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Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity

  • * Corresponding author: P. Poláčik

    * Corresponding author: P. Poláčik 

The first author was supported in part by the NSF Grant DMS-1565388. The second author was supported in part by CONICYT-Chile Becas Chile, Convocatoria 2010

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  • We consider the equation

    $\Delta u+{{u}_{yy}}+f(x,u) = 0,\quad (x,y)\in {{\mathbb{R}}^{N}}\times \mathbb{R}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

    where $ f $ is sufficiently regular, radially symmetric in $ x $, and $ f(\cdot,0)\equiv 0 $. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in $ y $ and decaying as $ |x|\to\infty $ uniformly in $ y $. Such solutions are found using a center manifold reduction and results from the KAM theory. A required nondegeneracy condition is stated in terms of $ f_u(x,0) $ and $ f_{uu}(x,0) $, and is independent of higher-order terms in the Taylor expansion of $ f(x,\cdot) $. In particular, our results apply to some quadratic nonlinearities.

    Mathematics Subject Classification: Primary: 35B08, 35B15, 35J61; Secondary: 37J40, 37K55.

    Citation:

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