Positive periodic solution for indefinite singular Liénard equation with p-Laplacian

Abstract

Abstract The efficient conditions guaranteeing the existence of positive T-periodic solution to the p-Laplacian–Liénard equation (ϕp(x′(t)))′+f(x(t))x′(t)+α1(t)g(x(t))=α2(t)xμ(t), $$\bigl(\phi _{p}\bigl(x'(t)\bigr) \bigr)'+f \bigl(x(t)\bigr)x'(t)+\alpha _{1}(t)g\bigl(x(t)\bigr)= \frac{ \alpha _{2}(t)}{x^{\mu }(t)}, $$ are established in this paper. Here ϕp(s)=|s|p−2s $\phi _{p}(s)=|s|^{p-2}s$, p>1 $p>1$, α1,α2∈L([0,T],R) $\alpha _{1},\alpha _{2}\in L([0,T],{R}) $, f∈C(R+,R) $f\in C({R}_{+},{R})$ ( R+ ${R} _{+}$ stands for positive real numbers) with a singularity at x=0 $x=0$, g(x) $g(x)$ is continuous on (0;+∞) $(0;+\infty )$, μ is a constant with μ>0 $\mu >0$, the signs of α1 $\alpha _{1}$ and α2 $\alpha _{2} $ are allowed to change. The approach is based on the continuation theorem for p-Laplacian-like nonlinear systems obtained by Manásevich and Mawhin in (J. Differ. Equ. 145:367–393, 1998).

Keywords

• Singularity
• Continuation theorem
• Periodic solution