Abstract
We study the existence of positive and sign-changing solutions to the boundary value problem − Δ u = |u|p−1u in a bounded smooth domain Ω in ℝ2, with homogeneous Dirichlet boundary condition, when p is a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution concentrating at exactly m points as p→∞. In particular, for a non-simply connected domain such a solution exists for any given m ≥ 1. Moreover, for p large enough, we prove the existence of two pairs of solutions which change sign exactly once and whose nodal lines intersect the boundary of Ω.
The author is supported by M.U.R.S.T., project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.
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© 2006 Birkhäuser Verlag Basel/Switzerland
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Pistoia, A. (2006). Concentrating Solutions for a Two-dimensional Elliptic Problem with Large Exponent in Nonlinearity. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_34
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DOI: https://doi.org/10.1007/978-3-7643-7719-9_34
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