Existence Of Positive Ground States Research Articles

Concentrating ground state for linearly coupled Schrödinger systems involving critical exponent cases

We study the following linearly coupled Schrödinger system:{−Δu+(μV1(x)+a)u=f(x)|u|p−2u+β(x)v,x∈RN,−Δv+(μV2(x)+b)v=g(x)|v|2⁎−2v+β(x)u,x∈RN, where N≥3,2<p≤2⁎, V1,V2∈C(RN,R+) are potential wells with bottoms Ωi=intVi−1(0), the parameters a>−λ1(Ω1),b>−λ1(Ω2) and λ1(Ωi) is the first eigenvalue of −Δ in H01(Ωi). Under some suitable assumptions on β(x) which relate to the potentials V1,V2 and the parameters a,b, the existence of positive ground states is obtained by variational method. Some interesting phenomena are that we relax the upper control condition of the coupling function β(x) and we do not need the weight function f(x) to be integrable or bounded in the subcritical case 2<p<2⁎. Moreover, from the concentration phenomenon of solutions, we obtain some results of the existence of positive ground states to a linearly coupled Schrödinger system in a bounded domain, which extend the recent results of Peng, et al. (2017), [25].

Existence of Positive Ground States of Nonlocal Nonlinear Schrödinger Equations

We investigate ground states of a (nonlocal) nonlinear Schrödinger equation which generalizes classical (fractional, relativistic, etc.) Schrödinger equations, so that we extend relevant results and study common properties of these equations in a uniform way. To obtain the existence of ground states, we first solve a minimization problem and then prove that the solution of the minimization problem is a ground state of the equation. After examining the regularity of the solutions to the equation, we demonstrate that any ground state is sign-definite.

Groundstates of the planar Schrödinger–Poisson system with potential well and lack of symmetry

The Schrödinger–Poisson system describes standing waves for the nonlinear Schrödinger equation interacting with the electrostatic field. In this paper, we are concerned with the existence of positive ground states to the planar Schrödinger–Poisson system with a nonlinearity having either a subcritical or a critical exponential growth in the sense of Trudinger–Moser. A feature of this paper is that neither the finite steep potential nor the reaction satisfies any symmetry or periodicity hypotheses. The analysis developed in this paper seems to be the first attempt in the study of planar Schrödinger–Poisson systems with lack of symmetry.

On concentration of solutions for quasilinear Schrödinger equations with critical growth in the plane

In this work, we study the existence of positive ground states and concentration phenomena for the following class of quasilinear Schrödinger equations: where is a parameter, is a continuously differentiable function, and are positive and bounded continuous potentials and the nonlinearity can exhibit critical exponential growth. In order to prove our concentration result, we exploit variational arguments that take into account an interaction between the potentials V and Q, in combination with the Trudinger–Moser inequality and Moser iteration method.

Ground states for a linearly coupled indefinite Schrödinger system with steep potential well

In this paper, we study a class of linearly coupled Schrödinger systems with steep potential wells, which arises from Bose–Einstein condensates. The existence of positive ground states is investigated by exploiting the relation between the Nehari manifold and fiberring maps. Some interesting phenomena are that we do not need the weight functions in the nonlinear terms to be integrable or bounded and we can relax the upper control condition of the coupling function. Moreover, the decay rate and concentration phenomenon of positive ground states are also studied.

Choquard-type equation with competing coefficients

In this paper, we investigate the following Choquard-type equation−Δu+Vu=[Iα⁎(K|u|p)]K|u|p−2uinRN, where V,K∈C(RN) are two potential functions, and Iα denotes the Riesz potential. By comparing the decay rates of V and K, we obtain two results stating the existence of positive ground states to the above equation. Our proofs are heavily based on the fundamental properties of solutions. These properties are also important to study the Choquard-type equations.

A Positivity Preserving Iterative Method for Finding the Ground States of Saturable Nonlinear Schrödinger Equations

In this paper, we propose an iterative method to compute the positive ground states of saturable nonlinear Schrodinger equations. A discretization of the saturable nonlinear Schrodinger equation leads to a nonlinear algebraic eigenvalue problem (NAEP). For any initial positive vector, we prove that this method converges globally with a locally quadratic convergence rate to a positive solution of NAEP. During the iteration process, the method requires the selection of a positive parameter $$\theta _k$$ in the kth iteration, and generates a positive vector sequence approximating the eigenvector of NAEP and a scalar sequence approximating the corresponding eigenvalue. We also present a halving procedure to determine the parameters $$\theta _k$$ , starting with $$\theta _k=1$$ for each iteration, such that the scalar sequence is strictly monotonic increasing. This method can thus be used to illustrate the existence of positive ground states of saturable nonlinear Schrodinger equations. Numerical experiments are provided to support the theoretical results.

Nonlinear scalar field equations with general nonlinearity

Consider the nonlinear scalar field equation (0.1)−Δu=f(u)inRN,u∈H1(RN),where N≥3 and f satisfies the general Berestycki–Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski (2017) made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski’s results on (0.1). The keys to our approach are an extension to the symmetric mountain pass setting of the monotonicity trick, and a new decomposition result for bounded Palais–Smale sequences.

Existence and nonexistence of positive solutions for a static Schrödinger–Poisson–Slater equation

In this paper we study the following type of the Schrödinger–Poisson–Slater equation with critical growth−△u+(u2⋆1|4πx|)u=μ|u|p−1u+|u|4u,inR3, where μ>0 and p∈(11/7,5). For the case of p∈(2,5). We develop a novel perturbation approach, together with the well-known Mountion–Pass theorem, to prove the existence of positive ground states. For the case of p=2, we obtain the nonexistence of nontrivial solutions by restricting the range of μ and also study the existence of positive solutions by the constrained minimization method. For the case of p∈(11/7,2), we use a truncation technique developed by Brezis and Oswald [9] together with a measure representation concentration-compactness principle due to Lions [27] to prove the existence of radial symmetrical positive solutions for μ∈(0,μ⁎) with some μ⁎>0. The above results nontrivially extend some theorems on the subcritical case obtained by Ianni and Ruiz [18] to the critical case.

On some nonlocal equations with competing coefficients

In this paper we study the following nonlocal problems{−(a+b∫R3|∇u|2)Δu+V(x)u=Q(x)up,x∈R3,u∈H1(R3),u>0,x∈R3, where the constants a,b>0, 3<p<5, V(x) and Q(x) are two functions in Clocγ(R3)⋂L∞(R3). By comparing the decay rate of V(x) and Q(x), we first obtain two theorems stating the existence of positive ground states. Under certain assumptions on Q(x), we further prove the existence of positive bound states by using a linking argument with a barycenter map restricted on a Nehari manifold.

Ground states for pseudo-relativistic Choquard equations

We investigate the existence of positive ground states for pseudo-relativistic nonlinear Choquard equations. Our results are based on Nehari manifold technique and rearrangement methods. Furthermore, we also obtain the nonrelativistic limit of ground states in H1(RN) space.

Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent

In this paper, we consider the following semilinear Kirchhoff type equation $$\left\{\begin{array}{ll}-\left(\epsilon^2a + \epsilon b \int \limits_{\mathbb{R}^3}|\nabla u|^2 \right) \triangle {u}+V(x)u = \mu K(x)|u|^{p-1}u + Q(x)|u|^4u, \,\, \mathrm{in}\, \mathbb{R}^3, \\ u \in H^1(\mathbb{R}^3), \,\, u > 0,\end{array}\right.$$ where \({\epsilon > 0}\) is a small parameter, \({p \in [3,5)}\), a, b are positive constants, μ > 0 is a parameter, and the nonlinear growth of |u|4u reaches the Sobolev critical exponent since 2* = 6 for three spatial dimensions. We prove the existence of a positive ground state solution \({u_\epsilon}\) with exponential decay at infinity for μ > 0 and \({\epsilon}\) sufficiently small under some suitable conditions on the nonnegative functions V, K and Q. Moreover, \({u_\epsilon}\) concentrates around a global minimum point of V as \({\epsilon \rightarrow 0^+}\). The methods used here are based on the concentration-compactness principle of Lions.

Existence of positive ground states for some nonlinear Schrödinger systems

Abstract We prove the existence of positive ground states for the nonlinear Schrödinger system { − Δ u + ( 1 + a ( x ) ) u = F u ( u , v ) + λ v , − Δ v + ( 1 + b ( x ) ) v = F v ( u , v ) + λ u , where a, b are periodic or asymptotically periodic and F satisfies some superlinear conditions in ( u , v ) . The proof is based on the method of Nehari manifold and the concentration-compactness principle. MSC:35J05, 35J50, 35J61.

Positive ground states for asymptotically periodic Schrödinger–Poisson systems

In this paper, we establish the existence of positive ground states for asymptotically periodic Schrödinger–Poisson systems with general nonlinearities by the method of Nehari manifold. Copyright © 2012 John Wiley &amp; Sons, Ltd.