On the existence of periodic solutions for a class of \(p\)-Laplacian system
The existence of periodic solutions of the Liénard-type system \[ (\varphi_p(u'))' + \frac{d}{dt} \operatorname {grad} F(u) + \operatorname {grad} G(u) = e(t), \] where \(\varphi_p(u_1,\dots, u_n) = (| u_1| ^{p-2}u_1, \dots, | u_n| ^{p-2}u_n)\) with \( p>1 \), is proved in the following three cases: 1. \( \text{Hess} F\) is sign-definite and \(\text{grad} G\) satisfies a sign and growth condition, roughly saying that \(\text{grad} G\) lies in some sense ``to the right of the eigenvalue \(0\)''; 2. \(\text{grad} G\) satisfies a sign and growth condition, roughly saying that \(\text{grad} G\) lies ``to the left of the eigenvalue \(0\)''; 3. \(\text{grad} G\) satisfies a sign and growth condition, roughly saying that \(\text{grad} G\) lies ``to the left of the first positive eigenvalue'' and some components of \(\text{grad} G -\bar e\) (\(\bar e\) being the mean value of \(e\)) lie ``to the left of the eigenvalue \(0\)'' and other ones ``to the right of the eigenvalue \(0\)''.
scientific article; zbMATH DE number 5078917