Weth

Existence of solution for two classes of Schrödinger equations in RN\mathbb{R}^N with magnetic field and zero mass

In this paper, we consider the existence of a nontrivial solution for the following Schrödinger equations with a magnetic potential $A$ -\Delta_A u=K(x)f(|u|^2)u,~\quad \mbox{in}~\mathbb{R}^N where $N\geqslant3$, $K$ is a nonnegative function verifying two kinds of conditions and $f$ is continuous with subcritical growth

Homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign nonlinearities

In this paper, we obtain the multiplicity of homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign potentials. The concentration-compactness principle is applied to show the compactness. As a byproduct, we obtain the uniqueness of the positive ground state solution for a class of autonomous Hamiltonian systems and the best constant for Sobolev inequality which are of independent interests

Positive ground state of coupled planar systems of nonlinear Schrödinger equations with critical exponential growth

In this paper, we prove the existence of a positive ground state solution to the following coupled system involving nonlinear Schrödinger equations: −∆u + V1(x)u = f1(x, u) + λ(x)v, x ∈ R2 −∆v + V2(x)v = f2(x, v) + λ(x)u, x ∈ R2 where λ, V1, V2 ∈ C(R2 ,(0, +∞)) and f1, f2 : R2 × R → R have critical exponential growth in the sense of Trudinger–Moser inequality. The potentials V1(x) and V2(x) satisfy a condition involving the coupling term λ(x), namely 0 < λ(x) ≤ λ0 p V1(x)V2(x). We use non-Nehari manifold, Lions’s concentration compactness and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap regularity lifting argument and L q -estimates we get regularity and asymptotic behavior. Our results improve and extend the previous results