The behavior of positive solutions of the nonlocal elliptic problem \(\begin{aligned} {\left\{ \begin{array}{ll} \Delta u+\lambda u-[a(x)+\varepsilon ]u^p-b(x)u\int _\Omega c(y)u^r(y)\,dy=0,& x\in \Omega ,\\ u=0,& x\in \partial \Omega , \end{array}\right. } \end{aligned}\) is studied, where \(\Omega \subset {\mathbb {R}}^N (N\ge 1\) ) is a bounded domain with smooth boundary, \(\lambda >0\) , \(p>1\) and \(r\ge 1\) , a, \(b\in C^\alpha (\bar{\Omega })\) with \(0<\alpha <1\) are nonnegative functions, \(c\in L^\infty (\Omega )\) satisfies \(c>0\) , and \(\varepsilon >0\) is a small perturbation parameter. It is known for the case \(\varepsilon =0\) that spatial degeneracies in the functions a and b can fundamentally change the structure of positive solutions. In order to find the essential role of spatial degeneracies, we examine the asymptotic profiles as \(\varepsilon \rightarrow 0\) and show that the degeneracy of a and b induces sharp patterns of positive solutions in the region where both coefficients a and b exhibit degeneracies. In sharp contrast to classical reaction-diffusion equations, the presence of the nonlocal term causes the positive solution to decay to zero as \(\varepsilon \rightarrow 0\) in the region where \(b>0\) .
