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首页> 外文期刊>Differential and integral equations >ASYMPTOTICS AND SYMMETRIES OF GROUND-STATE AND LEAST ENERGY NODAL SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS WITH SLOWLY GROWING SUPERLINEARITIES
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ASYMPTOTICS AND SYMMETRIES OF GROUND-STATE AND LEAST ENERGY NODAL SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS WITH SLOWLY GROWING SUPERLINEARITIES

机译：超线性缓慢增长的边值问题的基态和最小能量节点解的渐近性和对称性

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摘要

We study the problems -△u =f_θ(u) in Ω, u = 0 on _Ω， -△u+u =f_θ(u)in Ω，__v u = 0 on _Ω，wheref/θ is a slowly superlinearly growing nonlinearity, and Ω is a bounded domain. Namely, we are interested in generalizing the results obtained in [4]， where the model nonlinearity f_θ(u) = |u|~(θ- 2)u was considered in the case of Dirichlet boundary conditions. We derive the asymptotic behaviour of ground state and least energy nodal solutions when θ→2, leading to symmetry results for e small. Our assumptions permit us to study some typical nonlinearities such as a superlinear perturbation of a small pure power or the sum of small powers and slowly exponentialy growing nonlinearities in dimension 2.
机译：我们研究了以下问题：-△u =f_θ（u）Ω，u = 0在_Ω，-△u + u =f_θ（u）Ω，__ v u = 0在_Ω，其中f /θ是一个缓慢超线性增长的非线性，并且Ω是有界域。即，我们有兴趣对在[4]中获得的结果进行概括，其中在Dirichlet边界条件下考虑模型非线性f_θ（u）= | u |〜（θ-2）u。当θ→2时，我们推导出基态和最小能量节点解的渐近行为，导致e的对称结果很小。我们的假设使我们能够研究一些典型的非线性，例如小纯功率的超线性摄动或小功率的总和以及在2维上缓慢指数增长的非线性。

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