Quasilinear Elliptic Equations on Half- and Quarter-spaces

Advanced Nonlinear Studies - Tập 13 Số 1 - Trang 115-136 - 2013
E. N. Dancer1, Yihong Du2, Messoud Efendiev3
1School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
2School of Science and Technology, University of New England, Armidale, NSW, 2351, Australia
3Helmholtz Zentrum München, Institute für Biomathematik und Biometrie Ingostädter Landstrasse 1, D-85764 Neuherberg, Germany

Tóm tắt

Abstract We consider quasilinear elliptic problems of the form Δpu + f(u) = 0 over the half-space H = {x ∈ ℝN : x1 > 0} and over the quarter-space Q = {x ∈ ℝN : x1 > 0, xN > 0}. In the half-space case we assume u ≥ 0 on ∂H, and in the quarter-space case we assume that u ≥ 0 on {x1 = 0} and u = 0 on {xN = 0}. Let u ≢ 0 be a bounded nonnegative solution. For some general classes of nonlinearities f , we show that, in the half-space case, limx1→∞ u(x1, x2, ..., xN) always exists and is a positive zero of f ; and in the quarter-space case, where V is a solution of the one-dimensional problem ΔpV + f(V) = 0 in ℝ+, V(0) = 0, V(t) > 0 for t > 0, V(+∞) = z, with z a positive zero of f . Our results extend most of those in the recent paper of Efendiev and Hamel [6] for the special case p = 2 to the general case p > 1. Moreover, by making use of a sharper Liouville type theorem, some of the results in [6] are improved. To overcome the difficulty of the lack of a strong comparison principle for p-Laplacian problems, we employ a weak sweeping principle.

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Tài liệu tham khảo

Efendiev, 2011, Asymptotic beavior of solutions of semilinear elliptic equations in unbounded domains : two approaches, Math, 228

Dancer, 2003, Some remarks on Louville type results for quasilinear elliptic equations, Proc Math Soc, 131

Serrin, 2002, Cauchy - Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math, 189

Bidaut, 2001, Non - existence results and estimates for some nonlinear elliptic equa - tions, Anal Math, 84

Berestycki, 1997, Monotonicity for elliptic equations in unbounded Lipschitz domains Pure, Appl Math, 50

Du, 2004, Symmetry for elliptic equations in a half - space without strong maximum principle, Proc Soc, 134

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Thông tin xuất bản

Advanced Nonlinear Studies

Tập 13 Số 1

115-136

Thông tin tác giả