Multiple solutions for a class of superquadratic fractional Hamiltonian systems

Yıl 2018, , 186 - 195, 30.09.2018

Mohsen Timoumi

https://doi.org/10.32323/ujma.388067

Cited By: 2

Öz

In this paper, we are concerned with the existence of solutions for a class of fractional Hamiltonian systems \[\left\{ \begin{array}{l} _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in\mathbb{R}\\ u\in H^{\alpha}(\mathbb{R},\ \mathbb{R}^{N}), \end{array}\right. \] where $_{t}D_{\infty}^{\alpha}$ and $_{-\infty}D^{\alpha}_{t}$ are the Liouville-Weyl fractional derivatives of order $\frac{1}{2}<\alpha<1$, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric matrix-valued function and $W(t,x)\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. Applying a Symmetric Mountain Pass Theorem, we prove the existence of infinitely many solutions for (1) when $L$ is not required to be either uniformly positive definite or coercive and $W(t,x)$ satisfies some weaker superquadratic conditions at infinity in the second variable but does not satisfy the well-known Ambrosetti-Rabinowitz superquadratic growth condition.

Anahtar Kelimeler

Fractional Hamiltonian systems, Variational methods, Symmetric Mountain Pass Theorem

Kaynakça

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