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The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains

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Abstract

In this paper, we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem {fx1-1} where Ω is a domain in ℝN, possibly unbounded, with empty or smooth boundary, ɛ is a small positive parameter, f ∈ C 1(ℝ+, ℝ) is of subcritical and V: ℝN → ℝ is a locally Hölder continuous function which is bounded from below, away from zero, such that infΛ V < min∂Λ V for some open bounded subset Λ of Ω. We prove that there is an ɛ 0 > 0 such that for any ɛ ∈ (0, ɛ 0], the above mentioned problem possesses a weak solution u ε with exponential decay. Moreover, u ε concentrates around a minimum point of the potential V in Λ. Our result generalizes a similar result by del Pino and Felmer (1996) for semilinear elliptic equations to the p-Laplacian type problem.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Yi He & GongBao Li

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  1. Yi He
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  2. GongBao Li
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Correspondence to GongBao Li.

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He, Y., Li, G. The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains. Sci. China Math. 57, 1927–1952 (2014). https://doi.org/10.1007/s11425-014-4830-2

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