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Oscillation of half-linear partial differential equations with first order terms. (English) Zbl 1052.35013
Summary: The half-linear partial differential equation with a first order term \[ P_\alpha[v]\equiv \nabla\cdot \bigl(A(x)|\nabla v |^{\alpha-1}\nabla v\bigr)+(\alpha+1)|\nabla v|^{\alpha-1}B(x)\cdot\nabla v+C(x)| v|^{\alpha-1}v=0 \] is studied, and sufficient conditions are derived for every solution \(v\) of \(P_\alpha [v]=0\) to be oscillatory in an unbounded domain \(\Omega\subset \mathbb R^n\). Reducing the oscillation problem for \(P_\alpha [v]=0\) to a one-dimensional oscillation problem for half-linear ordinary differential equations of the form \[ \bigl(r^{n-1}a(r)| y'|^{\alpha-1}y' \bigr)'+ r^{n-1}c(r)| y|^{\alpha-1} y=0, \] we obtain various oscillation results for \(P_\alpha[v]=0\).

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35G15 Boundary value problems for linear higher-order PDEs
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