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Nonexistence results for elliptic equations with gradient terms. (English) Zbl 1335.35028

This paper considers the elliptic problem \(-\Delta u+|\nabla u|^q=\lambda f(u)\) in the complement of a ball in \(\mathbb{R}^N\), \(N\geq 2\), with Dirichlet conditions on the boundary of the ball. The exponent \(q\) is in \((1,\infty)\), the nonlinearity \(f\) is defined in \((0,\infty)\), is continuous, nondecreasing and positive, while \(\lambda >0\) is a parameter. Under suitable assumptions on \(f\) near zero or infinity, the authors obtain some nonexistence results for positive super-solutions (in the weak sense), depending on the relative values of \(q\) and \(N/(N-1)\), and on the parameter \(\lambda\). In particular, in contrast to previous results which required \(f\) to behave like a power near zero, they allow logarithmic perturbations of them. Their analysis relies in reducing the problem to a radially symmetric situation by means of the method of upper and lower solutions.

MSC:

35J15 Second-order elliptic equations
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35B33 Critical exponents in context of PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
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