Stationary Sign Changing Solutions for an Inhomogeneous Nonlocal Problem
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Date
2011
Journal Title
Journal ISSN
Volume Title
Publisher
INDIANA UNIV MATH JOURNAL
Abstract
We consider the following nonlocal equation:
integral(R)J (x - y/g(y)) u(y)/g(y) dy - u(x) = 0 x epsilon R,
where J is an even, compactly supported, Holder continuous probability kernel, g is a continuous function, bounded and bounded away from zero in R. We prove the existence of a sign changing solution q(x) which is strictly positive when x > K and strictly negative for x < -K, provided that K is chosen large enough. The solution q(x) so constructed verifies a(1) <= q(x)/x <= a(2) for positive constants a(1), a(2) and large vertical bar x vertical bar. In addition, we show that all solutions with polynomial growth are of the form Aq(x) + Bp (x), where p is the unique normalized positive (bounded) solution of the equation. In the particular case where g = 1 we also construct solutions with exponential growth.
integral(R)J (x - y/g(y)) u(y)/g(y) dy - u(x) = 0 x epsilon R,
where J is an even, compactly supported, Holder continuous probability kernel, g is a continuous function, bounded and bounded away from zero in R. We prove the existence of a sign changing solution q(x) which is strictly positive when x > K and strictly negative for x < -K, provided that K is chosen large enough. The solution q(x) so constructed verifies a(1) <= q(x)/x <= a(2) for positive constants a(1), a(2) and large vertical bar x vertical bar. In addition, we show that all solutions with polynomial growth are of the form Aq(x) + Bp (x), where p is the unique normalized positive (bounded) solution of the equation. In the particular case where g = 1 we also construct solutions with exponential growth.
Description
Keywords
nonlocal diffusion, sign changing solution, uniqueness, ASYMPTOTIC-BEHAVIOR, DIRICHLET PROBLEM, TRAVELING-WAVES, EQUATION, UNIQUENESS, EXISTENCE, MODEL, DISPERSAL