We study the following Schrodinger-Poisson system where （Pλ）{-△u＋ V（x）u＋λФ（x）u^p=x∈R^3,-△Ф=u^2,lim│x│→∞Ф（x） =0,u〉0,where λ≥0 is a parameter,1 〈 p 〈 ＋∞, V（x） and Q（x）=1 ,D.Ruiz[19] prov...We study the following Schrodinger-Poisson system where （Pλ）{-△u＋ V（x）u＋λФ（x）u^p=x∈R^3,-△Ф=u^2,lim│x│→∞Ф（x） =0,u〉0,where λ≥0 is a parameter,1 〈 p 〈 ＋∞, V（x） and Q（x）=1 ,D.Ruiz[19] proved that（Pλ）with p∈ （2, 5） has always a positive radial solution, but （Pλ） with p E （1, 2] has solution only if λ 〉 0 small enough and no any nontrivial solution if λ≥1/4.By using sub-supersolution method,we prove that there exists λ0〉0 such that（Pλ）with p ∈（1＋∞）has alaways a bound state（H^1（R^3）solution for λ∈[0,λ0）and certain functions V（x）and Q（x）in L^∞（R^3）.Moreover,for every λ∈[0,λ0）,the solutions uλ of （Pλ）converges,along a subsequence,to a solution of （P0）in H^1 as λ→0展开更多
Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we pro...Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.展开更多
We prove the existence of positive solutions for the system {-△pu=λa（x）f（v）u^-a x∈Ω -△qv=λb（x）g（u）v^-β,x∈Ω u=v=0xEf,x∈Ωwhere △rz =-div（｜z｜^r-2 z）, for r 〉1 denotes the r-Laplacian operator and...We prove the existence of positive solutions for the system {-△pu=λa（x）f（v）u^-a x∈Ω -△qv=λb（x）g（u）v^-β,x∈Ω u=v=0xEf,x∈Ωwhere △rz =-div（｜z｜^r-2 z）, for r 〉1 denotes the r-Laplacian operator and λ is a positive parameter, Ω is a bounded domain in R^n, n≥ 1 with sufficiently smooth boundary and a, E （0,1）. Here a（x） and b（x） are C1 sign-changing functions that maybe negative near the boundary and f,g are C1 nondecreasing functions, such that f,g： [0,∞） → [0,∞）; f（s） 〉,0,g（s） 〉0 for s〉0, lims→∞g（s） =∞ and lim s→∞ f（Mg（s）1/q-1）/s^p-1＋a=0 M〉0We discuss the existence of positive weak solutions when f, g, a（x） and b（x） satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.展开更多
In this work, we are interested to obtain some result of existence and nonex- istence of positive weak solution for the following p-Laplacian system {-△piui=λifi（u1,^…,um）,inΩ, i=1,...,m, ui=0,on δΩ,Vi=1,…,...In this work, we are interested to obtain some result of existence and nonex- istence of positive weak solution for the following p-Laplacian system {-△piui=λifi（u1,^…,um）,inΩ, i=1,...,m, ui=0,on δΩ,Vi=1,…,m,where △piz = div（｜△z｜^pi-2△Z）, Pi ≥ 1,λi,1 ≤ i ≤ m are a positive parameter, and Ω is a bounded domain in IR^N with smooth boundary δΩ. The proof of the main results is based to the method of sub-supersolutions.展开更多
We show that there exist saddle solutions of the nonlinear elliptic equation involving the p-Laplacian, p 〉 2, -△p^u -= f（u） in R^2m for all dimensions satisfying 2m ≥ p, by using sub-supersolution method. The ex...We show that there exist saddle solutions of the nonlinear elliptic equation involving the p-Laplacian, p 〉 2, -△p^u -= f（u） in R^2m for all dimensions satisfying 2m ≥ p, by using sub-supersolution method. The existence of saddle solutions of the above problem was known only in dimensions 2m≥ 2p.展开更多
In this paper,our main purpose is to establish the existence of positive solution of the following system{−△ p(x)u=F(x,u,v),x∈W,−D q(x)v=H(x,u,v),x∈W,u=v=0,x∈∂W,where W=B(0,r)⊂RN or W=B(0,r2)\B(0,r1)⊂RN,0<r,0&l...In this paper,our main purpose is to establish the existence of positive solution of the following system{−△ p(x)u=F(x,u,v),x∈W,−D q(x)v=H(x,u,v),x∈W,u=v=0,x∈∂W,where W=B(0,r)⊂RN or W=B(0,r2)\B(0,r1)⊂RN,0<r,0<r1<r2 are constants.F(x,u,v)=λp(x)[g(x)a(u)+f(v)],H(x,u,v)=θq(x)[g1(x)b(v)+h(u)],λ,θ>0 are parameters,p(x),q(x)are radial symmetric functions,−D p(x)=−div(|∇u|p(x)−2∇u)is called p(x)-Laplacian.We give the existence results and consider the asymptotic behavior of the solutions.In particular,we do not assume any symmetric condition,and we do not assume any sign condition on F(x,0,0)and H(x,0,0)either.展开更多
基金Supported by NSFC(10631030) and CAS-KJCX3-SYW-S03
文摘We study the following Schrodinger-Poisson system where （Pλ）{-△u＋ V（x）u＋λФ（x）u^p=x∈R^3,-△Ф=u^2,lim│x│→∞Ф（x） =0,u〉0,where λ≥0 is a parameter,1 〈 p 〈 ＋∞, V（x） and Q（x）=1 ,D.Ruiz[19] proved that（Pλ）with p∈ （2, 5） has always a positive radial solution, but （Pλ） with p E （1, 2] has solution only if λ 〉 0 small enough and no any nontrivial solution if λ≥1/4.By using sub-supersolution method,we prove that there exists λ0〉0 such that（Pλ）with p ∈（1＋∞）has alaways a bound state（H^1（R^3）solution for λ∈[0,λ0）and certain functions V（x）and Q（x）in L^∞（R^3）.Moreover,for every λ∈[0,λ0）,the solutions uλ of （Pλ）converges,along a subsequence,to a solution of （P0）in H^1 as λ→0
基金supported by the Beijing Natural Science Foundation(1212003)。
文摘Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.
文摘We prove the existence of positive solutions for the system {-△pu=λa（x）f（v）u^-a x∈Ω -△qv=λb（x）g（u）v^-β,x∈Ω u=v=0xEf,x∈Ωwhere △rz =-div（｜z｜^r-2 z）, for r 〉1 denotes the r-Laplacian operator and λ is a positive parameter, Ω is a bounded domain in R^n, n≥ 1 with sufficiently smooth boundary and a, E （0,1）. Here a（x） and b（x） are C1 sign-changing functions that maybe negative near the boundary and f,g are C1 nondecreasing functions, such that f,g： [0,∞） → [0,∞）; f（s） 〉,0,g（s） 〉0 for s〉0, lims→∞g（s） =∞ and lim s→∞ f（Mg（s）1/q-1）/s^p-1＋a=0 M〉0We discuss the existence of positive weak solutions when f, g, a（x） and b（x） satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.
文摘In this work, we are interested to obtain some result of existence and nonex- istence of positive weak solution for the following p-Laplacian system {-△piui=λifi（u1,^…,um）,inΩ, i=1,...,m, ui=0,on δΩ,Vi=1,…,m,where △piz = div（｜△z｜^pi-2△Z）, Pi ≥ 1,λi,1 ≤ i ≤ m are a positive parameter, and Ω is a bounded domain in IR^N with smooth boundary δΩ. The proof of the main results is based to the method of sub-supersolutions.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11101134, 11371128） and the Young Teachers Program of Hunan University. The authors thank the anonymous referees for their valuable comments and suggestions.
文摘We show that there exist saddle solutions of the nonlinear elliptic equation involving the p-Laplacian, p 〉 2, -△p^u -= f（u） in R^2m for all dimensions satisfying 2m ≥ p, by using sub-supersolution method. The existence of saddle solutions of the above problem was known only in dimensions 2m≥ 2p.
基金supported by the National Natural Science Foundation of China(No.11171092 and No.11471164)the Natural Science Foundation of Jiangsu Education Office(No.12KJB110002).
文摘In this paper,our main purpose is to establish the existence of positive solution of the following system{−△ p(x)u=F(x,u,v),x∈W,−D q(x)v=H(x,u,v),x∈W,u=v=0,x∈∂W,where W=B(0,r)⊂RN or W=B(0,r2)\B(0,r1)⊂RN,0<r,0<r1<r2 are constants.F(x,u,v)=λp(x)[g(x)a(u)+f(v)],H(x,u,v)=θq(x)[g1(x)b(v)+h(u)],λ,θ>0 are parameters,p(x),q(x)are radial symmetric functions,−D p(x)=−div(|∇u|p(x)−2∇u)is called p(x)-Laplacian.We give the existence results and consider the asymptotic behavior of the solutions.In particular,we do not assume any symmetric condition,and we do not assume any sign condition on F(x,0,0)and H(x,0,0)either.
