In this paper,we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms:{(−Δ)_(a/1)^(α/2)u1(x)=u_(1)^(q11)(x)+u_(2...In this paper,we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms:{(−Δ)_(a/1)^(α/2)u1(x)=u_(1)^(q11)(x)+u_(2)^(q12)(x)+h_(1)(x,u_(1)(x),u_(2)(x),∇u_(1)(x),∇u_(2)(x)),x∈Ω,(−Δ)_(a2)^(β/2)u2(x)=u_(1)^(q21)(x)+u_(2)^(q22)(x)+h_(2)(x,u_(1)(x),u_(2)(x),∇u_(1)(x),∇u_(2)(x)),x∈Ω,u_(1)(x)=0,u_(2)(x)=0,x∈R^(n)∖Ω.Here(−Δ)_(a1)^(α/2) and(−Δ)_(a2)^(β/2) denote weighted fractional Laplacians andΩ⊂R^(n) is a C^(2) bounded domain.It is shown that under some assumptions on h_(i)(i=1,2),the problem admits at least one positive solution(u_(1)(x),u_(2)(x)).We first obtain the{a priori}bounds of solutions to the system by using the direct blow-up method of Chen,Li and Li.Then the proof of existence is based on a topological degree theory.展开更多
In this paper, we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions. The proof is obtained by a bootstrap argument and maximal...In this paper, we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions. The proof is obtained by a bootstrap argument and maximal regularity estimates. If n≥ 10/3m, we also give another proof which does not use maximal regularity estimates.展开更多
In this article, the authors establish some new nonlinear difference inequalities in two independent variables, which generalize some existing results and can be used as handy tools in the study of qualitative as well...In this article, the authors establish some new nonlinear difference inequalities in two independent variables, which generalize some existing results and can be used as handy tools in the study of qualitative as well as quantitative properties of solutions of certain classes of difference equations.展开更多
Some new inequalities involving improper integrals are established in the paper which generalize the related results due to Pachpatte and Rodrigues.Discrete analogues of the integral inequalities obtained are also der...Some new inequalities involving improper integrals are established in the paper which generalize the related results due to Pachpatte and Rodrigues.Discrete analogues of the integral inequalities obtained are also derived.An example is given to show that the bound in Theorem 1 is not improvable.展开更多
LetΩbe a bounded smooth domain in RN(N≥3).Assuming that 0<s<1,1<p,q≤N+2s/N-2s with(p,q)≠(N+2s/N-2s,N+2s/N-2s),and a,b>0 are constants,we consider the existence results for positive solutions of a class...LetΩbe a bounded smooth domain in RN(N≥3).Assuming that 0<s<1,1<p,q≤N+2s/N-2s with(p,q)≠(N+2s/N-2s,N+2s/N-2s),and a,b>0 are constants,we consider the existence results for positive solutions of a class of fractional elliptic system below,{(a+b[u]^(2)_(s))(-Δ)^(s)u=vp+h_(1)(x,u,v,▽u,▽v),x∈Ω,(-Δ)^(s)v=u^(q)+h_(2)(x,u,▽,▽u,▽v),x∈Q,u,v>0,x∈Ω,u=v=0,x∈RN\Ω.Under some assumptions of hi(x,u,v,▽u,▽v)(i=1,2),we get a priori bounds of the positive solutions to the problem(1.1)by the blow-up methods and rescaling argument.Based on these estimates and degree theory,we establish the existence of positive solutions to problem(1.1).展开更多
Some new integral and discrete inequalities with power nonlinearity are established. Two recent results of B.G. Pachpatte are improved by some particularcorollaries of our results. Application examples are also indica...Some new integral and discrete inequalities with power nonlinearity are established. Two recent results of B.G. Pachpatte are improved by some particularcorollaries of our results. Application examples are also indicated.展开更多
Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure...Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.展开更多
Nonlinear integral and discrete inequalities are obtained. which are related to some recent results of B. C. Pachpatte in [1] given therein as generalizations of Ou-Iang’s integral inequality[2]. As special cases, so...Nonlinear integral and discrete inequalities are obtained. which are related to some recent results of B. C. Pachpatte in [1] given therein as generalizations of Ou-Iang’s integral inequality[2]. As special cases, some new inequalities with the power nonlinearity are derived from the main results. To show the contribution of our results, boundedness of solutions to certain nonlinear difference equation is also considered.展开更多
A new discrete inequality with the power nonlinearity is obtained which unifies and generalizes some known results due to B.G.Pachpatte. A certain initial value problem of a sum-difference equation is also given to co...A new discrete inequality with the power nonlinearity is obtained which unifies and generalizes some known results due to B.G.Pachpatte. A certain initial value problem of a sum-difference equation is also given to convey the usefulness of the inequality obtained.展开更多
We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniquen...We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniqueness and nonnegativeness of solutions when the principal operator is monotone and the nonlinearity is nonincreasing. Our operator is of the most general form containing all previous ones and we also weaken assumptions on the operator and the nonlinearity to get the above results. Moreover, we do not impose the restricted condition on p(x) and the uniform monotonicity of the operator to show the existence of three distinct solutions.展开更多
Abstract We consider the defocusing nonlinear SchrSdinger equations iut ＋△u = lulPu with p being an even integer in dimensions d ≥ 5. We prove that an a priori bound of critical norm implies global well-posedness a...Abstract We consider the defocusing nonlinear SchrSdinger equations iut ＋△u = lulPu with p being an even integer in dimensions d ≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution.展开更多
Some new N-independent-variable discrete inequalities of Gronwall-On-fang type are established. Application examples to certain multivariate summary-difference equations are also sketched.
Pointwise a priori bounds on solutions to some linear and nonlinear hyperbolic partial-integral Volterra inequalities are obtained. They are handy tools in the study of some partial differential equations with boundar...Pointwise a priori bounds on solutions to some linear and nonlinear hyperbolic partial-integral Volterra inequalities are obtained. They are handy tools in the study of some partial differential equations with boundary value conditions. An example of such application is also indicated.展开更多
文摘In this paper,we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms:{(−Δ)_(a/1)^(α/2)u1(x)=u_(1)^(q11)(x)+u_(2)^(q12)(x)+h_(1)(x,u_(1)(x),u_(2)(x),∇u_(1)(x),∇u_(2)(x)),x∈Ω,(−Δ)_(a2)^(β/2)u2(x)=u_(1)^(q21)(x)+u_(2)^(q22)(x)+h_(2)(x,u_(1)(x),u_(2)(x),∇u_(1)(x),∇u_(2)(x)),x∈Ω,u_(1)(x)=0,u_(2)(x)=0,x∈R^(n)∖Ω.Here(−Δ)_(a1)^(α/2) and(−Δ)_(a2)^(β/2) denote weighted fractional Laplacians andΩ⊂R^(n) is a C^(2) bounded domain.It is shown that under some assumptions on h_(i)(i=1,2),the problem admits at least one positive solution(u_(1)(x),u_(2)(x)).We first obtain the{a priori}bounds of solutions to the system by using the direct blow-up method of Chen,Li and Li.Then the proof of existence is based on a topological degree theory.
文摘In this paper, we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions. The proof is obtained by a bootstrap argument and maximal regularity estimates. If n≥ 10/3m, we also give another proof which does not use maximal regularity estimates.
基金a HKU Seed grant the Research Grants Council of the Hong Kong SAR(HKU7016/07P)
文摘In this article, the authors establish some new nonlinear difference inequalities in two independent variables, which generalize some existing results and can be used as handy tools in the study of qualitative as well as quantitative properties of solutions of certain classes of difference equations.
基金Supported by the Natural Science Foundation of Guangdong Pronvince( 0 1 1 471 ) and Education Bu-reau( 0 1 76)
文摘Some new inequalities involving improper integrals are established in the paper which generalize the related results due to Pachpatte and Rodrigues.Discrete analogues of the integral inequalities obtained are also derived.An example is given to show that the bound in Theorem 1 is not improvable.
基金supported by National Natural Science Foundation of China (No.11761030)Hubei Provincial Natural Science Foundation of China (No.2022CFC016)Cultivation Project for High-Level Scientific Research Achievements of Hubei Minzu University (No.PY20002)。
文摘LetΩbe a bounded smooth domain in RN(N≥3).Assuming that 0<s<1,1<p,q≤N+2s/N-2s with(p,q)≠(N+2s/N-2s,N+2s/N-2s),and a,b>0 are constants,we consider the existence results for positive solutions of a class of fractional elliptic system below,{(a+b[u]^(2)_(s))(-Δ)^(s)u=vp+h_(1)(x,u,v,▽u,▽v),x∈Ω,(-Δ)^(s)v=u^(q)+h_(2)(x,u,▽,▽u,▽v),x∈Q,u,v>0,x∈Ω,u=v=0,x∈RN\Ω.Under some assumptions of hi(x,u,v,▽u,▽v)(i=1,2),we get a priori bounds of the positive solutions to the problem(1.1)by the blow-up methods and rescaling argument.Based on these estimates and degree theory,we establish the existence of positive solutions to problem(1.1).
文摘Some new integral and discrete inequalities with power nonlinearity are established. Two recent results of B.G. Pachpatte are improved by some particularcorollaries of our results. Application examples are also indicated.
基金the Australian Research Council's Discovery Projects(DP0450752)Linkage International(LX0561259)
文摘Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.
基金The project is supported in part by the NSF of Guangdong Province (Grnat No. 940651) the SF of Key Discipline of the State Council Office of Overseas Chinese Affairs of China (Grant No.93-93-6)
文摘Nonlinear integral and discrete inequalities are obtained. which are related to some recent results of B. C. Pachpatte in [1] given therein as generalizations of Ou-Iang’s integral inequality[2]. As special cases, some new inequalities with the power nonlinearity are derived from the main results. To show the contribution of our results, boundedness of solutions to certain nonlinear difference equation is also considered.
文摘A new discrete inequality with the power nonlinearity is obtained which unifies and generalizes some known results due to B.G.Pachpatte. A certain initial value problem of a sum-difference equation is also given to convey the usefulness of the inequality obtained.
基金supported by the National Research Foundation of Korea Grant Funded by the Korea Government (Grant No. NRF-2015R1D1A3A01019789)
文摘We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniqueness and nonnegativeness of solutions when the principal operator is monotone and the nonlinearity is nonincreasing. Our operator is of the most general form containing all previous ones and we also weaken assumptions on the operator and the nonlinearity to get the above results. Moreover, we do not impose the restricted condition on p(x) and the uniform monotonicity of the operator to show the existence of three distinct solutions.
文摘Abstract We consider the defocusing nonlinear SchrSdinger equations iut ＋△u = lulPu with p being an even integer in dimensions d ≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution.
文摘Some new N-independent-variable discrete inequalities of Gronwall-On-fang type are established. Application examples to certain multivariate summary-difference equations are also sketched.
文摘Pointwise a priori bounds on solutions to some linear and nonlinear hyperbolic partial-integral Volterra inequalities are obtained. They are handy tools in the study of some partial differential equations with boundary value conditions. An example of such application is also indicated.
