On the structure of positive radial solutions to an equation containing a p-Laplacian with weight
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Date
2006
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ACADEMIC PRESS INC ELSEVIER SCIENCE
Abstract
Let A. B : (0, infinity) -> (0 infinity) be two given weight functions and consider the equation (P) - div (A(vertical bar x vertical bar)vertical bar del u vertical bar(p-2)del u) = B (vertical bar x vertical bar)vertical bar u vertical bar(q-2)u, x is an element of R-n, where q > p > 1. By considering positive radial solutions to this equation that are bounded, we are led to study the initial value problem
{-(a(r)vertical bar u'vertical bar(p-2) u') = b(r) (u(+))(q-1), r is an element of (0, infinity) u(0) = alpha > 0, lim(r -> 0) a (r)vertical bar u'(r)vertical bar(p-1)=0,
where a(r) = r((N-1)) A(r) and b(r) = r((N-1)) B(r). By means of two key functions in and B-q defined below, we obtain several new results that allow us to classify solutions to this initial value problem as being respectively crossing, slowly decaying, or rapidly decaying. We also generalize several results in Clement et al. (Asymptotic Anal. 17 (1998) 13-29), Kawano et al. (Funkcial. Ekvac 36 (1993) 121-145), Yanagida and Yotsutani (Arch. Rational Mech. Anal. 124 (1993) 239-259), Yanagida and Yotsutani (J. Differential Equations 115 (1995) 477-502), Yanagida and Yotsutani.
{-(a(r)vertical bar u'vertical bar(p-2) u') = b(r) (u(+))(q-1), r is an element of (0, infinity) u(0) = alpha > 0, lim(r -> 0) a (r)vertical bar u'(r)vertical bar(p-1)=0,
where a(r) = r((N-1)) A(r) and b(r) = r((N-1)) B(r). By means of two key functions in and B-q defined below, we obtain several new results that allow us to classify solutions to this initial value problem as being respectively crossing, slowly decaying, or rapidly decaying. We also generalize several results in Clement et al. (Asymptotic Anal. 17 (1998) 13-29), Kawano et al. (Funkcial. Ekvac 36 (1993) 121-145), Yanagida and Yotsutani (Arch. Rational Mech. Anal. 124 (1993) 239-259), Yanagida and Yotsutani (J. Differential Equations 115 (1995) 477-502), Yanagida and Yotsutani.
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Keywords
p-Laplacian, weight functions, energy identities, SEMILINEAR ELLIPTIC-EQUATIONS, DELTA-U+K(VERTICAL-BAR-X-VERTICAL-BAR) U(P)=0, ASYMPTOTIC-BEHAVIOR, SINGULAR SOLUTIONS, EXISTENCE, SYMMETRY, THEOREMS, R(N), RN