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Bartsch, Thomas; Peng, Shuangjie

Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations. II. (English) Zbl 1195.35143

J. Differ. Equations 248, No. 11, 2746-2767 (2010).
Summary: We construct spike layered solutions for the semilinear elliptic equation \(-\varepsilon ^{2}\Delta u+V(x)u=K(x)u^{p-1}\) on a domain \(\Omega \subset \mathbb R^N\) which may be bounded or unbounded. The solutions concentrate simultaneously on a finite number of \(m\)-dimensional spheres in \(\Omega \). These spheres accumulate as \(\varepsilon \rightarrow 0\) at a prescribed sphere in \(\Omega \) whose location is determined by the potential functions \(V,K\).
For part I, cf. the authors, Indiana Univ. Math. J. 57, No. 4, 1599–1631 (2008; Zbl 1155.35026).

Cited in 6 Documents

MSC:

35J61 Semilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

singularly perturbed elliptic equation; variational methods; critical point theory; concentrating solutions; spike layered solutions

Citations:

Zbl 1155.35026
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Full Text: DOI

References:

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