In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p＆q-Laplacian type involving the critical Sobolev exponent：{-△pu-△qu=│u│^p＊-2u＋μ│u│^r-2u in Ω u...In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p＆q-Laplacian type involving the critical Sobolev exponent：{-△pu-△qu=│u│^p＊-2u＋μ│u│^r-2u in Ω u│δΩ=0,where Ω belong to R^N is a bounded domain,N〉p,p^＊=Np/N-p is the critical Sobolev exponent and μ 〉0. We prove that if 1 〈 r 〈 q 〈 p 〈 N, then there is a μ0 〉 0, such that for any μ∈ （0, μ0）, the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.展开更多
In this paper, we proof some properties of the space of bounded p(·)-variation in Wiener’s sense. We show that a functions is of bounded p(·)-variation in Wiener’s sense with variable exponent if and only ...In this paper, we proof some properties of the space of bounded p(·)-variation in Wiener’s sense. We show that a functions is of bounded p(·)-variation in Wiener’s sense with variable exponent if and only if it is the composition of a bounded nondecreasing functions and h?lderian maps of the variable exponent. We show that the composition operator H, associated with , maps the spaces into itself if and only if h is locally Lipschitz. Also, we prove that if the composition operator generated by maps this space into itself and is uniformly bounded, then the regularization of h is affine in the second variable.展开更多
This paper is concerned with the evolutionary p-Laplacian with interior and boundary sources.The critical exponents for the nonlinear sources are determined.
In this paper, we deal with the following problem：By variational method, we prove the existenceof a nontrivial weak solution whenand the existence of a cylindricalweak solution when
In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N i...In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N is a smooth and bounded domain,λ,μ>0,0<s 2<s 1<1,1<q<p<Ns 1.We establish the existence of a non-negative nontrivial weak solution to(Pμ,λ)by using the Mountain Pass Theorem.The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.展开更多
Based on the unified Hauser–Feshbach and exciton model,which can describe the particle emission processes between discrete energy levels with energy,angular momentum,and parity conservations,a statistical theory of l...Based on the unified Hauser–Feshbach and exciton model,which can describe the particle emission processes between discrete energy levels with energy,angular momentum,and parity conservations,a statistical theory of light nucleus reaction(STLN)is developed to calculate the double-differential cross-sections of the outgoing neutron and light charged particles for the proton-induced^(6) Li reaction.A significant difference is observed between the p+^(6) Li and p+^(7) Li reactions owing to the discrepancies in the energy-level structures of the targets.The reaction channels,including sequential and simultaneous emission processes,are analyzed in detail.Taking the double-differential cross-sections of the outgoing proton as an example,the influence of contaminations(such as^(1) H,^(7)Li,^(12)C,and^(16)O)on the target is identified in terms of the kinetic energy of the first emitted particles.The optical potential parameters of the proton are obtained by fitting the elastic scattering differential cross-sections.The calculated total double-differential cross-sections of the outgoing proton and deuteron at E_(p)=14 MeV agree well with the experimental data for different outgoing angles.Simultaneously,the mixed double differential cross-sections of^(3) He andαare in good agreement with the measurements.The agreement between the measured data and calculated results indicates that the two-body and three-body breakup reactions need to be considered,and the pre-equilibrium reaction mechanism dominates the reaction processes.Based on the STLN model,a PLUNF code for the p+^(6) Li reaction is developed to obtain an ENDF-6-formatted file of the double-differential cross-sections of the nucleon and light composite charged particles.展开更多
In this paper we present the notion of the space of bounded p(·)-variation in the sense of Wiener-Korenblum with variable exponent. We prove some properties of this space and we show that the composition operator...In this paper we present the notion of the space of bounded p(·)-variation in the sense of Wiener-Korenblum with variable exponent. We prove some properties of this space and we show that the composition operator H, associated with , maps the into itself, if and only if h is locally Lipschitz. Also, we prove that if the composition operator generated by maps this space into itself and is uniformly bounded, then the regularization of h is affine in the second variable, i.e. satisfies the Matkowski’s weak condition.展开更多
Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical ...Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients：{-div（｜△u｜p-2△u）=｜x｜s｜u｜p＊（s）-2u＋λ｜x｜t｜u｜p-2u, x∈B1, u｜σB1 =0, where t, s〉-p, 2≤p〈N, p＊（s）= （N＋s）pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p（p-1）t＋p（p2-p＋1） andλ∈（0,λ1,t）, whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ （ps＋p） min{1, p＋t/p＋s}＋p2p-（p-1） min{1, p＋tp＋s} andλ〉0 is small.展开更多
基金Supported by NSFC (10571069 and 10631030) the Lap of Mathematical Sciences, CCNU, Hubei Province, China
文摘In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p＆q-Laplacian type involving the critical Sobolev exponent：{-△pu-△qu=│u│^p＊-2u＋μ│u│^r-2u in Ω u│δΩ=0,where Ω belong to R^N is a bounded domain,N〉p,p^＊=Np/N-p is the critical Sobolev exponent and μ 〉0. We prove that if 1 〈 r 〈 q 〈 p 〈 N, then there is a μ0 〉 0, such that for any μ∈ （0, μ0）, the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.
基金supported by the Key P rogram of National Natural Science Foundation of China(No.51931003)the National Natural Science Foundation of China(Nos.52271033,52071179)+1 种基金the Natural Science Foundation of Jiangsu Province,China(No.BK20221493)the Foundation of Qinglan Project for Colleges and Universities in Jiangsu Province,China.
文摘In this paper, we proof some properties of the space of bounded p(·)-variation in Wiener’s sense. We show that a functions is of bounded p(·)-variation in Wiener’s sense with variable exponent if and only if it is the composition of a bounded nondecreasing functions and h?lderian maps of the variable exponent. We show that the composition operator H, associated with , maps the spaces into itself if and only if h is locally Lipschitz. Also, we prove that if the composition operator generated by maps this space into itself and is uniformly bounded, then the regularization of h is affine in the second variable.
基金supported by NSFCResearch Fundfor the Doctoral Program of Higher Education of China,Fundamental Research Project of Jilin University(200903284)Graduate Innovation Fund of Jilin University(20101045)
文摘This paper is concerned with the evolutionary p-Laplacian with interior and boundary sources.The critical exponents for the nonlinear sources are determined.
基金Supported by the National Science Foundation of China(11071245 and 11101418)
文摘In this paper, we deal with the following problem：By variational method, we prove the existenceof a nontrivial weak solution whenand the existence of a cylindricalweak solution when
基金National Natural Science Foundation of China(11501252 and 11571176)。
文摘In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N is a smooth and bounded domain,λ,μ>0,0<s 2<s 1<1,1<q<p<Ns 1.We establish the existence of a non-negative nontrivial weak solution to(Pμ,λ)by using the Mountain Pass Theorem.The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.
基金supported by the National Natural Science Foundation of China(No.12065003)the Guangxi Key R&D Project(2023AB07029)+1 种基金the Scientific Research and Technology Development Project of Guilin(20210104-2)the Central Government Guides Local Scientific and Technological Development Funds of China(Guike ZY22096024)。
文摘Based on the unified Hauser–Feshbach and exciton model,which can describe the particle emission processes between discrete energy levels with energy,angular momentum,and parity conservations,a statistical theory of light nucleus reaction(STLN)is developed to calculate the double-differential cross-sections of the outgoing neutron and light charged particles for the proton-induced^(6) Li reaction.A significant difference is observed between the p+^(6) Li and p+^(7) Li reactions owing to the discrepancies in the energy-level structures of the targets.The reaction channels,including sequential and simultaneous emission processes,are analyzed in detail.Taking the double-differential cross-sections of the outgoing proton as an example,the influence of contaminations(such as^(1) H,^(7)Li,^(12)C,and^(16)O)on the target is identified in terms of the kinetic energy of the first emitted particles.The optical potential parameters of the proton are obtained by fitting the elastic scattering differential cross-sections.The calculated total double-differential cross-sections of the outgoing proton and deuteron at E_(p)=14 MeV agree well with the experimental data for different outgoing angles.Simultaneously,the mixed double differential cross-sections of^(3) He andαare in good agreement with the measurements.The agreement between the measured data and calculated results indicates that the two-body and three-body breakup reactions need to be considered,and the pre-equilibrium reaction mechanism dominates the reaction processes.Based on the STLN model,a PLUNF code for the p+^(6) Li reaction is developed to obtain an ENDF-6-formatted file of the double-differential cross-sections of the nucleon and light composite charged particles.
文摘In this paper we present the notion of the space of bounded p(·)-variation in the sense of Wiener-Korenblum with variable exponent. We prove some properties of this space and we show that the composition operator H, associated with , maps the into itself, if and only if h is locally Lipschitz. Also, we prove that if the composition operator generated by maps this space into itself and is uniformly bounded, then the regularization of h is affine in the second variable, i.e. satisfies the Matkowski’s weak condition.
基金supported by the National Natural Science Foundation of China(11326139,11326145)Tian Yuan Foundation(KJLD12067)Hubei Provincial Department of Education(Q20122504)
文摘Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients：{-div（｜△u｜p-2△u）=｜x｜s｜u｜p＊（s）-2u＋λ｜x｜t｜u｜p-2u, x∈B1, u｜σB1 =0, where t, s〉-p, 2≤p〈N, p＊（s）= （N＋s）pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p（p-1）t＋p（p2-p＋1） andλ∈（0,λ1,t）, whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ （ps＋p） min{1, p＋t/p＋s}＋p2p-（p-1） min{1, p＋tp＋s} andλ〉0 is small.
