In this paper, we consider a class of Kirchhoff type problem with superlinear nonlinearity. A sign-changing solution with exactly two nodal domains will be obtained by combining the Nehari method and an iterative tech...In this paper, we consider a class of Kirchhoff type problem with superlinear nonlinearity. A sign-changing solution with exactly two nodal domains will be obtained by combining the Nehari method and an iterative technique.展开更多
研究了一类广义Kirchhoff方程-a+b∫R^(3)|u|2 d x△u+V(x)u=g(u)其中a,b>0是常数.由于在方程中出现了非局部项b∫R^(3)|u|2 d x△u,所以,方程的变分泛函与b=0时方程的变分泛函具有不同的性质.与相关文献相比,g不需要满足单调性条件...研究了一类广义Kirchhoff方程-a+b∫R^(3)|u|2 d x△u+V(x)u=g(u)其中a,b>0是常数.由于在方程中出现了非局部项b∫R^(3)|u|2 d x△u,所以,方程的变分泛函与b=0时方程的变分泛函具有不同的性质.与相关文献相比,g不需要满足单调性条件,并且非线性项g包含g(t)=|t|^(p-2) t(2<p≤4)这种情况,V也不需要满足强制性条件.首先引入辅助算子,构造伪梯度向量场,证明了下降流不变集的存在性.其次,由于4超线性AR条件不成立,所以引入了一种非局部扰动方法,即增加了一个高阶项β|u|^(r-2)u和另一个非局部扰动.对于扰动问题,通过改进的AR条件和下降流不变集下的极大极小参数得到了扰动问题的变号解,进而得到了原方程的变号解.最后,证明了该变号解是原方程的基态变号解.展开更多
In this paper, we study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making s...In this paper, we study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making some assumptions about the nonlinear function term . The existence of the family of global attractor and its Hausdorff dimension and Fractal dimension estimation are proved.展开更多
In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial condition...In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial conditions and boundary conditions, using the previous research results for reference. Firstly, the existence of bounded absorption set is proved by using a prior estimation, then the existence and uniqueness of the global solution of the problem is proved by using the classical Galerkin’s method. Finally, Housdorff dimension and fractal dimension of the family of global attractors are estimated by linear variational method and generalized Sobolev-Lieb-Thirring inequality.展开更多
文摘In this paper, we consider a class of Kirchhoff type problem with superlinear nonlinearity. A sign-changing solution with exactly two nodal domains will be obtained by combining the Nehari method and an iterative technique.
文摘研究了一类广义Kirchhoff方程-a+b∫R^(3)|u|2 d x△u+V(x)u=g(u)其中a,b>0是常数.由于在方程中出现了非局部项b∫R^(3)|u|2 d x△u,所以,方程的变分泛函与b=0时方程的变分泛函具有不同的性质.与相关文献相比,g不需要满足单调性条件,并且非线性项g包含g(t)=|t|^(p-2) t(2<p≤4)这种情况,V也不需要满足强制性条件.首先引入辅助算子,构造伪梯度向量场,证明了下降流不变集的存在性.其次,由于4超线性AR条件不成立,所以引入了一种非局部扰动方法,即增加了一个高阶项β|u|^(r-2)u和另一个非局部扰动.对于扰动问题,通过改进的AR条件和下降流不变集下的极大极小参数得到了扰动问题的变号解,进而得到了原方程的变号解.最后,证明了该变号解是原方程的基态变号解.
文摘In this paper, we study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making some assumptions about the nonlinear function term . The existence of the family of global attractor and its Hausdorff dimension and Fractal dimension estimation are proved.
文摘In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial conditions and boundary conditions, using the previous research results for reference. Firstly, the existence of bounded absorption set is proved by using a prior estimation, then the existence and uniqueness of the global solution of the problem is proved by using the classical Galerkin’s method. Finally, Housdorff dimension and fractal dimension of the family of global attractors are estimated by linear variational method and generalized Sobolev-Lieb-Thirring inequality.