Multiple interior and boundary peak solutions to singularly perturbed nonlinear Neumann problems under the Berestycki

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Multiple interior and boundary peak solutions to singularly perturbed nonlinear Neumann problems under the Berestycki–Lions condition

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作者：Youngae Lee ; Jinmyoung Seok
关键词：Mathematics Subject Classification35J20 ; 35J25 ; 35J61
刊名：Mathematische Annalen
出版年：2017
出版时间：February 2017
年：2017
卷：367
期：1-2
页码：881-928
全文大小：
刊物类别：Mathematics and Statistics
刊物主题：Mathematics, general;
出版者：Springer Berlin Heidelberg
ISSN：1432-1807
卷排序：367

文摘

Let $\varOmega $ be a smooth bounded domain in $\mathbb {R}^N$, $N\ge 3$. We consider the following singularly perturbed nonlinear elliptic problem on $\varOmega $, $$\begin{aligned} \varepsilon ^2\varDelta v-v+f(v)=0,\quad v>0\ \text {on}\ \varOmega ,\qquad \frac{\partial v}{\partial \nu }=0\quad \text {on}\ \partial \varOmega , \end{aligned}$$where $\nu $ is an exterior unit normal vector to $\partial \varOmega $ and a nonlinearity f satisfies subcritical growth condition. It has been known that for any $l_0, l_1 \in \mathbb {N} \cup \{ 0 \}$, $l_0+l_1>0$, there exists a solution $v_\varepsilon $ of the above problem which exhibits $l_0$-boundary peaks and $l_1$-interior peaks for small $\varepsilon >0$ under rather strong conditions on f, such as the linearized non-degeneracy condition for a limiting problem. 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