Sharp patterns of positive solutions for some weighted semilinear elliptic problems

Abstract

This paper deals with the semilinear elliptic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda m(x)u-[a(x)+\varepsilon b(x)]u^p &{}\text { in } \Omega ,\\ Bu=0 &{}\text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where $p>1$, $\lambda >0$, $m,a,b\in C({\bar{\Omega }})$, with $a \gneq 0$, $b\gneq 0$, $\Omega$ is a bounded $C^{2}$ domain of ${\mathbb {R}}^N$ ($N\ge 1$), B is a general classical mixed boundary operator, and $\varepsilon \ge 0$. Thus, a(x) and b(x) can vanish on some subdomain of $\Omega$ and the weight function m(x) can change sign in $\Omega$. Through this paper we are always considering classical solutions. First, we characterize the existence of positive solutions of this problem in the special case when $\varepsilon =0$. Then, we investigate the sharp patterns of the positive solutions when $\varepsilon \downarrow 0$ and $\varepsilon \uparrow \infty$. Our study reveals how the existence of sharp profiles is determined by the behavior of b(x).