Anisotropic semipositone quasilinear problems (Q2396682)

The aim of this article is to prove the existence of positive solutions of the degenerate (or singular) quasilinear boundary value problem \[ \left\{ {{\displaystyle -\text{div}(a(x)|\nabla u|^{p-2}\nabla u) = \lambda b(x)f(u)\qquad\text{in}\quad B_R,}\atop{ \displaystyle \lim_{|x|\to R^-}u(x) = 0, }}\right. \eqno{(1.1)} \] where \(N \geq 2\), \(p \in (1, N)\) and \(\lambda> 0\) is a positive parameter. For every \(R \in (0, \infty],\, B_R \subset \mathbb R^N\) denotes the open ball of radius \(R\) centered at the origin with the convention that \(B_R := \mathbb R^N\) when \(R = \infty\). The properties and the behavior of the functions \(a\) and \(b\) in the domain \(B_R\) are described by the following hypotheses: (H1) The function \(a:B_R \to\mathbb R\) is continuous, radially symmetric and positive. (H2) For any \(r \in (0, R)\), the function \(|x|^{\frac{-p(N-1)}{p-1}} a^{\frac{- 1}{p-1}} (x)\), \(x \in B_R\), belongs to \(L^1(B_R \setminus B_r)\). (H3) The function \(b:B_R \to \mathbb R\) is measurable and there exist radially symmetric functions \(\underline{b} \in C(B_R \setminus\{0\})\) and \(\overline{b} \in L^m_{loc}(B_R)\), for some \(m > \frac{N }{p}>1\), such that \[ 0<\underline{b}(x) \leq b(x) \leq \overline{b}(x) \qquad\text{for \,\, a.e.}\quad x \in B_R, x \not= 0\,. \] (H4) There exist \(R_0 \in (0, R)\) and a constant \(K>0\) such that \[ \underline{b}(x) \leq K|x|^{\frac{-p(N-1)}{p-1}} a^{\frac{- 1}{p-1}} (x)\quad\text{for}\quad R_0 < |x| < R. \eqno{(1.2)} \] As for the nonlinearity \(f\), the authors assume the following: (H5) The function \(f:(0, \infty) \to R\) is continuous and there exists \(q \in (0, p - 1)\) such that \[ \mu_\infty := \lim_{\zeta\to \infty} \frac{f(\zeta)}{\zeta^q} \in (0,\infty). \] (H6) There exist \(\gamma \in (0, 1)\), \(K_0 > 0\) and \(\delta_0> 0 \) such that \[ \zeta^\gamma f(\zeta) \geq -K_0\quad\text{for\,\, all}\quad \zeta \in (0, \delta_0). \] The main result reads as follows: Theorem 1.1. Let \(a(x), b(x)\) satisfy (H1) - (H4). Assume in addition that \(f\) satisfies (H5) and (H6). Then, there exists \(\lambda_\star>0\) large such that, for every \(\lambda \geq \lambda_\star\), problem (1.1) has at least one solution \(u_\lambda \). The proof of this theorem is based on the monotone iterations technique and on the existence of a pair of sub- and supersolutions to (1.1).