Sublinear positone and semipositone problems on the exterior of a ball in R2

Abstract

We study positive solutions to problems of the form,(0.1)$\{\begin{array}{cc}-\Delta u& =\lambda K\left(x\right)f\left(u\right)\text{ in }{B}_1^c,\\ u\left(x\right)& =0\text{ on }\partial B_1,\end{array}$ where $B_1^c=\{x\in R^2:|x|>1\}$, λ is a positive parameter, $K:B_1^c\to R^{+}$ belongs to a class of Hölder continuous functions which satisfy certain decay assumptions and $f:(0,\infty )\to R$ belongs to a class of Hölder continuous functions which are sublinear. For a class of positone problems of the form (0.1), we establish the existence of multiple positive solutions for a range of the parameter λ and uniqueness of positive solutions for either sufficiently large or small values of λ. Additionally, we obtain an existence result for a semipositone problem of the form (0.1). Our results extend the study of similar problems on exterior domains in $R^n$, $n>2$.