Stationary sign changing solutions for an inhomogeneous nonlocal problem
The authors prove the existence of a sign changing solution \(q(x)\) of the nonlocal equation NEWLINE\[NEWLINE \int\limits_R J\left(\frac{x-y}{g(y)}\right)\frac{u(y)}{g(y)}dy - u(x) = 0, \, x \in R, \tag{1} NEWLINE\]NEWLINE where \(J\) is an even, compactly supported, Holder continuous probability kernel, \(g\) is a continuous function, bounded and bounded away from zero in \(R.\) The solution \(q(x)\) has the following properties: 1) it is strictly positive when \(x > k\) and strictly negative for \(x < -k,\) provided that \(k\) is chosen large enough, 2) the solution \(q(x)\) verifies the inequalities \(a_1 \leq q(x)/x \leq a_2\) for positive constants \(a_1,\) \(a_2\) and large \(|x|.\) The authors investigate the properties of solutions with exponensial and potential growth.
scientific article; zbMATH DE number 6039333