含非局部项的椭圆方程解的存在性和集中现象的变分方法研究
发布时间：2018-09-14 07:32
【摘要】：本文首先研究如下带有积分-微分算子的一般非局部问题其中Ω是RN中具有Lipschitz边界的有界区域,LK是积分-微分算子.利用约束极小和定量形变引理,我们证明了上述问题至少存在一个变号基态解(指所有变号解中具有最低能量的解)且其能量严格大于基态解的能量.其次,我们研究了上述问题带有凹凸非线性项即的情形,其中1q2p2s*:= N2,N2s,s ∈(0,1).利用喷泉定理及其对偶形式我们获得了无穷多解的存在性结果.最后,我们考虑如下分数阶Kirchhoff问题基态解的存在性和集中现象其中M(t)=ε2sa + ε4s-3bt是Kirchhoff函数,0s1,ε0是充分小的参数,V是能达到全局极小值、正的连续位势,f在无穷远处超3次但次临界增长.当ε0充分小时,我们证明了上述分数阶Kirchhoff问题基态解的存在性.其次,我们建立了在ε→0+时基态解的收敛性、集中性以及衰减估计.
[Abstract]:In this paper, we first study the following general nonlocal problems with integro-differential operators, where 惟 is a bounded domain of RN with Lipschitz boundary and LK is an integro-differential operator. By using constrained minima and quantitative deformation Lemma, we prove that there exists at least one solution of the ground state of the above problem (that is, the solution with the lowest energy) and that the energy of the solution is strictly larger than that of the solution of the ground state. Secondly, we study the case of the above problem with concave and convex nonlinear terms, where 1q2p2sn = N 2n 2s n 2s 鈭，
本文编号：2242002
[Abstract]:In this paper, we first study the following general nonlocal problems with integro-differential operators, where 惟 is a bounded domain of RN with Lipschitz boundary and LK is an integro-differential operator. By using constrained minima and quantitative deformation Lemma, we prove that there exists at least one solution of the ground state of the above problem (that is, the solution with the lowest energy) and that the energy of the solution is strictly larger than that of the solution of the ground state. Secondly, we study the case of the above problem with concave and convex nonlinear terms, where 1q2p2sn = N 2n 2s n 2s 鈭，
本文编号：2242002
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