An alternative theorem for gradient systems

Biagio Ricceri

Published 2020-02-04Version 1

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 $\xi\to {{F'(\xi)}\over {\xi}}$ is strictly decreasing in $]0,+\infty[$ ; $(iii)$ $G$ is positive and convex. Then, for every positive function $\alpha\in L^{\infty}(\Omega)$, the problem $$\cases {-\Delta u=\alpha(x) G(v(x))F'(u) & in $\Omega$ \cr & \cr -\Delta v=-\alpha(x) F(u(x))G'(v) & in $\Omega$ \cr & \cr u=v=0 & on $\partial\Omega$\cr} $$ has a non-zero weak solution belonging to $L^{\infty}(\Omega)\times L^{\infty}(\Omega)$.