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
In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u^3+ β(x)v^2u, x ∈R^N,-?v + V_2(x)v = μ_2(x)v^3+ β(x)u^2v, x ∈R^N,u, v ∈H^1(R^N),wher...In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u^3+ β(x)v^2u, x ∈R^N,-?v + V_2(x)v = μ_2(x)v^3+ β(x)u^2v, x ∈R^N,u, v ∈H^1(R^N),where N = 1, 2, 3; V_1(x) and V_2(x) are positive and continuous, but may not be well-shaped; and μ_1(x), μ_2(x)and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), μ_i(x)(i = 1, 2) and β(x) satisfy some additional conditions.展开更多
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supported by National Natural Science Foundation of China (Grant No. 11371159)the Excellent Doctorial Dissertation Cultivation Grant from Central China Normal University (Grant No. 2016YBZZ085)
文摘In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u^3+ β(x)v^2u, x ∈R^N,-?v + V_2(x)v = μ_2(x)v^3+ β(x)u^2v, x ∈R^N,u, v ∈H^1(R^N),where N = 1, 2, 3; V_1(x) and V_2(x) are positive and continuous, but may not be well-shaped; and μ_1(x), μ_2(x)and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), μ_i(x)(i = 1, 2) and β(x) satisfy some additional conditions.
