Sharp patterns of positive solutions for some weighted semilinear elliptic problems

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).