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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?

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