This article studies a class of nonlinear Kirchhoff equations with exponential critical growth,trapping potential,and perturbation.Under appropriate assumptions about f and h,the article obtained the existence of norm...This article studies a class of nonlinear Kirchhoff equations with exponential critical growth,trapping potential,and perturbation.Under appropriate assumptions about f and h,the article obtained the existence of normalized positive solutions for this equation via the Trudinger-Moser inequality and variational methods.Moreover,these solutions are also ground state solutions.Additionally,the article also characterized the asymptotic behavior of solutions.The results of this article expand the research of relevant literature.展开更多
This paper is concerned with the positive ground state solutions for a quasilinear Schrodinger equation with a Hardy-type term.We obtain positive ground state solutions for the given quasilinear Schrodinger equation b...This paper is concerned with the positive ground state solutions for a quasilinear Schrodinger equation with a Hardy-type term.We obtain positive ground state solutions for the given quasilinear Schrodinger equation by using a change of variables and variational method.展开更多
In this paper,we investigate a class of nonlinear Chern-Simons-Schr?dinger systems with a steep well potential.By using variational methods,the mountain pass theorem and Nehari manifold methods,we prove the existence ...In this paper,we investigate a class of nonlinear Chern-Simons-Schr?dinger systems with a steep well potential.By using variational methods,the mountain pass theorem and Nehari manifold methods,we prove the existence of a ground state solution forλ>0 large enough.Furthermore,we verify the asymptotic behavior of ground state solutions asλ→+∞.展开更多
This article is concerned with the nonlinear Dirac equations-iδtψ=ich ∑k=1^3 αkδkψ-mc^2βψ＋Rψ（x,ψ） in R^3.Under suitable assumptions on the nonlinearity, we establish the existence of ground state solution...This article is concerned with the nonlinear Dirac equations-iδtψ=ich ∑k=1^3 αkδkψ-mc^2βψ＋Rψ（x,ψ） in R^3.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.展开更多
We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical ...We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical Sobolev exponent in 1R3.Under some more general assumptions on f,we prove that(0.1)admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.展开更多
We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|^p-1u,where the value of |x|→∞ lim Q(x)may not exist and Q may change sign.This means that the problem may have no limit problem.The existence of...We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|^p-1u,where the value of |x|→∞ lim Q(x)may not exist and Q may change sign.This means that the problem may have no limit problem.The existence of nonnegative ground state solutions is established.Our method relies upon the variational method and some analysis tricks.展开更多
In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a ste...In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a steep potential well and the nonlinearity f∈C(R,R)satisfies certain assumptions.By applying a signchanging Nehari manifold combined with the method of constructing a sign-changing(PS)C sequence,we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when λ is large enough,and find that its energy is strictly larger than twice that of the ground state solutions.In addition,we also prove the concentration of ground state sign-changing solutions.展开更多
In this article,we study the following Klein-Gordon-Maxwell system involving critical exponent■where λ and w are two positive constants.We found the existence of positive ground state solutions(that is,for solutions...In this article,we study the following Klein-Gordon-Maxwell system involving critical exponent■where λ and w are two positive constants.We found the existence of positive ground state solutions(that is,for solutions which minimizes the action functional among all the solutions)of(KGM) which improves some previous existence result in Carriao et al.(2012) [8].展开更多
We study the Choquard equation-Δu+V(x)u-b(x)∫R3|u(y)|2/|x-y|dyu,x∈R3,where V(x)=V1(x),b(x)=b1(x)for x1>0 and V(x)=V2(x),b(x)=b2(x)for x1<0,and V1,V2,b1and b2are periodic in each coordinate direction.Under som...We study the Choquard equation-Δu+V(x)u-b(x)∫R3|u(y)|2/|x-y|dyu,x∈R3,where V(x)=V1(x),b(x)=b1(x)for x1>0 and V(x)=V2(x),b(x)=b2(x)for x1<0,and V1,V2,b1and b2are periodic in each coordinate direction.Under some suitable assumptions,we prove the existence of a ground state solution of the equation.Additionally,we find some sufficient conditions to guarantee the existence and nonexistence of a ground state solution of the equation.展开更多
In this article,we study the generalized quasilinear Schrodinger equation-div(ε^2g^2(u)▽u)+ε^2g(u)g′(u)|▽u|^2+V(x)u=K(x)|u|^p-2u,x∈R^N where A≥3,e>0,4<p<,22*,g∈C 1(R,R+),V∈C(R^N)∩L∞(R^N)has a posit...In this article,we study the generalized quasilinear Schrodinger equation-div(ε^2g^2(u)▽u)+ε^2g(u)g′(u)|▽u|^2+V(x)u=K(x)|u|^p-2u,x∈R^N where A≥3,e>0,4<p<,22*,g∈C 1(R,R+),V∈C(R^N)∩L∞(R^N)has a positive global minimum,and K∈C(R^N)∩L∞(R^N)has a positive global maximum.By using a change of variable,we obtain the existence and concentration behavior of ground state solutions for this problem and establish a phenomenon of exponential decay.展开更多
In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,w...In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in[L.Jeanjean,Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlinear Anal.(1997)].展开更多
In this paper,we consider the Chern-Simons-Schrodinger system{−Δu+[e^(2)|A|^(2)+(V(x)+2eA_(0))+2(1+κq/2)N]u+q|u|^(p−2)u=0,−ΔN+κ^(2)q^(2)N+q(1+κq2)u^(2)=0,κ(∂_(1)A_(2)−∂_(2)A_(1))=−eu^(2),∂_(1)A_(1)+∂_(2)A_(2)=0,...In this paper,we consider the Chern-Simons-Schrodinger system{−Δu+[e^(2)|A|^(2)+(V(x)+2eA_(0))+2(1+κq/2)N]u+q|u|^(p−2)u=0,−ΔN+κ^(2)q^(2)N+q(1+κq2)u^(2)=0,κ(∂_(1)A_(2)−∂_(2)A_(1))=−eu^(2),∂_(1)A_(1)+∂_(2)A_(2)=0,κ∂_(1)A_(0)=e^(2)A_(2)u^(2),κ∂_(2)A_(0)=−e^(2)A_(1)u^(2),where u∈H^(1)(R^(2)),p∈(2,4),Aα:R^(2)→R are the components of the gauge potential(α=0,1,2),N:R^(2)→R is a neutral scalar field,V(x)is a potential function,the parametersκ,q>0 represent the Chern-Simons coupling constant and the Maxwell coupling constant,respectively,and e>0 is the coupling constant.In this paper,the truncation function is used to deal with a neutral scalar field and a gauge field in the Chern-Simons-Schrödinger problem.The ground state solution of the problem(P)is obtained by using the variational method.展开更多
This paper deals with a class of Schr¨odinger-Poisson systems. Under some conditions, we prove that there exists a ground state solution of the system. The proof is based on the compactness lemma for the system. ...This paper deals with a class of Schr¨odinger-Poisson systems. Under some conditions, we prove that there exists a ground state solution of the system. The proof is based on the compactness lemma for the system. Our results here improve some existing results in the literature.展开更多
We present a scheme for the electromagnetically-induced-absorption(EIA)-like ground state cooling in a hybrid optomechanical system which is combined by two-level quantum systems(qubits)and a high-Q optomechanical cav...We present a scheme for the electromagnetically-induced-absorption(EIA)-like ground state cooling in a hybrid optomechanical system which is combined by two-level quantum systems(qubits)and a high-Q optomechanical cavity.Under the weak qubit-cavity coupling,the system exhibits an EIA-like effect and this effect is caused by quantum destructive interference that is distinct from the conventional EIA effect driven by quantum constructive interference.More importantly,the EIA-like cooling mechanism can significantly enhance the cooling rate of the hybrid system,enabling the final phonon number beyond the classical cooling limit in the strong optomechanical coupling regime.Meanwhile,the cooling effects of the EIA case is better than that of the normalmode splitting case under the same optomechanical coupling strength and qubit dissipation rate.展开更多
In this paper,we mainly focus on the following Choquard equation-{△u-V(x)(I_(a*)|u|^(p))|u|^(p-2)u=λu,x∈R^(N),u∈H^(1)(R^(N))where N≥1,λ∈R will arise as a Lagrange multiplier,0<a<N and N+a/N<p<N+a+2/...In this paper,we mainly focus on the following Choquard equation-{△u-V(x)(I_(a*)|u|^(p))|u|^(p-2)u=λu,x∈R^(N),u∈H^(1)(R^(N))where N≥1,λ∈R will arise as a Lagrange multiplier,0<a<N and N+a/N<p<N+a+2/N Under appropriate hypotheses on V(x),we prove that the above Choquard equation has a normalized ground state solution by utilizing variational methods.展开更多
Machine learning has rapidly become a powerful tool for addressing challenges in ultracold atomic systems;however,its application to intricate three-dimensional(3D)systems remains relatively underexplored.In this stud...Machine learning has rapidly become a powerful tool for addressing challenges in ultracold atomic systems;however,its application to intricate three-dimensional(3D)systems remains relatively underexplored.In this study,we introduce a3D residual network(3D Res Net)framework based on 3D convolutional neural networks(3D CNN)to predict ground states phases in 3D dipolar spinor Bose–Einstein condensates(BECs).Our results show that the 3D Res Net framework predicts ground states with high accuracy and efficiency across a broad parameter space.To enhance phase transition predictions,we incorporate data augmentation techniques,leading to a notable improvement in the model's performance.The method is further validated in more complex scenarios,particularly when transverse magnetic fields are introduced.Compared to conventional imaginary-time evolution methods(ITEM),the 3D Res Net drastically reduces computational costs,offering a rapid and scalable solution for complex 3D multi-parameter nonlinear systems.展开更多
The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potenti...The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potentials.As an example, the method is applied to discuss the ground state solution of Yukawa potential, using Hulthen solution as the trial function.展开更多
This paper mainly discusses the following equation: where the potential function V : R<sup>3</sup> → R, α ∈ (0,3), λ > 0 is a parameter and I<sub>α</sub> is the Riesz potential. We stud...This paper mainly discusses the following equation: where the potential function V : R<sup>3</sup> → R, α ∈ (0,3), λ > 0 is a parameter and I<sub>α</sub> is the Riesz potential. We study a class of Schrödinger-Poisson system with convolution term for upper critical exponent. By using some new tricks and Nehair-Pohožave manifold which is presented to overcome the difficulties due to the presence of upper critical exponential convolution term, we prove that the above problem admits a ground state solution.展开更多
In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger...In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.展开更多
In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infin...In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.展开更多
基金Supported by National Natural Science Foundation of China(11671403,11671236)Henan Provincial General Natural Science Foundation Project(232300420113)。
文摘This article studies a class of nonlinear Kirchhoff equations with exponential critical growth,trapping potential,and perturbation.Under appropriate assumptions about f and h,the article obtained the existence of normalized positive solutions for this equation via the Trudinger-Moser inequality and variational methods.Moreover,these solutions are also ground state solutions.Additionally,the article also characterized the asymptotic behavior of solutions.The results of this article expand the research of relevant literature.
基金Supported by Research Start-up Fund of Jianghan University(06050001).
文摘This paper is concerned with the positive ground state solutions for a quasilinear Schrodinger equation with a Hardy-type term.We obtain positive ground state solutions for the given quasilinear Schrodinger equation by using a change of variables and variational method.
基金supported by National Natural Science Foundation of China(11971393)。
文摘In this paper,we investigate a class of nonlinear Chern-Simons-Schr?dinger systems with a steep well potential.By using variational methods,the mountain pass theorem and Nehari manifold methods,we prove the existence of a ground state solution forλ>0 large enough.Furthermore,we verify the asymptotic behavior of ground state solutions asλ→+∞.
基金supported by the Hunan Provincial Innovation Foundation for Postgraduate(CX2013A003)the NNSF(11171351,11361078)SRFDP(20120162110021)of China
文摘This article is concerned with the nonlinear Dirac equations-iδtψ=ich ∑k=1^3 αkδkψ-mc^2βψ＋Rψ（x,ψ） in R^3.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.
基金the Science and Technology Project of Education Department in Jiangxi Province(GJJ180357)the second author was supported by NSFC(11701178).
文摘We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical Sobolev exponent in 1R3.Under some more general assumptions on f,we prove that(0.1)admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.
基金National Natural Science Foundation of China(11471267)the first author was supported by Graduate Student Scientific Research Innovation Projects of Chongqing(CYS17084).
文摘We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|^p-1u,where the value of |x|→∞ lim Q(x)may not exist and Q may change sign.This means that the problem may have no limit problem.The existence of nonnegative ground state solutions is established.Our method relies upon the variational method and some analysis tricks.
基金the National Natural Science Foundation of China (11971393)。
文摘In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a steep potential well and the nonlinearity f∈C(R,R)satisfies certain assumptions.By applying a signchanging Nehari manifold combined with the method of constructing a sign-changing(PS)C sequence,we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when λ is large enough,and find that its energy is strictly larger than twice that of the ground state solutions.In addition,we also prove the concentration of ground state sign-changing solutions.
基金partially supported by the National Natural Science Foundation of China(11801400,11571187)partially supported by the National Natural Science Foundation of China(11861053)the Postdoctoral Science Foundation of China(2017M611159)
文摘In this article,we study the following Klein-Gordon-Maxwell system involving critical exponent■where λ and w are two positive constants.We found the existence of positive ground state solutions(that is,for solutions which minimizes the action functional among all the solutions)of(KGM) which improves some previous existence result in Carriao et al.(2012) [8].
基金supported by National Natural Science Foundation of China(11971202)Outstanding Young foundation of Jiangsu Province(BK20200042)。
文摘We study the Choquard equation-Δu+V(x)u-b(x)∫R3|u(y)|2/|x-y|dyu,x∈R3,where V(x)=V1(x),b(x)=b1(x)for x1>0 and V(x)=V2(x),b(x)=b2(x)for x1<0,and V1,V2,b1and b2are periodic in each coordinate direction.Under some suitable assumptions,we prove the existence of a ground state solution of the equation.Additionally,we find some sufficient conditions to guarantee the existence and nonexistence of a ground state solution of the equation.
基金supported by the National Natural Science Foundation of China(11661053,11771198,11901345,11901276,11961045 and 11971485)partly by the Provincial Natural Science Foundation of Jiangxi,China(20161BAB201009 and 20181BAB201003)+1 种基金the Outstanding Youth Scientist Foundation Plan of Jiangxi(20171BCB23004)the Yunnan Local Colleges Applied Basic Research Projects(2017FH001-011).
文摘In this article,we study the generalized quasilinear Schrodinger equation-div(ε^2g^2(u)▽u)+ε^2g(u)g′(u)|▽u|^2+V(x)u=K(x)|u|^p-2u,x∈R^N where A≥3,e>0,4<p<,22*,g∈C 1(R,R+),V∈C(R^N)∩L∞(R^N)has a positive global minimum,and K∈C(R^N)∩L∞(R^N)has a positive global maximum.By using a change of variable,we obtain the existence and concentration behavior of ground state solutions for this problem and establish a phenomenon of exponential decay.
基金Supported by the National Natural Science Foundation of China (11971393).
文摘In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in[L.Jeanjean,Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlinear Anal.(1997)].
基金partially supported by NSFC (12161044)Natural Science Foundation of Jiangxi Province (20212BAB211013)+1 种基金Benniao Li was partially supported by NSFC (12101274)Doctoral Research Startup Foundation of Jiangxi Normal University (12020927)
文摘In this paper,we consider the Chern-Simons-Schrodinger system{−Δu+[e^(2)|A|^(2)+(V(x)+2eA_(0))+2(1+κq/2)N]u+q|u|^(p−2)u=0,−ΔN+κ^(2)q^(2)N+q(1+κq2)u^(2)=0,κ(∂_(1)A_(2)−∂_(2)A_(1))=−eu^(2),∂_(1)A_(1)+∂_(2)A_(2)=0,κ∂_(1)A_(0)=e^(2)A_(2)u^(2),κ∂_(2)A_(0)=−e^(2)A_(1)u^(2),where u∈H^(1)(R^(2)),p∈(2,4),Aα:R^(2)→R are the components of the gauge potential(α=0,1,2),N:R^(2)→R is a neutral scalar field,V(x)is a potential function,the parametersκ,q>0 represent the Chern-Simons coupling constant and the Maxwell coupling constant,respectively,and e>0 is the coupling constant.In this paper,the truncation function is used to deal with a neutral scalar field and a gauge field in the Chern-Simons-Schrödinger problem.The ground state solution of the problem(P)is obtained by using the variational method.
文摘This paper deals with a class of Schr¨odinger-Poisson systems. Under some conditions, we prove that there exists a ground state solution of the system. The proof is based on the compactness lemma for the system. Our results here improve some existing results in the literature.
基金Project supported by the National Natural Science Foundation of China(Grant No.11832016)the National Key Research and Development Program of China(Grant No.2021YFB4000802)the Steady Support Fund for the State Key Laboratory(Grant No.JBS242800180).
文摘We present a scheme for the electromagnetically-induced-absorption(EIA)-like ground state cooling in a hybrid optomechanical system which is combined by two-level quantum systems(qubits)and a high-Q optomechanical cavity.Under the weak qubit-cavity coupling,the system exhibits an EIA-like effect and this effect is caused by quantum destructive interference that is distinct from the conventional EIA effect driven by quantum constructive interference.More importantly,the EIA-like cooling mechanism can significantly enhance the cooling rate of the hybrid system,enabling the final phonon number beyond the classical cooling limit in the strong optomechanical coupling regime.Meanwhile,the cooling effects of the EIA case is better than that of the normalmode splitting case under the same optomechanical coupling strength and qubit dissipation rate.
基金Supported by National Natural Science Foundation of China(Grant Nos.11671403 and 11671236)Henan Provincial General Natural Science Foundation Project(Grant No.232300420113)National Natural Science Foundation of China Youth Foud of China Youth Foud(Grant No.12101192).
文摘In this paper,we mainly focus on the following Choquard equation-{△u-V(x)(I_(a*)|u|^(p))|u|^(p-2)u=λu,x∈R^(N),u∈H^(1)(R^(N))where N≥1,λ∈R will arise as a Lagrange multiplier,0<a<N and N+a/N<p<N+a+2/N Under appropriate hypotheses on V(x),we prove that the above Choquard equation has a normalized ground state solution by utilizing variational methods.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11904309 and 12305015)the Natural Science Foundation of Hunan Province,China(Grant No.2020JJ5528)the Natural Science Foundation of Hebei Province,China(Grant No.A2024205027)。
文摘Machine learning has rapidly become a powerful tool for addressing challenges in ultracold atomic systems;however,its application to intricate three-dimensional(3D)systems remains relatively underexplored.In this study,we introduce a3D residual network(3D Res Net)framework based on 3D convolutional neural networks(3D CNN)to predict ground states phases in 3D dipolar spinor Bose–Einstein condensates(BECs).Our results show that the 3D Res Net framework predicts ground states with high accuracy and efficiency across a broad parameter space.To enhance phase transition predictions,we incorporate data augmentation techniques,leading to a notable improvement in the model's performance.The method is further validated in more complex scenarios,particularly when transverse magnetic fields are introduced.Compared to conventional imaginary-time evolution methods(ITEM),the 3D Res Net drastically reduces computational costs,offering a rapid and scalable solution for complex 3D multi-parameter nonlinear systems.
文摘The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potentials.As an example, the method is applied to discuss the ground state solution of Yukawa potential, using Hulthen solution as the trial function.
文摘This paper mainly discusses the following equation: where the potential function V : R<sup>3</sup> → R, α ∈ (0,3), λ > 0 is a parameter and I<sub>α</sub> is the Riesz potential. We study a class of Schrödinger-Poisson system with convolution term for upper critical exponent. By using some new tricks and Nehair-Pohožave manifold which is presented to overcome the difficulties due to the presence of upper critical exponential convolution term, we prove that the above problem admits a ground state solution.
文摘In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.
文摘In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.
