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Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian

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This work was supported by National Natural Science Foundation of China (No.11671287, No.11771105) and Guangxi Natural Science Foundation (No.2017GXNSFFA198012)

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  • In this paper we prove the existence and multiplicity of subharmonic solutions for nonlinear equations involving the singular $ \phi $-Laplacian. Such equations are in particular motivated by the one-dimensional mean curvature problems and by the acceleration of a relativistic particle of mass one at rest moving on a straight line. Our approach is based on phase-plane analysis and an application of the Poincaré-Birkhoff twist theorem.

    Mathematics Subject Classification: Primary: 34C25, 34B15; Secondary: 37C25.

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  • Figure 1.  The relations between the fundamental period $ T_h $ and "energy" $ h $ with various potentials: (a) Toda potential $ G(x) = k(x+{\mathrm{e}}^{-x}) $ with $ k = 1 $; (b) Sublinear potential $ G(x) = \frac{4}{5}|x|^{5/4} $; (c) Harmonic potential $ G(x) = \frac{1}{2}x^{2} $; (d) Superlinear potential $ G(x) = \frac{2}{5}|x|^{5/2} $

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