Mohsen Timoumi, Homoclinics of superquadratic or asymptotically quadratic fourth order differential equations, Vol. 2022 (2022), Article ID 5, pp. 1-15

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DOI: 10.23952/cot.2022.5

Received March 10, 2022; Accepted May 24, 2022; Published June 11, 2022

Abstract. In this paper, we study the existence of homoclinic and ground state homoclinic solutions for the fourth order differential equation $u^{(4)}(x)+2q(x)u^{(3)}(x)+\big(q^{2}(x)+q'(x)+\omega\big)u''(x)+\omega q(x)u'(x)+a(x)u(x)=f(x,u(x))$

when the potential $F(x,u)=\int^{u}_{0}f(x,v)dv$ is superquadratic or asymptotically quadratic in the second variable. We apply the critical point theory and variational methods. To the best of our knowledge, the existence of homoclinic solutions of this type of equations was not previously studied.

How to Cite this Article:
M. Timoumi, Homoclinics of superquadratic or asymptotically quadratic fourth order differential equations, Commun. Optim. Theory 2022 (2022) 5.