A note on the nonexistence of solutions for prescribed mean curvature equations on a ball. (English) Zbl 1235.35144

Summary: We prove the nonexistence of solutions for a prescribed mean curvature equation \[ \begin{cases} -\text{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=\lambda|u|^{p-1}u,\quad & x\in{\mathcal B}_R\subseteq\mathbb{R}^n,\\ u=0,\quad & x\in\partial{\mathcal B}_R,\end{cases} \] when \(p\geq 1\) and the positive parameter \(\lambda\) is small. The result extends theorems of K. Narukawa and T. Suzuki [Boll. Unione Mat. Ital., VII. Ser., B 4, No. 1, 223–241 (1990; Zbl 0702.76024)], and R. Finn [in: Continuum mechanics and related problems of analysis. Proceedings of the international symposium. Dedicated to the centenary of academician N. Muskhelishvili. June 6-11, 1991, Tbilisi (Georgia). Tbilisi: Metsniereba. 127–139 (1993; Zbl 0884.35017)], from the case of \(n=2\), \(p=1\) to all \(n\geq 2\), \(p\geq 1\). Moreover, our proof is very simple and the result is not limited to positive (and negative) solutions. We also show that a similar result for positive solutions is still true if \(|u|^{p-1}u\) is replaced by the exponential nonlinearity \(e^u-1\).


35J93 Quasilinear elliptic equations with mean curvature operator
35B32 Bifurcations in context of PDEs
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