• Corpus ID: 211020583

An alternative theorem for gradient systems.

  title={An alternative theorem for gradient systems.},
  author={Biagio Ricceri},
  journal={arXiv: Analysis of PDEs},
  • B. Ricceri
  • Published 4 February 2020
  • Mathematics, Physics
  • arXiv: Analysis of PDEs
Here is one of the result obtained in this paper: Let $\Omega\subset {\bf R}^2$ be a smooth bounded domain and let $F, G : {\bf R}\to {\bf R}$ be two $C^1$ functions satisfying the following conditions: $(i)$ for some $p>0$, one has $$\limsup_{|\xi|\to +\infty}{{|F'(\xi)|+|G'(\xi)|}\over {|\xi|^p}}<+\infty\ ;$$ $(ii)$ $F$ is non-negative, non-decreasing, $\lim_{\xi\to +\infty}{{F(\xi)}\over {\xi^2}}=0$, $\lim_{\xi\to 0^+}{{F(\xi)}\over {\xi^2}}=+\infty$ and the function $\xi\to {{F'(\xi)}\over… 
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