On the uniqueness of positive solutions of a quasilinear equation containing a weighted p-Laplacian, the superlinear case
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Date
2008
Journal Title
Journal ISSN
Volume Title
Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
Abstract
We consider the quasilinear equation of the form
-Delta(p)u = K(vertical bar x vertical bar) f(u), x epsilon R-n, n > p > 1, (P)
where Delta(p)u := div(|del u|(p- 2)del u) is the degenerate p-Laplace operator and the weight K is a positive C-1 function defined in R+. We deal with the case in which f epsilon C[0, infinity) has one zero at u(0) > 0, is non positive and not identically 0 in (0, u(0)), and is locally Lipschitz, positive and satisfies some superlinear growth assumption in (u(0), infinity). We carefully study the behavior of the solution of the corresponding initial value problem for the radial version of the quasilinear equation, as well as the behavior of its derivative with respect to the initial value. Combining, as Cortazar, Felmer and Elgueta, comparison arguments due to Coffman and Kwong, with some separation results, we show that any zero of the solutions of the initial value problem is monotone decreasing with respect to the initial value, which leads immediately the uniqueness of positive radial ground states, and the uniqueness of positive radial solutions of the Dirichlet problem on a ball.
-Delta(p)u = K(vertical bar x vertical bar) f(u), x epsilon R-n, n > p > 1, (P)
where Delta(p)u := div(|del u|(p- 2)del u) is the degenerate p-Laplace operator and the weight K is a positive C-1 function defined in R+. We deal with the case in which f epsilon C[0, infinity) has one zero at u(0) > 0, is non positive and not identically 0 in (0, u(0)), and is locally Lipschitz, positive and satisfies some superlinear growth assumption in (u(0), infinity). We carefully study the behavior of the solution of the corresponding initial value problem for the radial version of the quasilinear equation, as well as the behavior of its derivative with respect to the initial value. Combining, as Cortazar, Felmer and Elgueta, comparison arguments due to Coffman and Kwong, with some separation results, we show that any zero of the solutions of the initial value problem is monotone decreasing with respect to the initial value, which leads immediately the uniqueness of positive radial ground states, and the uniqueness of positive radial solutions of the Dirichlet problem on a ball.
Description
Keywords
positive solutions, uniqueness, superlinear, separation, SEMILINEAR ELLIPTIC-EQUATIONS, GROUND-STATE SOLUTIONS, RADIAL SOLUTIONS, NONNEGATIVE SOLUTIONS, ASYMPTOTIC-BEHAVIOR, RN, DELTA-U+F(U)=0, EXISTENCE, R(N), SYMMETRY