We give a definition of relative entropy with respect to a sublinear expectation and establish large deviation principle for the empirical measures for independent random variables under the sublinear expectation.
This paper is concerned with the nonlinear Schrodinger-Kirchhoff system-(a+b∫R^3/u/^2dx)△u+λV(x)u=f(x,u)in R^3,where constants a>0,6≥0 andλ>0 is a parameter.We require that V(x)∈C(R^3)and has a potential w...This paper is concerned with the nonlinear Schrodinger-Kirchhoff system-(a+b∫R^3/u/^2dx)△u+λV(x)u=f(x,u)in R^3,where constants a>0,6≥0 andλ>0 is a parameter.We require that V(x)∈C(R^3)and has a potential well V^-1(0).Combining this with other suitable assumptions on K and f,the existence of nontrivial solutions is obtained via variational methods.Furthermore,the concentration behavior of the nontrivial solution is also explored on the set V^-1(0)asλ→+∞as A—>H-oo as well.It is worth noting that the(PS)-condition can not be directly got as done in the literature,which makes the problem more complicated.To overcome this difficulty,we adopt different method.展开更多
It is proved that the semilinear elliptic problem with zero boundary value -Δ u=λu-|u| q-1 u has a changing sign solution, as q∈(0,1) and λ>λ 2 , where λ 2 is the second eigenvalue of the ...It is proved that the semilinear elliptic problem with zero boundary value -Δ u=λu-|u| q-1 u has a changing sign solution, as q∈(0,1) and λ>λ 2 , where λ 2 is the second eigenvalue of the operator -Δ in the space H 1 0(Ω).展开更多
Two results about the multiplicity of nontrivial periodic bouncing solutions for sublinear damped vibration systems-x=g(t)x+f(t,x) are obtained via the Generalized Nonsmooth Saddle Point Theorem and a technique establ...Two results about the multiplicity of nontrivial periodic bouncing solutions for sublinear damped vibration systems-x=g(t)x+f(t,x) are obtained via the Generalized Nonsmooth Saddle Point Theorem and a technique established by Wu Xian and Wang Shaomin.Both of them imply the condition "f≥0" required in some previous papers can be weakened,furthermore,one of them also implies the condition about ■F(t,x)/■t required in some previous papers,such as "|■F(t,x)/■t|=σ_(0)F(t,x)" and "|■F(t,x)/■t|≤C(1+F(t,x))", is unnecessary,where F(t,x):=∫_(0)~xf(t,x)ds,and σ_(0),C are positive constants.展开更多
In this paper,we study a class of sublinear Kirchhoff equations:-(a+b∫R_(N)|■u|^(2)dx)+△u+V(x)u=f(x,u)in R^(N),where a,b>0,V:R^(N)→R can be sign-changing,and f:R^(N)×R→R.Under some conditions on V and f,w...In this paper,we study a class of sublinear Kirchhoff equations:-(a+b∫R_(N)|■u|^(2)dx)+△u+V(x)u=f(x,u)in R^(N),where a,b>0,V:R^(N)→R can be sign-changing,and f:R^(N)×R→R.Under some conditions on V and f,we verify that the problem possesses at least one energy solution by using variational method.展开更多
基金supported by the National Natural Science Foundation of China(11171262)the Specialized Research Fund for the Doctoral Program of Higher Education (200804860048)
文摘We give a definition of relative entropy with respect to a sublinear expectation and establish large deviation principle for the empirical measures for independent random variables under the sublinear expectation.
基金Supported by the Youth Foundation of Shangqiu Institute of Technology(2018XKQ01)。
文摘This paper is concerned with the nonlinear Schrodinger-Kirchhoff system-(a+b∫R^3/u/^2dx)△u+λV(x)u=f(x,u)in R^3,where constants a>0,6≥0 andλ>0 is a parameter.We require that V(x)∈C(R^3)and has a potential well V^-1(0).Combining this with other suitable assumptions on K and f,the existence of nontrivial solutions is obtained via variational methods.Furthermore,the concentration behavior of the nontrivial solution is also explored on the set V^-1(0)asλ→+∞as A—>H-oo as well.It is worth noting that the(PS)-condition can not be directly got as done in the literature,which makes the problem more complicated.To overcome this difficulty,we adopt different method.
基金This research is supported by NNSFC(1 9771 0 72 ) and ZNSF.And thanks to JNCASR in India Fortheir host when the firstauthor is
文摘It is proved that the semilinear elliptic problem with zero boundary value -Δ u=λu-|u| q-1 u has a changing sign solution, as q∈(0,1) and λ>λ 2 , where λ 2 is the second eigenvalue of the operator -Δ in the space H 1 0(Ω).
基金Supported by the National Natural Science Foundation of China (Grant No. 12171355)Elite Scholar Program in Tianjin University,P. R. China。
文摘Two results about the multiplicity of nontrivial periodic bouncing solutions for sublinear damped vibration systems-x=g(t)x+f(t,x) are obtained via the Generalized Nonsmooth Saddle Point Theorem and a technique established by Wu Xian and Wang Shaomin.Both of them imply the condition "f≥0" required in some previous papers can be weakened,furthermore,one of them also implies the condition about ■F(t,x)/■t required in some previous papers,such as "|■F(t,x)/■t|=σ_(0)F(t,x)" and "|■F(t,x)/■t|≤C(1+F(t,x))", is unnecessary,where F(t,x):=∫_(0)~xf(t,x)ds,and σ_(0),C are positive constants.
文摘In this paper,we study a class of sublinear Kirchhoff equations:-(a+b∫R_(N)|■u|^(2)dx)+△u+V(x)u=f(x,u)in R^(N),where a,b>0,V:R^(N)→R can be sign-changing,and f:R^(N)×R→R.Under some conditions on V and f,we verify that the problem possesses at least one energy solution by using variational method.
