Positive profiles for the perturbed nonlocal dispersal equations (Q6622647)

In this paper, the authors study the sharp patterns of the perturbed nonlocal dispersal equation\N\[\N \int_{\Omega}J(x-y)u(y)dy -u(x)+\lambda u(x)+ \epsilon b(x)u(x) - a(x)u^{p}(x)=0,\qquad x\in \bar{\Omega},\tag{1}\N\]\Nwhere \(\Omega \subset \mathbb R^{N}(N \geqslant 1)\) is a smooth bounded domain, \(p>1, \lambda\in \mathbb R\) and \(\epsilon > 0\) are parameters. Moreover, \(J\in C(\mathbb R^{n})\) such that \(J >0\) in \(B_{1}\) (the unit ball), \(J=0\) in \(\mathbb R^{N} \setminus B_{1} , J(x) =J(-x)\) in \(B_{1}\) with \(\int_{\mathbb R^{N}} J(x)dx = 1\) and \(a,b\in C(\bar{\Omega}),a(x)> 0\) and \(b(x)\neq 0\) for \(x\in\bar{\Omega}\). The nonlocal eigenvalue problem\N\[\N \int _{\Omega}J (x-y)u(y)dy-u(x) + \epsilon b(x)u(x)=-\lambda u(x),\qquad x\in \bar{\Omega}\tag{2}\N\]\Nplays an important role in the study of positive solutions for \((1)\).\N\NA real number \(\lambda \in \mathbb R\) is called a principal eigenvalue of \((2)\) if there is \(u\in C(\bar{\Omega})\) with \(u(x)> 0\) for \(x\in \bar{\Omega}\) such that \((2)\) holds. The principal spectrum point of \((2)\), denoted by \(\lambda_{PS}(\epsilon,b,\Omega)\), is defined by \(\lambda_{PS}(\epsilon,b,\Omega) =\operatorname{Sup} \{ \mathrm{Re} \lambda: \lambda\in \sigma(\varPsi)\}\), where \(\sigma(\varPsi)\) stands for the spectrum set of \(\varPsi\) acting on \(C(\bar{\Omega}),\) where \(\varPsi u=\int_{\Omega} J(x-y)u(y)dy-u(x)+\epsilon b(x)u(x).\) The generalized principal eigenvalue of \(\varPsi\) or \((2)\), denoted by \(\lambda_{GP} (\epsilon,b,\Omega)\) is defined by \(\lambda_{GP}(\epsilon,b,\Omega) = \sup\{ \lambda\in \mathbb R:\exists u\in C (\bar{\Omega}),u > 0\text{ such that }\varPsi u\leq -\lambda u(x)\}\).\N\NIn the first part of this paper, the authors investigate the asymptotic behaviour of the principal spectrum point or generalized principal eigenvalue of \((2)\) with respect to parameter \(\epsilon\). The main conclusion will be used in the study of asymptotic profiles for the positive solutions of \((1)\). However, in this paper, no example is given to illustrate the results obtained.