Abstract
The main purpose of this paper is to investigate the existence of a positive periodic solution for a prescribed mean curvature generalized Liénard equation with a singularity (weak and strong singularities of attractive type, or weak and strong singularities of repulsive type). Our proof is based on an extension of Mawhin’s continuation theorem.
Highlights
In [9], Zhang discussed the existence of a positive periodic solution to equation (1.1), where √ u (t) = u (t), p = 2, f (t, u) = f (u) and e(t) ≡ 0, g satisfies a semilinear condition and 1+(u (t))2 has a strong singularity of repulsive type, i.e., lim g(u) = –∞ and lim g(ν) dν = +∞
In the following, applying Lemma 2.1, we prove the existence of a positive periodic solution for equation (1.1) with a singularity of repulsive type
3 Positive periodic solution for equation (1.1) when p > 1 and p = 2 In the following, by Lemma 2.1 and Theorem 2.1, we prove the existence of a positive periodic solution for equation (1.1) with a singularity of repulsive type
Summary
We consider the following p-Laplacian prescribed mean curvature Liénard equation:. During the past 30 years, the problem of existence of positive periodic solutions to Liénard equations with singularity was extensively studied by many researchers [1,2,3,4,5,6,7,8,9]. In [9], Zhang discussed the existence of a positive periodic solution to equation (1.1), where √ u (t) = u (t), p = 2, f (t, u) = f (u) and e(t) ≡ 0, g satisfies a semilinear condition and 1+(u (t)) has a strong singularity of repulsive type, i.e., lim g(u) = –∞ and lim g(ν) dν = +∞. As far as we know, prescribed mean curvature √ u (t) of u(t) appears in different
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