On semipositone problems over RN for the fractional p-Laplace operator

Abstract

For N >= 1, s is an element of (0, 1), and p is an element of (1, Ns) we find a positive solution to the following class of semipositone problems associated with the fractional p-Laplace operator:(SP) (-Delta)psu = g(x)f(a)(u) in R-N, where g is an element of L-1(R-N) boolean AND L infinity(R-N) is a positive function, a > 0 is a parameter and fa is an element of C(R) is defined as fa(t) = f(t)-a for t >= 0, fa(t) = -a(t+1) for t is an element of [-1, 0], and f(a)(t) = 0 for t <= -1, where f is a non-negative continuous function on [0, infinity) satisfies f(0) = 0 with subcritical and Ambrosetti-Rabinowitz type growth. Depending on the range of a, we obtain the existence of a mountain pass solution to (SP) in Ds, p(R-N). Then, we prove mountain pass solutions are uniformly bounded with respect to a, over L (R)(R-N) for every r is an element of [Np/N-sp, infinity]. In addition, if p > 2N/N+2s, we establish that (SP) admits a non-negative mountain pass solution for each a near zero. Finally, under the assumption g(x) <= B/|x|(beta(p-1)+s)p for B > 0, x not equal 0, and beta is an element of (N-sp/p-1, N/p-1), we derive an explicit positive radial subsolution to (SP) and show that the non-negative solution is positive a.e. in R-N. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.