Periodic solutions of forced dissipative \(p\)-Liénard equations with singularities
This paper deals with the periodic problem for \(p\)-Laplacian equations with a singularity \[ (| x'| ^{p-2}x')'+f(x)x'+g(x)=h(t),\;\;\; x(0)=x(T),\;\;x'(0)=x'(T),\tag{1} \] where \(p>1\), \(f\) is continuous, \(| f(x)| \geq c>0\) \(\forall x>0\), \(g\) is continuous, defined in \(]0,\infty[\) and singular at \(0\), and \(h\in L^2(0,T)\). The authors extend a result du to \textit{P. Habets} and \textit{L. Sanchez} [Proc. Am. Math. Soc. (1990; Zbl 0695.34036)] for the case \(p=2\): they prove the existence of a positive periodic solution of problem (1) under the conditions that \[ \limsup_{x\to0^+}g(x)<\bar h<\liminf_{x\to\infty}g(x), \] where \(\bar h\) is the mean value of \(h\) in \([0,T]\), and \[ \int_0^1g(u)\,du=-\infty. \] The proof essentially relies on apriori estimates and a continuation lemma for \(p\)-Laplacian equations proved by \textit{R. Manásévich} and \textit{J. Mawhin} [J. Differ. Equations 145 (1998; Zbl 0910.34051)].
scientific article; zbMATH DE number 5009726