Positive Ground State Solution Research Articles

Positive ground states for coupled nonlinear Choquard equations involving Hardy–Littlewood–Sobolev critical exponent

In this paper, we consider the following coupled nonlinear Schrödinger equations with Choquard type nonlinearities: −Δu+λ1u=μ1(1|x|μ∗u2μ∗)u2μ∗−1+β(1|x|μ∗v2μ∗)u2μ∗−1,x∈Ω,−Δv+λ2v=μ2(1|x|μ∗v2μ∗)v2μ∗−1+β(1|x|μ∗u2μ∗)v2μ∗−1,x∈Ω,u,v≥0inΩ,u=v=0on∂Ω,where Ω⊂RN is a smooth bounded domain, 2μ∗≔2N−μN−2 is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, −λ1(Ω)<λ1,λ2<0,λ1(Ω) is the first eigenvalue of (−Δ,H01(Ω)), μ1,μ2>0 and β≠0 is a coupling constant. We show that the critical nonlocal system has a positive ground state solution for negative β and positive large β via variational methods. Moreover, we study the limit behavior of the positive ground state solution (uβ,vβ) as β→−∞ and some different phenomenon arises comparing with the local Schrödinger system.

Multiplicity of positive solutions for Kirchhoff type problems with nonlinear boundary condition

In this paper, we study the existence of multiple positive solutions to problem \[\left \{\begin{aligned} &\bigg (a+b \int _\Omega (|\nabla u|^2+|u|^2)\,dx\bigg )(-\Delta u+u)=|u|^{4}u &&\mbox {in } \Omega, \\ &\frac {\partial u}{\partial \nu }=\lambda |u|^{q-2}u &&\mbox {on } \partial \Omega,\end{aligned} \right . \] where $\Omega \subset \mathbb {R}^{3}$ is a smooth bounded domain, $a, b \gt 0$, $\lambda \gt 0$ and $1\lt q\lt 2$. Based on the Nehari manifold and variational methods, we prove that the problem has at least two positive solutions, and one of the solutions is a positive ground state solution.

Groundstates for Kirchhoff-Type Equations with Hartree-Type Nonlinearities

In this paper, we consider the following nonlinear problem of Kirchhoff-type with Hartree-type nonlinearities: $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\mathbb {R}^N}|Du|^2\right) \Delta u+V(x)u=(I_{\alpha }*|u|^{p})|u|^{p-2}u,&{}\quad x\in \mathbb {R}^N,\\ \\ u\in H^1(\mathbb {R}^N),\quad u>0,&{}\quad x\in \mathbb {R}^N, \end{array}\right. \end{aligned}$$ where $$N\ge 3$$ , $$\max \{0,N-4\}<\alpha <N$$ , $$2<p<\frac{N+\alpha }{N-2}$$ , $$a>0,b\ge 0$$ are constants, $$I_{\alpha }$$ is the Riesz potential and $$V{:}\,\mathbb {R}^N\rightarrow \mathbb {R}$$ is a potential function. Under certain assumptions on V, we prove that the problem has a positive ground state solution by using global compactness lemma, monotonicity technique and some new tricks recently given in the literature.

Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent

In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent \begin{document}$ \begin{cases} (-\Delta)^{s}u+\mu u = |u|^{p-1}u+\lambda v,&amp; x\in\mathbb{R}^{N},\\ (-\Delta)^{s}v+\nu v = |v|^{2^{\ast}-2}v+\lambda u,&amp; x\in\mathbb{R}^{N},\\ \end{cases} $\end{document} where $ (-\Delta)^{s} $ is the fractional Laplacian, $ 0&lt;s&lt;1,\ N&gt;2s, \ \lambda &lt;\sqrt{\mu\nu },\ 1&lt;p&lt;2^{\ast}-1\; \text{and}\; \ 2^{\ast} = \frac{2N}{N-2s} $ is the Sobolev critical exponent. By using the Nehari\ manifold, we show that there exists a $ \mu_{0}\in(0,1) $, such that when $ 0&lt;\mu\leq\mu_{0} $, the system has a positive ground state solution. When $ \mu&gt;\mu_{0} $, there exists a $ \lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\sqrt{\mu\nu}) $ such that if $ \lambda&gt;\lambda_{\mu,\nu} $, the system has a positive ground state solution, if $ \lambda&lt;\lambda_{\mu,\nu} $, the system has no ground state solution.

Existence of positive ground state solutions for Choquard equation with variable exponent growth

In this paper, we investigate the following Choquard equation \begin{document}$ \begin{equation*} -\Delta u = (I_\alpha*|u|^{f(x)})|u|^{f(x)-2}u {\rm in} \mathbb{R}^N, \end{equation*} $\end{document} where \begin{document}$ N\geq 3 $\end{document} , \begin{document}$ \alpha\in (0,N) $\end{document} and \begin{document}$ I_\alpha $\end{document} is the Riesz potential. If \begin{document}$ \begin{equation*} f(x) = \begin{cases} p, x'>where \begin{document}$ 1 and \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} is a bounded set with nonempty, we obtain the existence of positive ground state solutions by using the Nehari manifold.

Existence of Positive Ground State Solution for the Nonlinear Schrödinger-Poisson System with Potentials

In the present paper we study the existence of positive ground state solutions for the nonautonomous Schrödinger-Poisson system with competing potentials. Under some assumptions for the potentials we prove the existence of positive ground state solution.

On positive ground state solutions for the nonlinear Kirchhoff type equations in ℝ3

In this article, we first give a much more concise proof on the existence of ground state solutions of Kirchhoff equation by rearrangement than that given in Li and Ye [Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3. J. Differ. Equat. 2014;257(2):566-600] and Ye [Positive high energy solution for Kirchhoff equation in R3 with superlinear nonlinearities via Nehari–Pohozaev manifold. Discrete Contin. Dyn. Syst. 2015;35(8):3857-3877], where a,b>0,2<p<5. And then, we prove a nonexistence result on ground state solutions of the non-autonomous Kirchhoff equation which can be considered as a nice complement to Zhang et al. [Semiclassical ground states for a class of nonlinear Kirchhoff-type problems. Appl. Anal. 2016;96(13):2267-2284].

Asymptotic behavior of ground state solutions for nonlinear Schrödinger systems

Consider the nonlinear Schrödinger system −Δu1+V1(x)u1=μ1(x)u13+β(x)u1u22inRN,−Δu2+V2(x)u2=β(x)u12u2+μ2(x)u23inRN,uj∈H1(RN),j=1,2,where N=1,2,3, and the potentials Vj,μj,β are periodic or Vj are well-shaped and μj,β are anti-well-shaped. When the coupling coefficient β is either small or large in terms of Vj and μj, existence of a positive ground state solution was proved in Liu and Liu (2015). In this paper, we describe the asymptotic behavior of ground state solutions when |β|L∞(RN) tends to zero or minRNβ tends to +∞.

Positive ground state solutions for fractional Kirchhoff type equations with critical growth

We study the existence of positive ground state solutions for the following fractional Kirchhoff type equation urn:x-wiley:mma:media:mma5411:mma5411-math-0001 where a,b &gt; 0 are constants, μ is a positive parameter, with and s ∈ (0,1). Under suitable assumptions on V(x), by using a monotonicity trick and a global compactness principle, we prove that the equation admits a positive ground state solution if and μ &gt; 0 large enough.

Nonlocal scalar field equations: Qualitative properties, asymptotic profiles and local uniqueness of solutions

We study the nonlocal scalar field equation with a vanishing parameter:(Pϵ){(−Δ)su+ϵu=|u|p−2u−|u|q−2uinRNu∈Hs(RN), where s∈(0,1), N>2s, q>p>2 are fixed parameters and ϵ>0 is a vanishing parameter. For ϵ small, we prove the existence and qualitative properties of positive solutions. Next, we study the asymptotic behavior of ground state solutions when p is subcritical, supercritical or critical Sobolev exponent 2⁎=2NN−2s. For p<2⁎, the ground state solution asymptotically coincides with unique positive ground state solution of (−Δ)su+u=up, whereas for p=2⁎ the asymptotic behavior of the solutions is given by the unique positive solution of the nonlocal critical Emden–Fowler type equation. For p>2⁎, the solution asymptotically coincides with a ground-state solution of (−Δ)su=up−uq. Furthermore, using these asymptotic profile of positive solutions, we establish the local uniqueness of positive solution.

Existence of positive solutions for a Schrödinger-Poisson system with critical growth

ABSTRACT In this paper, we are concerned with the existence, multiplicity and concentration of positive solutions for the semilinear Schrödinger-Poisson system where (small) and λ are real parameters, a,b and c are continuous functions, and f is a continuous superlinear and subcritical nonlinearity. First, we prove that there are two families of semiclassical positive solutions for small. Then we show that these solutions are concentrating on some sets which is generated by the minimal points of a and maximal points of b and c. In addition, we investigate the relation between the number of positive solutions and the topology of the set of the global minima(or maxima) of the potentials by minimax theorems and the Ljusternik-Schnirelmann theory. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger‐Poisson system with critical growth

In this paper, we study the following fractional Schrödinger‐Poisson system involving competing potential functions urn:x-wiley:mma:media:mma5289:mma5289-math-0001 where ϵ &gt; 0 is a small parameter, f is a function of C1 class, superlinear and subcritical nonlinearity, , , t ∈ (0,1), V(x), K(x), and Q(x) are positive continuous functions. Under some suitable assumptions on V, K, and Q, we prove that there is a family of positive ground state solutions with polynomial growth for sufficiently small ϵ &gt; 0, of which it is concentrating on the set of minimal points of V(x) and the sets of maximal points of K(x) and Q(x).

Existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional Kirchhoff-type systems

This paper is concerned with the following linearly coupled fractional Kirchhoff-type system a+b∫R3|(−△)α2u|2dx(−△)αu+λu=f(u)+γv,inR3,c+d∫R3|(−△)α2v|2dx(−△)αv+μv=g(v)+γu,inR3,u,v∈Hα(R3),where a,c,λ,μ>0, b,d≥0 are constants, α∈[34,1) and γ>0 is a coupling parameter. Under the general Berestycki–Lions conditions on the nonlinear terms f and g, we prove the existence of positive vector ground state solutions of Pohožaev type for the above system via variational methods. Moreover, the asymptotic behavior of these solutions as γ→0+ is explored as well. Recent results from the literature are generally improved and extended.

Existence of a ground state solution for Choquard equation with the upper critical exponent

In this paper, we investigate the following Choquard equation −Δu+u=(Iα∗|u|N+αN−2)|u|N+αN−2−2u+g(u)inRN,where N⩾3,α∈(0,N), Iα is the Riesz potential and g is a given function. If g satisfies the general subcritical growth conditions, we obtain the existence of a positive ground state solution.

Ground State Solutions for Quasilinear Schrödinger Equations with Critical Growth and Lower Power Subcritical Perturbation

Abstract We study the following generalized quasilinear Schrödinger equation: - ( g 2 ⁢ ( u ) ⁢ ∇ ⁡ u ) + g ⁢ ( u ) ⁢ g ′ ⁢ ( u ) ⁢ | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ u = h ⁢ ( u ) , x ∈ ℝ N , -(g^{2}(u)\nabla u)+g(u)g^{\prime}(u)|\nabla u|^{2}+V(x)u=h(u),\quad x\in% \mathbb{R}^{N}, where N ≥ 3 {N\geq 3} , g : ℝ → ℝ + {g\colon\mathbb{R}\rightarrow\mathbb{R}^{+}} is an even differentiable function such that g ′ ⁢ ( t ) ≥ 0 {g^{\prime}(t)\geq 0} for all t ≥ 0 {t\geq 0} , h ∈ C 1 ⁢ ( ℝ , ℝ ) {h\in C^{1}(\mathbb{R},\mathbb{R})} is a nonlinear function including critical growth and lower power subcritical perturbation, and the potential V ⁢ ( x ) : ℝ N → ℝ {V(x)\colon\mathbb{R}^{N}\rightarrow\mathbb{R}} is positive. Since the subcritical perturbation does not satisfy the (AR) condition, the standard variational method cannot be used directly. Combining the change of variables and the monotone method developed by Jeanjean in [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on 𝐑 N {\mathbf{R}}^{N} , Proc. Roy. Soc. Edinburgh Sect. A 129 1999, 4, 787–809], we obtain the existence of positive ground state solutions for the given problem.

On existence and concentration behavior of positive ground state solutions for a class of fractional Schrödinger–Choquard equations

In this paper, we first employ variational methods to show the existence of positive ground state solutions for fractional Schrodinger–Choquard equations $$\begin{aligned} \varepsilon ^{2s}(-\Delta )^{s}u+Vu=(I_{\alpha }*|u|^p)|u|^{p-2}u,~~~\mathrm {in}~~~\mathbb {R}^{N}, \end{aligned}$$ where potential $$V(x)\in C(\mathbb {R}^{N})$$ is nonnegative and bounded away from 0 as $$|x|\rightarrow \infty $$ , $$I_{\alpha }$$ is the Riesz potential of order $$\alpha \in (0,N)$$ and $$\varepsilon >0$$ is a parameter small enough. When $$V(x)\in C(\mathbb {R}^{N})$$ achieves 0 with a homogeneous behavior, we then investigate the concentration behavior of positive ground state solutions as $$\varepsilon \rightarrow 0^{+}$$ .

A perturbation of nonlinear scalar field equations

In the paper, we consider the following Schrödinger equations −△u+V(x)u=g(u),x∈RN,N≥3,where g satisfies Berestycki–Lions conditions and V is a small perturbation. The assumptions on V are different from the ones in Azzollini and Pomponio (2009). By using the variational methods, we show the equation has a positive ground state solution.

Existence and qualitative properties of solutions for Choquard equations with a local term

In this paper, a nontrivial solution u∈H1(RN) to the autonomous Choquard equation with a local term −Δu+λu=(Iα∗|u|p)|u|p−2u+|u|q−2uinRNis obtained, where N≥3, α∈(0,N), λ>0 is a constant, Iα is the Riesz potential, N+αN<p<N+αN−2 and q∈(2,2∗=2NN−2). Under some further assumptions on p and q, the regularity and the Pohožaev identity of the solution are established, and then it is shown that the obtained solution is a groundstate of mountain pass type. Moreover, the positivity and symmetry of the groundstate are also considered. By using the results obtained for the autonomous equation, a positive groundstate solution for the nonautonomous equation −Δu+V(x)u=(Iα∗|u|p)|u|p−2u+|u|q−2uinRNis also found under some assumptions on V(x).

Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent

In this paper, we study the following critical system with fractional Laplacian: \t\t\t{(−Δ)su+λ1u=μ1|u|2∗−2u+αγ2∗|u|α−2u|v|βin Ω,(−Δ)sv+λ2v=μ2|v|2∗−2v+βγ2∗|u|α|v|β−2vin Ω,u=v=0in RN∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} (-\\Delta)^{s}u+\\lambda_{1}u=\\mu_{1}|u|^{2^{\\ast}-2}u+\\frac{\\alpha \\gamma}{2^{\\ast}}|u|^{\\alpha-2}u|v|^{\\beta} & \\text{in } \\Omega, \\\\ (-\\Delta)^{s}v+\\lambda_{2}v= \\mu_{2}|v|^{2^{\\ast}-2}v+\\frac{\\beta \\gamma}{2^{\\ast}}|u|^{\\alpha}|v|^{\\beta-2}v & \\text{in } \\Omega, \\\\ u=v=0 & \\text{in } \\mathbb{R}^{N}\\setminus\\Omega, \\end{cases} $$\\end{document} where (-Delta)^{s} is the fractional Laplacian, 0< s<1, mu_{1},mu_{2}>0, 2^{ast}=frac{2N}{N-2s} is a fractional critical Sobolev exponent, N>2s, 1<alpha, beta<2, alpha+beta=2^{ast}, Ω is an open bounded set of mathbb{R}^{N} with Lipschitz boundary and lambda_{1},lambda_{2}>-lambda_{1,s}(Omega), lambda_{1,s}(Omega) is the first eigenvalue of the non-local operator (-Delta)^{s} with homogeneous Dirichlet boundary datum. By using the Nehari manifold, we prove the existence of a positive ground state solution of the system for all gamma>0. Via a perturbation argument and using the topological degree and a pseudo-gradient vector field, we show that this system has a positive higher energy solution. Then the asymptotic behaviors of the positive ground state solutions are analyzed when gammarightarrow0.

A result on a non-autonomous Kirchhoff type equation involving critical term

By using the method of Nehari manifold, we obtain the existence of a positive ground state solution for the following non-autonomous Kirchhoff type equation −(a+b∫RN|∇u|2dx)△u+V(x)u=uN+2N−2,x∈RN,u∈D1,2(RN),N=3,4,where a>0,b>0 and V satisfies some appropriate assumptions.