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Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations. (English) Zbl 1161.35388

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)

MSC:

35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
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