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Existence of solutions for the $p-$Laplacian with crossing nonlinearity
Higher order Melnikov function for a quartic hamiltonian with cuspidal loop
| 1. | Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China |
| 2. | Department of Mathematics, Zhongshan (Sun Yat-sen) University, 510275, Guangzhou, China |
$X_\epsilon=(H_y+\epsilon f(x,y,\epsilon))\frac{\partial}{\partial x}+ (-H_x+\epsilon g(x,y,\epsilon))\frac{\partial}{\partial y},$
where the Hamiltonian $H(x,y)=\frac{1}{2}y^2+U(x)$ has one center and one cuspidal loop, $deg U(x)=4$. In present paper we find an upper bound for the number of zeros of the $k$th order Melnikov function $M_k(h)$ for arbitrary polynomials $f(x,y,\epsilon)$ and $g(x,y,\epsilon)$.
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