Positive solutions for a class of equations with a p-Laplace like operator and weights
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Date
2009
Journal Title
Journal ISSN
Volume Title
Publisher
PERGAMON-ELSEVIER SCIENCE LTD
Abstract
In this paper we will study the problem of existence of positive solutions to the problem
(D) {(a(r)phi(u'))' + b(r)g(u) = 0. a.e. in (0, R), lim a(r)phi(u'(r)) = 0, u(R) = 0, r -> 0
where phi is an odd increasing homeomorphism of R and g is an element of C(R) is such that g(z) > 0 for all z > 0 with g(0) = 0. The functions a and b, that we will refer to as weight functions, satisfy a(r) > 0, b(r) > 0 for all r is an element of (0, R] and are such that a, b is an element of C-1(0, R) boolean AND L-1(0, R). If phi has the form phi(z) = zm(vertical bar z vertical bar), and
a(r) = r(N-1) (a) over tilde (r), b(r) = r(N-1) (b) over tilde (r), N >= 2
then solutions of problem (D) provide solutions with radial symmetry for the problem
(P) {div((a) over tilde(vertical bar x vertical bar)m(vertical bar del u vertical bar)del u) + (b) over tilde(vertical bar x vertical bar)g(u) = 0, x is an element of Omega, u = 0, x is an element of partial derivative Omega,
where Omega = B(0, R) denotes the ball with center 0 and radius R > 0 in R-N. (C) 2008 Elsevier Ltd. All rights reserved.
(D) {(a(r)phi(u'))' + b(r)g(u) = 0. a.e. in (0, R), lim a(r)phi(u'(r)) = 0, u(R) = 0, r -> 0
where phi is an odd increasing homeomorphism of R and g is an element of C(R) is such that g(z) > 0 for all z > 0 with g(0) = 0. The functions a and b, that we will refer to as weight functions, satisfy a(r) > 0, b(r) > 0 for all r is an element of (0, R] and are such that a, b is an element of C-1(0, R) boolean AND L-1(0, R). If phi has the form phi(z) = zm(vertical bar z vertical bar), and
a(r) = r(N-1) (a) over tilde (r), b(r) = r(N-1) (b) over tilde (r), N >= 2
then solutions of problem (D) provide solutions with radial symmetry for the problem
(P) {div((a) over tilde(vertical bar x vertical bar)m(vertical bar del u vertical bar)del u) + (b) over tilde(vertical bar x vertical bar)g(u) = 0, x is an element of Omega, u = 0, x is an element of partial derivative Omega,
where Omega = B(0, R) denotes the ball with center 0 and radius R > 0 in R-N. (C) 2008 Elsevier Ltd. All rights reserved.
Description
Keywords
phi-Laplacian, Pohozaev identities, Generalized Matukuma equations, Criticality, RADIAL SOLUTIONS