article

Title: Stationary sign changing solutions for an inhomogeneous nonlocal problem

Authors: Carmen Cortazar, Manuel Elgueta, Jorge Garcia-Melian and Salome Martinez

Issue: Volume 60 (2011), Issue 1, 209-232

Abstract:

$We consider the following nonlocal equation: $\int_{\mathbb{R}}J\left(\frac{x-y}{g(y)}\right)\frac{u(y)}{g(y)}\dy-u(x) = 0\quad x\in\mathbb{R}$, where $J$ is an even, compactly supported, H\"older continuous probability kernel, $g$ is a continuous function, bounded and bounded away from zero in $\mathbb{R}$. 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 $a_1\leq q(x)/x\leq a_2$ for positive constants $a_1$, $a_2$ and large $|x|$. 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.$