PUB - Publikationen an der Universität Bielefeld

We study pointwise behavior of positive solutions to nonlinear integral equations, and related inequalities, of the type u(x) - integral(Omega) G(x, y) g(u(y))d sigma(y) = h, where (Omega, sigma) is a locally compact measure space, G(x, y) : Omega x Omega -> [0, +infinity] is a kernel that satisfies a weak form of the maximum principle, h >= 0 is a measurable function, and g : [0, infinity) -> [0, infinity) is a monotone increasing function. In the special case where G is Green's function of the Laplacian (or fractional Laplacian) that satisfies the maximum principle and h = 1, a typical global pointwise bound for any supersolution u > 0 is given by u(x) >= F-1(G sigma(x)), x is an element of Omega, where F(t) := integral(t)(1) ds/g(s), t >= 1, and necessarily G sigma(x) < F(infinity) = integral(+infinity)(1) ds/g(s), for every x is an element of Omega such that u(x) < infinity. This problem is motivated by the semilinear fractional Laplace equation (-Delta)(alpha/2) u - g(u)sigma = mu in Omega, u = 0 in Omega(c), with measure coefficients sigma, mu, where g(u) = u(q), q > 0, and 0 < alpha < n, in domains Omega subset of R-n, or Riemannian manifolds, with positive Green's function G. In a similar way, we treat positive solutions to the equation u(x) + integral(Omega) G(x, y) g(u(y))d sigma(y) = h, and the corresponding fractional Laplace equation (-Delta)(alpha/2) u + g(u)sigma = mu, with a monotone decreasing function g, in particular g(u) = u(q), q < 0.