Multiple periodic solutions of some Liénard equations with \(p\)-Laplacian
Consider the periodic boundary value problems of the form \[ (| u'| ^{p-2}u')'+f(u)u'+ g(u)=e(t)+s,\quad u(0)-u(T)=0=u'(0)-u'(T), \tag{1} \] where \(p>1,\;f: \mathbb{R}\to \mathbb{R}\) is a continuous function. Assume that \(g,\;e,\;s\) satisfy the following conditions: (H1) \(\int_0^T e(t)dt=0;\) (H2) \(g(u)>0\) for all \(u\in \mathbb{R};\) (H3) \(g(\pm\infty)=\lim_{u\to\pm\infty} g(u)=0.\) By the properties of the Leray-Schauder degree, existence, non-existence and multiplicity of solutions of ({1}) are obtained. The main result is the following theorem. Theorem. If conditions (H1)-(H3) hold, then there exists \(s^*(e)\in (0, \sup_{\mathbb{R}} g]\) such that problem ({1}) has no, at least one or at least two solutions according to \(s\not\in(0, s^*(e)],\;s=s^*(e)\) or \(s\in (0, s^*(e))\).
scientific article; zbMATH DE number 5309613