Concentration Of Ground State Solutions Research Articles

Existence and Concentration of Ground State Solutions for a Schrödinger–Poisson-Type System with Steep Potential Well

Existence and Concentration of Ground State Solutions for a Schrödinger–Poisson-Type System with Steep Potential Well

On nonlinear fractional Choquard equation with indefinite potential and general nonlinearity

In this paper, we consider a class of fractional Choquard equations with indefinite potential (−Δ)αu+V(x)u=[∫RNM(ϵy)G(u)|x−y|μdy]M(ϵx)g(u),x∈RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ (-\\Delta )^{\\alpha}u+V(x)u= \\biggl[ \\int _{{\\mathbb{R}}^{N}} \\frac{M(\\epsilon y)G(u)}{ \\vert x-y \\vert ^{\\mu}}\\,\\mathrm{d}y \\biggr]M( \\epsilon x)g(u), \\quad x\\in {\\mathbb{R}}^{N}, $$\\end{document} where alpha in (0,1), N> 2alpha , 0<mu <2alpha , ϵ is a positive parameter. Here (-Delta )^{alpha} stands for the fractional Laplacian, V is a linear potential with periodicity condition, and M is a nonlinear reaction potential with a global condition. We establish the existence and concentration of ground state solutions under general nonlinearity by using variational methods.

Existence and Concentration of Ground State Solutions for Chern–Simons–Schrödinger System with General Nonlinearity

Existence and Concentration of Ground State Solutions for Chern–Simons–Schrödinger System with General Nonlinearity

Existence and concentration of ground state solutions for a class of subcritical, critical or supercritical problems with steep potential well

Existence and concentration of ground state solutions for a class of subcritical, critical or supercritical problems with steep potential well

Existence and concentration of ground state solutions for an equation with steep potential well and exponential critical growth

In this paper we study the existence and concentration of ground state solution for the following class of elliptic equations{−div(a(|∇u|p)|∇u|p−2∇u)+(1+μV(x))b(|u|p)|u|p−2u=f(u)in RN,u∈W1,p(RN)∩W1,N(RN), where 1<p≤N, N≥2, μ>0 is a parameter, the functions a,b∈C1(R) and the nonlinearity f has an exponential critical growth. To show the existence and concentration results we obtain interesting estimates which do not depend on μ.

On existence and concentration of solutions for Hamiltonian systems involving fractional operator with critical exponential growth

AbstractThis paper is concerned with the existence and concentration of ground state solutions for the following class of fractional Schrödinger system where H has exponential critical growth, λ is a positive parameter and has a potential well with consisting of k disjoint components . The proof relies on variational methods and combines truncation arguments and the Moser iteration technique.

Existence and concentration of ground state solutions for a critical Kirchhoff type equation in $\mathbb R^2$

In this paper, we dedicate to the following Kirchhoff type equation with critical exponential growth \begin{equation*} -\varepsilon^2M \Big ( \int_{\mathbb R^2}|\nabla u|^2\mathrm{d} x \Big ) \Delta u+V(x)u=f(u),\ \ x\in\mathbb R^2, \end{equation*} where $\varepsilon>0$ is a parameter, $M\in\mathcal{C}(\mathbb R^+,\mathbb R^+), V\in\mathcal{C}(\mathbb R^2,\mathbb R)$ and $f\in\mathcal{C}(\mathbb R,\mathbb R)$. Under some suitable conditions, we obtain that there exists a ground state solution of the equation concentrating at a global minimum of $V$ in the semi-classical limit by using a homeomorphism between the ground states of Kirchhoff equation and a related semilinear local elliptic equation combined with variational methods.

Existence and concentration of ground state solutions for critical Kirchhoff-type equation involving Hartree-type nonlinearities

Existence and concentration of ground state solutions for critical Kirchhoff-type equation involving Hartree-type nonlinearities

The Existence and Concentration of Ground State Solutions for Chern-Simons-schrödinger Systems with a Steep Well Potential

The Existence and Concentration of Ground State Solutions for Chern-Simons-schrödinger Systems with a Steep Well Potential

Existence and concentration of ground state solutions for alogarithmic Schr&amp;ouml;dinger equation with the critical exponent

We study a class of logarithmic Schrödinger equations involving competing for weight potentials and critical nonlinearities on an unbounded domain. The corresponding energy functional has no continuous first derivative in its natural Sobolev space. Firstly, using the non-smooth critical point theory, we show that the 问题 admits one ground state positive 解. Then, we relate the concentration of solutions to a global minimum set of a suitable ground energy function via a Nehari manifold method. Finally, we give a sufficient condition forthe nonexistence of ground state solutions.

Existence and concentration of ground state solutions for Kirchhoff type equations with general nonlinearities

In this paper, we study the following singularly perturbed Kirchhoff type equation in subcritical case where ε &gt; 0 is a parameter, and . Under some suitable conditions, we obtain that there exists a ground state solution of the equation concentrating at a global minimum of V in the semiclassical limit by applying some new techniques and subtle analyses. Moreover, we use Moser iteration to conquer the difficulty caused by the lack of Sobolev inequality.

Infinitely many solutions and concentration of ground state solutions for the Klein-Gordon-Maxwell system

In this paper, we consider a model which describes solitary waves of the nonlinear Klein-Gordon equation interacting with the electromagnetic field, namely, the following Klein-Gordon-Maxwell system{−Δu+(m2−ω2)u−(2ω+ϕ)ϕu=f(u),inR3,−Δϕ=−(ω+ϕ)u2,inR3, where 0<ω<m are constants and the nonlinearity f satisfies the superlinear condition. Here m and ω denote the mass and the phase respectively, while u and ϕ are unknowns. By using variational methods and some technique related to Pohozǎev identity, we construct bounded Palais-Smale sequences to obtain the existence of infinitely many high energy radial solutions. Moreover, we show that the ground state solutions of the Klein-Gordon-Maxwell system tend to the ground state solutions of the classical Schrödinger equation as ω→0.

Existence and concentration of ground state solutions for critical Kirchhoff-type equation with steep potential well

In this paper, we study the nonlinear Kirchhoff type equation where a, b are positive constants and . Suppose that the nonnegative continuous potential V represents a potential well with the bottom and satisfies certain assumptions. We obtain the existence of ground state solutions by using the variational methods. Moreover, the concentration behavior of the ground state solutions as is also worth considering.

Existence and concentration of ground state solutions for a class of fractional Schrödinger equations

In this paper, by using variational methods, we study the existence and concentration of ground state solutions for the following fractional Schrödinger equation ( − Δ ) α u + V ( x ) u = A ( ϵ x ) f ( u ) , x ∈ R N , where α ∈ ( 0 , 1 ), ϵ is a positive parameter, N &gt; 2 α, ( − Δ ) α stands for the fractional Laplacian, f is a continuous function with subcritical growth, V ∈ C ( R N , R ) is a Z N -periodic function and A ∈ C ( R N , R ) satisfies some appropriate assumptions.

Existence and concentration of ground state solutions for a class of nonlocal problem in [formula omitted

In this paper, we study the following nonlocal problem in RN−a−ϵ∫RN|Δu|2dxΔu+(1+λf(x))u=|u|p−2u,where a>0, 2<p<2∗=2N∕(N−2) with N≥3, ϵ>0 is small enough, and the parameter λ>0. Under some assumptions on f(x), we prove the existence of ground state solutions for the problem when λ is large enough via variational methods. In addition, we obtain some concentration behaviors of these solutions as λ→+∞.

Existence and concentration of ground state solutions for doubly critical Schrödinger–Poisson-type systems

In this paper, we are concerned with the existence and concentration of ground state solutions for the following nonlinear Schrodinger–Poisson-type system with doubly critical growth $$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta u+V(x)u-\phi |u|^3u=|u|^4u+f(u),&{}\mathrm{in}\; \mathbb {R}^3,\\ -\varepsilon ^2\Delta \phi =|u|^5,&{}\mathrm{in}\; \mathbb {R}^3, \end{array}\right. } \end{aligned}$$ where $$\varepsilon >0$$ is a small parameter. By employing the concentration-compactness principle and mountain pass theorem, we prove the existence of positive ground state solutions $$v_\varepsilon $$ with exponential decay at infinity for $$\varepsilon $$ sufficiently small under some suitable assumptions on the potential V and nonlinearity f. Moreover, as $$\varepsilon \rightarrow 0^+$$ , $$v_\varepsilon $$ concentrates around a global minimum point of V.

Existence and concentration of ground state solutions for Choquard equations involving critical growth and steep potential well

In the present paper, we are interested in the following Choquard type equation −Δu+(λV(x)−μ)u=(Iα∗|u|2α∗)|u|2α∗−2u+|u|p−2uinR3,where p∈(4,6), λ∈R+, μ∈R is a constant such that the operator Lλ≔−Δ+λV(x)−μ is non-degenerate, Iα is a Riesz potential of order α∈(0,3), 2α∗=3+α is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality, the function V∈C(R3,R) is nonnegative and has a potential well Ω≔intV−1(0), additionally, the operator −Δ has a sequence of Dirichlet eigenvalues in H01(Ω) expressed as 0<μ1<μ2<⋯<μn⟶n+∞. If μ<μ1, via the Nehari manifold techniques and the Ekeland variational principle, we prove the existence and concentration of positive ground state solutions for sufficiently large λ. If μ>μ1 and μ≠μj for all j∈N+, employing the Nehari–Pankov manifold methods and the constrained minimization arguments, we obtain the existence of ground state solution for λ large enough and verify the asymptotic behaviour of ground state solutions as λ→+∞.

Existence and concentration of ground state solutions for critical Schrödinger–Poisson system with steep potential well

In this paper, we consider the following Schrödinger–Poisson system{−Δu+(1+μg(x))u+ϕu=|u|4u+λ|u|q−2u,inR3,−Δϕ=u2,inR3,where q ∈ (3, 6) and λ, μ > 0 are positive parameters. Since f(u):=|u|4u+λ|u|q−2u with q ∈ (3, 4] does not satisfy the (AR) condition. Thus, we construct Nehari-Pohožaev-Palais-Smale sequence to overcome the boundedness of sequence. As q ∈ (4, 6), the boundedness of sequence is easily obtained. We need (g1) and (g2) to prove that cμ<13S32 independent of μ. Furthermore, we utilize the definition of the set of solutions to seek a ground state solution. Besides, the concentration behavior of the ground state solution is also described as μ → ∞.

Existence and concentration of ground state solutions for a class of Kirchhoff-type problems

This paper is concerned with the following singularly perturbed Kirchhoff-type problem −ε2a+εb∫R3|∇u|2dx△u+V(x)u=f(u),x∈R3;u∈H1(R3),where ε>0 is a small parameter, a,b>0 are two constants, V∈C(R3,R), and f∈C(R,R) is of super-linear growth at infinity and satisfies neither the usual Ambrosetti–Rabinowitz type condition nor monotonicity condition on f(u)∕u3. By using some new techniques and subtle analyses, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0], the above problem has a ground state solution concentrating around global minimum of V in the semi-classical limit. Our results are available to the case that f(u)∼|u|s−2u for s∈(2,6), and extend the existing results concerning the case that f(u)∼|u|s−2u for s∈[4,6).

Existence and concentration of ground state solutions for critical Schrödinger equation with steep potential well

In this paper, we study the following Schrödinger equation −Δu+(1+μa(x))u=f(u)+|u|2∗−2u,inRN,where N≥3 and μ>0. We obtain a ground state solution for the above equation. Moreover, the concentration behavior of the ground state solution is also described as μ→∞.