In this paper,we consider the following Kirchhoff-Schrodinger-Poisson system:{−(a+b∫_(R^(3))|∇u|^(2))△u+u+ϕu=μQ(x)|u|^(q-2)u+K(x)|u|^(4)u,in R^(3),−△ϕ=u^(2) the nonlinear growth of|u|^(4)u reaches the Sobolev crit...In this paper,we consider the following Kirchhoff-Schrodinger-Poisson system:{−(a+b∫_(R^(3))|∇u|^(2))△u+u+ϕu=μQ(x)|u|^(q-2)u+K(x)|u|^(4)u,in R^(3),−△ϕ=u^(2) the nonlinear growth of|u|^(4)u reaches the Sobolev critical exponent.By combining the variational method with the concentration-compactness principle of Lions,we establish the existence of a positive solution and a positive radial solution to this problem under some suitable conditions.The nonlinear term includes the nonlinearity f(u)~|u|^(q-2)u for the well-studied case q∈[4,6),and the less-studied case q∈(2,3),we adopt two different strategies to handle these cases.Our result improves and extends some related works in the literature.展开更多
In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequal...In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequality.展开更多
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
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 paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point ...In this paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point in the domain, which is also a critical point of the Robin’s function corresponding to the Green’s function of biharmonic operator with the same boundary condition. Similar conclusion has been obtained in [6] under the condition that the domain is strictly convex.展开更多
We consider the following eigenvalue problem: [GRAPHICS] Where f(x, t) is a continuous function with critical growth. We prove the existence of nontrivial solutions.
This paper deals with the Cauchy problem for a doubly singular parabolic equation with a weighted source ut=div|u|p-2um)+|x|α uq ,(x,t)∈RN×(0,t),where N ≥ 1, 1 〈 p 〈 2, m 〉 max(0,3 -p/N} satisfyi...This paper deals with the Cauchy problem for a doubly singular parabolic equation with a weighted source ut=div|u|p-2um)+|x|α uq ,(x,t)∈RN×(0,t),where N ≥ 1, 1 〈 p 〈 2, m 〉 max(0,3 -p/N} satisfying 2 〈 p+m 〈 3, q 〉 1, and(α 〉 N(3 - p - m) - p. We give the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and non-existence of global solutions of the Cauchy problem. Moreover, the life span of solutions is also studied.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
The ferroelectric transitions of several SrTiO3-based ferroelectrics are investigated experimentally and theoretically, with special attention to the critical scaling exponents associated with the phase transitions, i...The ferroelectric transitions of several SrTiO3-based ferroelectrics are investigated experimentally and theoretically, with special attention to the critical scaling exponents associated with the phase transitions, in order to understand the competition among quantum fluctuations (QFs), quenched disorder, and ferroelectric ordering. Two representative systems with sufficiently strong QFs and quenched disorders in competition with the ferroelectric ordering are investigated. We start from non-stoichiometric SrTiO3(STO) with the Sr/Ti ratio deviating slightly from one, which is believed to maintain strong QFs. Then, we address Ba/Ca co-doped Sr1-x(Ca0.6389Ba0.3611)xTiO3(SCBT) with the averaged Sr-site ionic radius identical to the Sr2+ ionic radius, which is believed to offer remarkable quenched disorder associated with the Sr-site ionic mismatch. The critical exponents associated with polarization P and dielectric susceptibility ε, respectively, as functions of temperature T close to the critical point Tc, are evaluated. It is revealed that both non-stoichiometric SrTiO3 and SCBT exhibit much bigger critical exponents than the Landau mean-field theory predictions. These critical exponents then decrease gradually with increasing doping level or deviation of Sr/Ti ratio from one. A transverse Ising model applicable to the Sr-site doped STO (e.g., Sr1-xCaxTiO3) at low level is used to explain the observed experimental data. It is suggested that the serious deviation of these critical exponents from the Landau theory predictions in these STO-based systems is ascribed to the significant QFs and quenched disorder by partially suppressing the long-range spatial correlation of electric dipoles around the transitions. The present work thus sheds light on our understanding of the critical behaviors of ferroelectric transitions in STO in the presence of quantum fluctuations and quenched disorder, whose effects have been demonstrated to be remarkable.展开更多
This paper mainly discusses the following equation: where the potential function V : R<sup>3</sup> → R, α ∈ (0,3), λ > 0 is a parameter and I<sub>α</sub> is the Riesz potential. We stud...This paper mainly discusses the following equation: where the potential function V : R<sup>3</sup> → R, α ∈ (0,3), λ > 0 is a parameter and I<sub>α</sub> is the Riesz potential. We study a class of Schrödinger-Poisson system with convolution term for upper critical exponent. By using some new tricks and Nehair-Pohožave manifold which is presented to overcome the difficulties due to the presence of upper critical exponential convolution term, we prove that the above problem admits a ground state solution.展开更多
This paper is concerned with the following nonlinear Dirichlet problem:where △pu = div(| ▽u|p- 2 ▽u) is the p-Laplacian of u, Ω is a bounded domain in Rn (n > 3), 1 < p < n, p = -pn/n-p is the critical ex...This paper is concerned with the following nonlinear Dirichlet problem:where △pu = div(| ▽u|p- 2 ▽u) is the p-Laplacian of u, Ω is a bounded domain in Rn (n > 3), 1 < p < n, p = -pn/n-p is the critical exponent for the Sobolev imbedding, λ > 0 and f(x, u) satisfies some conditions. It reaches the conclusion that this problem has infinitely many solutions. Some results as p = 2 or f(x,u) = |u|q-2u, where 1 < q < p, are generalized.展开更多
In this paper,we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball.We are interested in the critical global expon...In this paper,we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball.We are interested in the critical global exponent q_o and the critical Fujita exponent q_c for the problem considered,and show that q_o=q_c for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources,which is quite different from the known results that q_o〈q_c for the onedimensional case;moreover,the value is different from the slow case.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
In this paper,by an approximating argument,we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents■whereΩis a smooth bounded domain in RN with ...In this paper,by an approximating argument,we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents■whereΩis a smooth bounded domain in RN with 0∈?Ωand all the principle curvatures of?Ωat 0 are negative,a∈C1(Ω,R*+),μ>0,0<s<2,1<q<2 and N>2(q+1)/(q-1).By2*:=2N/(N-2)and 2*(s):(2(N-s))/(N-2)we denote the critical Sobolev exponent and Hardy-Sobolev exponent,respectively.展开更多
In this article, we study the quasilinear elliptic problem involving critical Hardy Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign-changing solutions...In this article, we study the quasilinear elliptic problem involving critical Hardy Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign-changing solutions to the problem.展开更多
In this article, by applying the super-solution and sub-solution methods, instead of energy estimate methods, the authors investigate the critical extinction exponents for a polytropic filtration equation with a nonlo...In this article, by applying the super-solution and sub-solution methods, instead of energy estimate methods, the authors investigate the critical extinction exponents for a polytropic filtration equation with a nonlocal source and an absorption term, and give a classification of the exponents and coefficients for the solutions to vanish in finite time or not, which improve one of our results (Applicable Analysis, 92(2013), 636-650) and the results of Zheng et al (Math. Meth. Appl. Sci., 36(2013), 730-743).展开更多
In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms.Under mild conditions,the critical exponent of Besov...In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms.Under mild conditions,the critical exponent of Besov spaces of certain selfsimilar sets coincides with the walk dimension,which plays an important role in the analysis on fractals.As an application,we show examples having different critical exponents are not Lipschitz equivalent.展开更多
In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε≤...In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε≤ε; for any m ∈ N, it has m pairs solutions if ε≤εm, where ε,εm are sufficiently small positive numbers. Moreover, these solutions axe closed to zero in W1,p(RN) as ε→0.展开更多
基金Supported by NSFC(12171014,ZR2020MA005,ZR2021MA096)。
文摘In this paper,we consider the following Kirchhoff-Schrodinger-Poisson system:{−(a+b∫_(R^(3))|∇u|^(2))△u+u+ϕu=μQ(x)|u|^(q-2)u+K(x)|u|^(4)u,in R^(3),−△ϕ=u^(2) the nonlinear growth of|u|^(4)u reaches the Sobolev critical exponent.By combining the variational method with the concentration-compactness principle of Lions,we establish the existence of a positive solution and a positive radial solution to this problem under some suitable conditions.The nonlinear term includes the nonlinearity f(u)~|u|^(q-2)u for the well-studied case q∈[4,6),and the less-studied case q∈(2,3),we adopt two different strategies to handle these cases.Our result improves and extends some related works in the literature.
基金Supported by NSFC(10471047)NSF Guangdong Province(05300159).
文摘In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequality.
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 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The research work was supported by the National Natural Foundation of China (10371045)Guangdong Provincial Natural Science Foundation of China (000671).
文摘In this paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point in the domain, which is also a critical point of the Robin’s function corresponding to the Green’s function of biharmonic operator with the same boundary condition. Similar conclusion has been obtained in [6] under the condition that the domain is strictly convex.
文摘We consider the following eigenvalue problem: [GRAPHICS] Where f(x, t) is a continuous function with critical growth. We prove the existence of nontrivial solutions.
基金partially supported by the Doctor Start-up Funding and Natural Science Foundation of Chongqing University of Posts and Telecommunications(A2014-25 and A2014-106)partially supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission(KJ1500403)+3 种基金the Basic and Advanced Research Project of CQCSTC(cstc2015jcyj A00008)partially supported by NSFC(11371384),partially supported by NSFC(11426047)the Basic and Advanced Research Project of CQCSTC(cstc2014jcyj A00040)the Research Fund of Chongqing Technology and Business University(2014-56-11)
文摘This paper deals with the Cauchy problem for a doubly singular parabolic equation with a weighted source ut=div|u|p-2um)+|x|α uq ,(x,t)∈RN×(0,t),where N ≥ 1, 1 〈 p 〈 2, m 〉 max(0,3 -p/N} satisfying 2 〈 p+m 〈 3, q 〉 1, and(α 〉 N(3 - p - m) - p. We give the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and non-existence of global solutions of the Cauchy problem. Moreover, the life span of solutions is also studied.
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.
基金the National Basic Research Program of China(Grant Nos.2011CB922101 and 2009CB623303)the National Natural Science Foundation of China(Grant Nos.11234005 and 11074113)the Priority Academic Development Program of Jiangsu Higher Education Institutions,China
文摘The ferroelectric transitions of several SrTiO3-based ferroelectrics are investigated experimentally and theoretically, with special attention to the critical scaling exponents associated with the phase transitions, in order to understand the competition among quantum fluctuations (QFs), quenched disorder, and ferroelectric ordering. Two representative systems with sufficiently strong QFs and quenched disorders in competition with the ferroelectric ordering are investigated. We start from non-stoichiometric SrTiO3(STO) with the Sr/Ti ratio deviating slightly from one, which is believed to maintain strong QFs. Then, we address Ba/Ca co-doped Sr1-x(Ca0.6389Ba0.3611)xTiO3(SCBT) with the averaged Sr-site ionic radius identical to the Sr2+ ionic radius, which is believed to offer remarkable quenched disorder associated with the Sr-site ionic mismatch. The critical exponents associated with polarization P and dielectric susceptibility ε, respectively, as functions of temperature T close to the critical point Tc, are evaluated. It is revealed that both non-stoichiometric SrTiO3 and SCBT exhibit much bigger critical exponents than the Landau mean-field theory predictions. These critical exponents then decrease gradually with increasing doping level or deviation of Sr/Ti ratio from one. A transverse Ising model applicable to the Sr-site doped STO (e.g., Sr1-xCaxTiO3) at low level is used to explain the observed experimental data. It is suggested that the serious deviation of these critical exponents from the Landau theory predictions in these STO-based systems is ascribed to the significant QFs and quenched disorder by partially suppressing the long-range spatial correlation of electric dipoles around the transitions. The present work thus sheds light on our understanding of the critical behaviors of ferroelectric transitions in STO in the presence of quantum fluctuations and quenched disorder, whose effects have been demonstrated to be remarkable.
文摘This paper mainly discusses the following equation: where the potential function V : R<sup>3</sup> → R, α ∈ (0,3), λ > 0 is a parameter and I<sub>α</sub> is the Riesz potential. We study a class of Schrödinger-Poisson system with convolution term for upper critical exponent. By using some new tricks and Nehair-Pohožave manifold which is presented to overcome the difficulties due to the presence of upper critical exponential convolution term, we prove that the above problem admits a ground state solution.
基金Supported by NSFC(10171032) NSF of Guangdong Proviance (011606)
文摘This paper is concerned with the following nonlinear Dirichlet problem:where △pu = div(| ▽u|p- 2 ▽u) is the p-Laplacian of u, Ω is a bounded domain in Rn (n > 3), 1 < p < n, p = -pn/n-p is the critical exponent for the Sobolev imbedding, λ > 0 and f(x, u) satisfies some conditions. It reaches the conclusion that this problem has infinitely many solutions. Some results as p = 2 or f(x,u) = |u|q-2u, where 1 < q < p, are generalized.
基金The Fundamental Research Funds for the Central Universities and the NSF(11071100) of China
文摘In this paper,we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball.We are interested in the critical global exponent q_o and the critical Fujita exponent q_c for the problem considered,and show that q_o=q_c for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources,which is quite different from the known results that q_o〈q_c for the onedimensional case;moreover,the value is different from the slow case.
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.
文摘In this paper,by an approximating argument,we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents■whereΩis a smooth bounded domain in RN with 0∈?Ωand all the principle curvatures of?Ωat 0 are negative,a∈C1(Ω,R*+),μ>0,0<s<2,1<q<2 and N>2(q+1)/(q-1).By2*:=2N/(N-2)and 2*(s):(2(N-s))/(N-2)we denote the critical Sobolev exponent and Hardy-Sobolev exponent,respectively.
基金supported partly by the National Natural Science Foundation of China (10771219)
文摘In this article, we study the quasilinear elliptic problem involving critical Hardy Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign-changing solutions to the problem.
基金supported by NSFC(11271154,11401252)Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education,the 985 program of Jilin University+1 种基金Fundamental Research Funds of Jilin University(450060501179)supported by Graduate Innovation Fund of Jilin University(2014084)
文摘In this article, by applying the super-solution and sub-solution methods, instead of energy estimate methods, the authors investigate the critical extinction exponents for a polytropic filtration equation with a nonlocal source and an absorption term, and give a classification of the exponents and coefficients for the solutions to vanish in finite time or not, which improve one of our results (Applicable Analysis, 92(2013), 636-650) and the results of Zheng et al (Math. Meth. Appl. Sci., 36(2013), 730-743).
基金The second author is supported by NSFC Nos.10631040 and 11471075。
文摘In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms.Under mild conditions,the critical exponent of Besov spaces of certain selfsimilar sets coincides with the walk dimension,which plays an important role in the analysis on fractals.As an application,we show examples having different critical exponents are not Lipschitz equivalent.
基金Supported by the National Natural Science Foundation of China(No.11501186,11326145,11526088)
文摘In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε≤ε; for any m ∈ N, it has m pairs solutions if ε≤εm, where ε,εm are sufficiently small positive numbers. Moreover, these solutions axe closed to zero in W1,p(RN) as ε→0.