Positive solutions for a nonhomogeneous nonlocal logistic equation

In this paper, we deal with the nonhomogeneous nonlocal dispersal equation integral(Omega)J(x-y)u(y)dy-u(x)+lambda u(x)-a(x)u(p)(x)+Mf(x)=0,x is an element of Omega<overline>, where Omega is a bounded and smooth domain, p>1 and the parameters M,lambda>0. The kernel function J(x) is nonnegative and symmetric, and the coefficients a,f is an element of C(Omega<overline>) are assumed to be nonnegative. We are interested in the existence, uniqueness and asymptotic profiles of positive solutions. It is shown that the structure of positive solutions makes a fundamental change due to the nonhomogeneous effect. Moreover, we study the sharp profiles of positive solutions when there is a spatial degeneracy of a(x). The result exhibits that the blow-up profiles are determined by the degeneracy of a(x).(c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.