【作者】 陈卓；

【导师】 姬超；

【作者基本信息】 华东理工大学 ， 数学， 2021， 硕士

【摘要】 在本文中,我们通过使用变分方法证明以下一类强不定问题的分数阶薛定谔方程基态解的存在性和集中性#12其中α ∈(0,1),ε是一个正参数,N>2α,(-△)α表示分数阶拉普拉斯算子。非线性项f:R→R是具有次临界增长的连续函数,函数A:RN→ R是正连续函数并满足一些适当的假设,位势函数V:RN→R是整数周期的连续函数。我们首先使用Szulkin和Weth首次提出的一类新的极小极大特征并结合广义Nehari流形的方法证明该问题的极限问题的基态解的存在性和解关于参数的单调性,之后通过比较极限问题和原问题之间的能量泛函,进而得到原方程基态解的存在性结果。最后在方程基态解的存在性结果的基础之上,考虑方程中参数ε趋于零时,对应的基态解的集中性现象,在这一问题上,我们先得到基态解关于参数在分数阶索伯列夫空间Hα(RN)上的强收敛性,再通过Bessel核卷积上非线性项得到基态解的积分表达式,通过Bessel核自身的性质和数学分析的办法估计出基态解在无穷远处的衰减性,最终得到基态解的最大值序列在函数A(x)的最大值点集合,即集合{x∈RN|A(x)=supz∈RNA(z)}上发生集中性现象。更多还原

【Abstract】 In this thesis,by using the variational method,we prove the existence and concentration of ground state solutions for the following indefinite strongly fractional Schrodinger equations#12 where α ∈(0,1),ε is a positive parameter,N>2α,(-△)α stands for the fractional Laplacian.f:R→R is a continuous function with subcritical growth.The function A:RN→R is a positive continuous function and satisfies some appropriate assumptions.The potential function V:RN→R is a ZN-periodic continuous function.To begin with,we use a new minimax character that was first found by Szulkin and Weth,which combine the generalized Nehari manifold method to prove the existence of the ground state solution of the limit equation and the monotonicity of the ground state solutions with re-spect to the parameter.Then,by comparing the ground state energy between the limit equation and the original equation,we obtain the existence of the ground state solution of the original equation.Finally,we consider the concentration phenomenon of the ground state solutions when the parameters approach to zero under the existence of the ground state solutions.In this problem,we first obtain the strong convergence of the ground state solutions with respect to the parame-ters in the fractional Sobolev space Hα(RN).Then,by using Bessel kernel convolution with the nonlinear terms,we get the integral expression of the ground state solutions.By using the prop-erties of the Bessel kernel and the mathematical analysis,we estimate the decay of the ground state solution at infinity.Finally,we get the result that the maximum points of the ground state solutions is concentrated on the maximum points of the function A(x).更多还原