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 paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on R<sup>N</sup> × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a sm...In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on R<sup>N</sup> × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a small parameter, I<sub>α</sub> is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique. .展开更多
In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-Δu+(λa(x)+1)u=(1/|x|α*F(u))f(u)in R^N,where N≥3,0<α<min{N,...In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-Δu+(λa(x)+1)u=(1/|x|α*F(u))f(u)in R^N,where N≥3,0<α<min{N,4},λis a positive parameter and the nonnegative potential function a(x)is continuous.Using variational methods,we prove that if the potential well int(a^-1(0))consists of k disjoint components,then there exist at least 2^k-1 multi-bump solutions.The asymptotic behavior of these solutions is also analyzed asλ→+∞.展开更多
In this article, we study the nonexistence of solution with finite Morse index for the following Choquaxd type equation -△u=∫Rn｜u（y）｜p/｜x-y｜αdy｜u（x）｜p-2u（x） in RN,where N≥3,0〈α〈min {4,N}.Suppose tha...In this article, we study the nonexistence of solution with finite Morse index for the following Choquaxd type equation -△u=∫Rn｜u（y）｜p/｜x-y｜αdy｜u（x）｜p-2u（x） in RN,where N≥3,0〈α〈min {4,N}.Suppose that 2 〈 p 〈2N-α/N-2,we will show that this problem does not possess nontrivial solution with finite Morse index. While for p =2N-α/N-2,if i（u） 〈∞, then we have ∫RN∫RN｜u（x）｜p｜u（y）｜p dxdy 〈∞ and ∫RN｜△u｜2 dx=｜∫RN∫RN｜u（x）｜p/｜x-y｜a dxdy.展开更多
In this paper,we are concerned with the autonomous Choquard equation−Δu+u=(Iα∗|u|^(α/N+1))|u|^(α/N−1)u+|u|^(2∗−2)u+f(u)inR^(N),where N≥3,Iαdenotes the Riesz potential of orderα∈(0,N),the exponentsα/N+1 and 2^...In this paper,we are concerned with the autonomous Choquard equation−Δu+u=(Iα∗|u|^(α/N+1))|u|^(α/N−1)u+|u|^(2∗−2)u+f(u)inR^(N),where N≥3,Iαdenotes the Riesz potential of orderα∈(0,N),the exponentsα/N+1 and 2^(∗)=2N/(N−2)are critical with respect to the Hardy-Littlewood-Sobolev inequality and Sobolev embedding,respectively.Based on the variational methods,by using the minimax principles and the Pohožaev manifold method,we prove the existence of ground state solution under some suitable assumptions on the perturbation f.展开更多
In this article, the authors consider the existence of a nontrivial solution for the following equation： -△u＋u=q（x）（｜u｜^2＊1/｜x｜）u, x∈R^3, where q（x） satisfies some conditions. Using a Min-Max method, th...In this article, the authors consider the existence of a nontrivial solution for the following equation： -△u＋u=q（x）（｜u｜^2＊1/｜x｜）u, x∈R^3, where q（x） satisfies some conditions. Using a Min-Max method, the authors prove that there is at least one nontrivial solution for the above equation.展开更多
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.
文摘In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on R<sup>N</sup> × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a small parameter, I<sub>α</sub> is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique. .
基金L.Guo is supported by the Fundamental Research Funds for the Central Universities(2662018QD039)T.Hu is supported by the Project funded by China Postdoctoral Science Foundation(2018M643389).
文摘In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-Δu+(λa(x)+1)u=(1/|x|α*F(u))f(u)in R^N,where N≥3,0<α<min{N,4},λis a positive parameter and the nonnegative potential function a(x)is continuous.Using variational methods,we prove that if the potential well int(a^-1(0))consists of k disjoint components,then there exist at least 2^k-1 multi-bump solutions.The asymptotic behavior of these solutions is also analyzed asλ→+∞.
文摘In this article, we study the nonexistence of solution with finite Morse index for the following Choquaxd type equation -△u=∫Rn｜u（y）｜p/｜x-y｜αdy｜u（x）｜p-2u（x） in RN,where N≥3,0〈α〈min {4,N}.Suppose that 2 〈 p 〈2N-α/N-2,we will show that this problem does not possess nontrivial solution with finite Morse index. While for p =2N-α/N-2,if i（u） 〈∞, then we have ∫RN∫RN｜u（x）｜p｜u（y）｜p dxdy 〈∞ and ∫RN｜△u｜2 dx=｜∫RN∫RN｜u（x）｜p/｜x-y｜a dxdy.
基金This paper is supported by the National Natural Science Foundation of China(No.11971393).
文摘In this paper,we are concerned with the autonomous Choquard equation−Δu+u=(Iα∗|u|^(α/N+1))|u|^(α/N−1)u+|u|^(2∗−2)u+f(u)inR^(N),where N≥3,Iαdenotes the Riesz potential of orderα∈(0,N),the exponentsα/N+1 and 2^(∗)=2N/(N−2)are critical with respect to the Hardy-Littlewood-Sobolev inequality and Sobolev embedding,respectively.Based on the variational methods,by using the minimax principles and the Pohožaev manifold method,we prove the existence of ground state solution under some suitable assumptions on the perturbation f.
基金Financial support in part by the Volkswagen Foundation of Germany and in part by NNSF of China
文摘In this article, the authors consider the existence of a nontrivial solution for the following equation： -△u＋u=q（x）（｜u｜^2＊1/｜x｜）u, x∈R^3, where q（x） satisfies some conditions. Using a Min-Max method, the authors prove that there is at least one nontrivial solution for the above equation.
