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Li, Gongbao; Zheng, Gao-Feng

The existence of positive solution to some asymptotically linear elliptic equations in exterior domains. (English) Zbl 1134.35038

Rev. Mat. Iberoam. 22, No. 2, 559-590 (2006).
Summary: We are concerned with the asymptotically linear elliptic problem \(-\Delta u+ \lambda_{0}u=f(u)\), \(u\in H_{0}^{1}(\Omega ) \) in an exterior domain \(\Omega= \mathbb{R}^{N}\setminus \overline{\mathcal O}\) \(( N\geqslant 3)\) with \({\mathcal O}\) a smooth bounded and star-shaped open set, and \(\lim_{t\rightarrow +\infty }\frac{ f(t)}{t}=l\), \(0<l<+\infty\). Using a precise deformation lemma and algebraic topology argument, we prove under our assumptions that the problem possesses at least one positive solution.

Cited in 9 Documents

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

asymptotically linear elliptic; exterior domain; algebraic topology argument; positive solution
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References:

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