EXPLICIT NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF NONNEGATIVE SOLUTIONS OF A p-LAPLACIAN BLOW-UP PROBLEM

Abstract

We establish explicit necessary and sufficient conditions for the existence of nonnegative solutions of the $p$-Laplacian boundary blow-up problem \begin{equation*} \left\{ \begin{array}{l} \left( \varphi _{p}(u^{\prime }(x))\right) ^{\prime }=\lambda f(u(x)), \, 0 \lt x \lt 1, \\ \lim\limits_{x\rightarrow 0^{+}}u(x)=\infty =\lim\limits_{x\rightarrow 1^{-}}u(x), \end{array} \right. \end{equation*} where $p>1$, $\varphi _{p}\left( y\right) =\left\vert y\right\vert ^{p-2}y$ and $\left( \varphi _{p}(u^{\prime })\right) ^{\prime }$ is the one-dimensional $p$-Laplacian, $\lambda $ is a positive bifurcation parameter and $f$ is a locally Lipschitz continuous function on $[0,\infty)$. The gap is extremely small between the explicit necessary condition and the explicit sufficient condition for the existence of nonnegative solutions. Our results improve and extend some main results of Anuradha, Brown and Shivaji [2] and of Wang [30] from $p=2$ to any $p\gt 1$.