For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy expon...For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy exponent, h（x） ∈ （D^1,2（R^N））^＊, the dual space of （D^1,2（R^N））, with h（x）≥（≠）0. By Ekeland＇s variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ｜｜h｜｜＊〈CN,sAs^N-s/4-2s（1-μ/μ）^1/2, CN,s=4-2s/N-2（N-2/N＋2-2s）^N＋2-2s/4-2s and As = inf u∈D^1,2（R^N）／{0}∫R^N（｜△↓u｜^2-μu^2/｜x｜^2）dx/（∫R^N｜u｜^2^＊（s）/｜x｜^sdx）^2/2^＊（s）.展开更多
In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the N...In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the Nehari manifold and mountain pass theorem.展开更多
The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational...The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.展开更多
In this paper,we consider the following Schrodinger-Poisson system{-Δu+ηΦu=f(x,μ)+μ^(5),x∈Ω,-ΔФ=μ^(2),x∈Ω,μ=Φ=0,x∈■Ω,whereΩis a smooth bounded domain in R^(3),η=±1 and the continuous function f...In this paper,we consider the following Schrodinger-Poisson system{-Δu+ηΦu=f(x,μ)+μ^(5),x∈Ω,-ΔФ=μ^(2),x∈Ω,μ=Φ=0,x∈■Ω,whereΩis a smooth bounded domain in R^(3),η=±1 and the continuous function f satisfies some suitable conditions.Based on the Mountain pass theorem,we prove the existence of positive ground state solutions.展开更多
文摘For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy exponent, h（x） ∈ （D^1,2（R^N））^＊, the dual space of （D^1,2（R^N））, with h（x）≥（≠）0. By Ekeland＇s variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ｜｜h｜｜＊〈CN,sAs^N-s/4-2s（1-μ/μ）^1/2, CN,s=4-2s/N-2（N-2/N＋2-2s）^N＋2-2s/4-2s and As = inf u∈D^1,2（R^N）／{0}∫R^N（｜△↓u｜^2-μu^2/｜x｜^2）dx/（∫R^N｜u｜^2^＊（s）/｜x｜^sdx）^2/2^＊（s）.
文摘In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the Nehari manifold and mountain pass theorem.
基金Supported by National Natural Science Foundation of China (11071198 11101347)+2 种基金Postdoctor Foundation of China (2012M510363)the Key Project in Science and Technology Research Plan of the Education Department of Hubei Province (D20112605 D20122501)
文摘The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.
基金Supported by the Fundamental Research Funds of China West Normal University(No.18B015)Natural Science Foundation of Sichuan(No.23NSFSC1720).
文摘In this paper,we consider the following Schrodinger-Poisson system{-Δu+ηΦu=f(x,μ)+μ^(5),x∈Ω,-ΔФ=μ^(2),x∈Ω,μ=Φ=0,x∈■Ω,whereΩis a smooth bounded domain in R^(3),η=±1 and the continuous function f satisfies some suitable conditions.Based on the Mountain pass theorem,we prove the existence of positive ground state solutions.
