Multiplicity of positive solutions for $N$-Laplace equation in a ball

Abstract

Let $N\ge 2,$ $ B_{R}\subset \mathbb R^{N}$ denote the open ball of radius $R$ about the origin. Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous map which behaves like $e^{s^{p}}$ for some $p\in [1,\frac{N}{N-1}]$ as $s\rightarrow \infty$ and like $s^{q}$ for some $q\in (0, N-1)$ as $s\rightarrow 0$. Then we show that there exists $\Lambda \in (0,\infty)$ such that the following problem, $$(P_{\lambda})\hspace{3cm}\left\{ \begin{array}{cllll}\left. \begin{array}{rllll} -\Delta_{N}u & = & \lambda f(u)\\ u & > & 0 \end{array}\right\} \;\; \text{in} \; B_{1}(0), \\[2mm] u\;\;=\;\; 0 \;\;\text{on}\;\; \partial B_{1}(0), \end{array}\right. $$ admits at least two solutions for all $\lambda \in (0,\Lambda)$ and no solution for $\lambda >\Lambda$.