Du, Yihong; Li, Shujie Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations. (English) Zbl 1161.35388 Adv. Differ. Equ. 10, No. 8, 841-860 (2005). This paper establishes general Liouville theorem for nonlinear elliptic equations of the type \[ -\Delta u=h(x_ 1)f(u) \quad \text{in}\;\mathbb R^ N \;\;(N\geq 2). \] In particular, if \(h(t)=t| t| ^{\alpha}\) or \(h(t)=(t^+)^{\alpha}\) for \(\alpha>0\), and \(f(u)=u^{p}\) for \(p>1\), then the bounded, nonnegative solution must be \(u\equiv 0\). The authors further show how the Liouville theorem can be used to derive a priori estimates for the nonlinear elliptic problem \[ -\Delta u=\lambda u +a(x) u^ p \;\;\text{in}\;\Omega, \quad u| _{\partial\Omega}=0, \] where \(\Omega\) is bounded domain in \(\mathbb R^ N\), \(1<p<N^{*}\), \(N^{*}=(N+2)/(N-2)\) if \(N\geq 3\), and \(N^{*}=\infty\) for \(N=2\), \(a(x)\) is a continuous sign-changing function, satisfying certain condition near its zero set. Reviewer: Ning Su (Beijing) Cited in 29 Documents MSC: 35J60 Nonlinear elliptic equations 35B45 A priori estimates in context of PDEs Keywords:Liouville theorem; elliptic equation; a priori estimate; moving plane method PDFBibTeX XMLCite \textit{Y. Du} and \textit{S. Li}, Adv. Differ. Equ. 10, No. 8, 841--860 (2005; Zbl 1161.35388)