#### Variational methods for singularly perturbed problems = 특이 섭동 문제들에 대한 변분법적 증명

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The present thesis studies existing issues for two singulary perturbed problems. First, we focus on the existence of multi-bump solutions of the singularly perturbed problem. $-\epsilon^2 \Delta v$ + V(x)v = f(v) in $R^N$. Extending previous results [11, 31, 69], we prove the existence of multi-bump solutions for an optimal class of nonlinearities f satisfying the Berestycki-Lions conditions and also for more general classes of potential wells than those previously studied. We devise two topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as $\epsilon \rightarrow 0$. Examples include a solid formed as the union of finitely many convex polyhedra or an embedded topological submanifold of $R^N$. Second, we study the existence of a least energy solution of the Henon equation with the homogeneous Neumann boundary condition $−\Delta u$ + u = $|x|^\alpha u^p$ , u > 0 in $\Omega$, $\partial u / \partial v = 0 on \partial\Omega$, where $n \geq 3, p = (n+2)/(n-2), \Omega \subset B(0, 1) \subset R^n, n \geq 3 and \partial^*\Omega \equiv \partial\Omega \cap \partial B(0, 1) \neq \varnothing$. It is well known that for $\alpha$ = 0, there exists a least energy solution of the problem. We are concerned with the existence of a least energy solution for $\alpha$ > 0 and its asymptotic behavior as the parameter $\alpha$ approaches from below to a threshold $\alpha_0 \in (0, \infty]$ for existence of a least energy solution.
Byeon, Jaeyoungresearcher변재형researcher
Description
한국과학기술원 :수리과학과,
Publisher
한국과학기술원
Issue Date
2020
Identifier
325007
Language
eng
Description

학위논문(박사) - 한국과학기술원 : 수리과학과, 2020.2,[ii, 84 p. :]

Keywords

Nonlinear Schrödinger equations▼aSingular perturbation▼asemi-classical standing waves▼aVariational methods▼aLeast energy solutions▼aHenon equation; limiting profile▼aNeumann boundary condition▼aweighted equations; 비선 슈레딩거 방정식▼a특이 섭동, 변분법적 방법▼a최소에너지해▼a극한 문제▼aNeumann(노 이만) 경계 조건

URI
http://hdl.handle.net/10203/283581