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Nonexistence of positive solutions of Neumann problems for elliptic inequalities of the mean curvature type. (English) Zbl 0957.35055
From the introduction: This paper concerns elliptic boundary value problems of the form $\begin{cases} Mu\equiv\text{div}\left[ {Du\over(1+|Du|^2)^\alpha} \right]\geq p(x)f(u), \quad & x\in\Omega,\\ D_vu\leq 0,\quad & x\in \partial\Omega, \end{cases} \tag{P}$ where $$x=(x_i)$$, $$Du=(D_iu)$$, $$D_iu= \partial u/\partial x_i$$ for $$i=1,2,\dots,N$$, $$N\geq 2$$, $$\Omega\subset \mathbb{R}^N$$ is an exterior domain whose boundary $$\partial\Omega$$ is of class $$C^2$$, $$v: \partial\Omega \to\mathbb{R}^N$$ is a vector field pointing outward with respect to $$\Omega$$, and $$D_vu$$ denotes the derivative of $$u$$ along the vector $$v$$. Throughout the paper we always assume the following: $$0\leq\alpha\leq 1/2$$; $$p: \overline\Omega \to(0,\infty)$$ is continuous; $$f:(0,\infty) \to(0,\infty)$$ is locally Lipschitz continuous and strictly increasing with $$\lim_{u\to \infty} f(u) =\infty$$. We intend to give several answers to the following questions affirmatively:
(I) When $$0\leq\alpha <1/2$$, it is possible to weaken the superlinear condition $\int^\infty_1 \left(\int^u_0 f(s)ds \right)^{-1/2}du <\infty \quad\text{and}\quad \int^1_0\left( \int^u_0f(s) ds\right)^{-1/2}du= \infty; \tag{1}$ and,
(II) When $$\alpha=1/2$$, are the superlinear conditions like (1) unnecessary to guarantee the nonexistence of positive solutions of (P) provided that $$p$$ is sufficiently large?

##### MSC:
 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs 35J60 Nonlinear elliptic equations 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
##### Keywords:
positive solution; Neumann problem; mean curvature