On the minimal solution for quasilinear degenerate elliptic equation and its blow-up

Abstract

In this article we shall investigate the minimal and the extremal solutions of quasilinear elliptic equation with a positive nonlinear term in the right hand side. More precisely we shall study the boundary value problem \[ \left\{ \begin{array}{lcl} L_{p}(u) = \lambda f(u),&\text{in}&\Omega ,\\ u = 0,&\text{on}&\partial\Omega , \end{array} \right. \] where $\lambda$ is a nonnegative parameter, $\Omega$ is a domain of $R^{N}$ and $L_{p}(\cdot )(p > 1)$ is the $p$-Laplace operator defined by $L_{p}(\cdot ) =-\mathrm{div}(|\nabla\cdot |^{p-2}\nabla\cdot))$. We assume that $f(t)$ is increasing on $[0,\infty )$ and strictly convex with $f(0) > 0$. Under some additional conditions, we first establish the existence of the minimal solution $u_{\lambda}$ and the extremal solution $u^{*}$ to this equation and study their behaviors in connection with the linearized operator given by $L_{p}'(u)(\cdot ) =-\mathrm{div}\left( |\nabla u|^{p-2}\left(\nabla\cdot +(p-2)\frac{(\nabla u, \nabla\cdot )}{|\nabla u|^{2}}\nabla u\right)\right)$. The minimal solution $u_{\lambda}\in C^{1,\sigma }(\Bar{\Omega })$ is defined as the smallest solution among all possible classical solutions, and the extremal solution is defined as an increasing limit of $u_{\lambda}$ in $W_{0}^{1,p}(\Omega )$ as $\lambda \to \lambda^{*}$ (the extremal value). Though $L_{p}'(u_{\lambda})(\cdot )$ is, roughly speaking, a degenerate elliptic operator, it is shown that $L_{p}'(u_{\lambda})(\cdot )$ has a compact inverse from $L^{2}(\Omega )$ to itself if $u_{\lambda}$ is minimal. Moreover the self-adjoint operator $L_{p}'(u_{\lambda})(\cdot )-\lambda f'(u_{\lambda})$ on $L^{2}(\Omega )$ has a positive first eigenvalue if $\lambda$ is sufficiently small and a nonnegative first eigenvalue for any $\lambda\in (0,\lambda ^{*})$. Finally in Section 10 we give the characterizations of the extremal solution which are essentially depend upon the value of $p$ and the topology of $\Omega$ (see Theorem 10.1 and subsequent Propositions). When $\Omega$ is a ball, we investigate these problems rather precisely using the weighted Hardy type inequality with a sharp missing term.