Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well
Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well
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DOI：
10.1007/s00526-014-0754-5
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发表时间：
2013-05
影响因子：
2.1
通讯作者：
S. Cingolani;L. Jeanjean;Kazunaga Tanaka
中科院分区：
文献类型：
--
作者：
S. Cingolani;L. Jeanjean;Kazunaga Tanaka
本文考虑奇摄动非线性薛定谔方程$$\开始{aligned} - \vareptag^2\Delta u + V（x）u = f（u），\，\，u > 0，\，\，v \in H^1（\mathbb {R}^N）\end{aligned}$$其中和是满足Berestycki-Lions条件的非线性项.我们假设存在一个有界域使得$$\开始{aligned} m_0 \equiv \inf _{x \in \Omega } V（x）< \inf _{x \in \partial \Omega } V（x）\end{aligned}$$并且我们设置。对于small，我们证明了（0.1）集中的至少个解的存在性.我们注意到，根据我们的假设，搜索解决方案（0.1）不能减少到研究的临界点的功能限制到一个Nehari流形。
We consider singularly perturbed nonlinear Schrödinger equations $$\begin{aligned} - \varepsilon ^2 \Delta u + V(x)u = f(u), \, \, u > 0, \, \, v \in H^1( \mathbb {R}^N) \end{aligned}$$whereandis a nonlinear term which satisfies the so-called Berestycki–Lions conditions. We assume that there exists a bounded domainsuch that $$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$and we set. Forsmall we prove the existence of at leastsolutions to (0.1) concentrating, asaround. We remark that, under our assumptions of, the search of solutions to (0.1) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.