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Infinitely many homoclinic solutions for damped vibration systems with locally defined potentials

Mohsen Timoumi|Wafa Selmi
대한수학회 (The Korean Mathematical Society)|2022.07|대한수학회논문집 (Communications of the KMS)|제37권 제3호|pp. 693-703|DOI 10.4134/CKMS.c210008

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키워드

Damped vibration systemsvariational methodshomoclinic solutionslocally defined potentialssymmetric mountain pass theorem
주제분류수학

초록

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$\ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\ \forall t\in\mathbb{R}, \leqno(1)$$ where $q\in C(\mathbb{R},\mathbb{R})$, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric and positive definite matix-valued function and $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. The novelty of this paper is that, assuming that $L$ is bounded from below unnecessarily coercive at infinity, and $W$ is only locally defined near the origin with respect to the second variable, we show that $(1)$ possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.

Abstract

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$\ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\ \forall t\in\mathbb{R}, \leqno(1)$$ where $q\in C(\mathbb{R},\mathbb{R})$, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric and positive definite matix-valued function and $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. The novelty of this paper is that, assuming that $L$ is bounded from below unnecessarily coercive at infinity, and $W$ is only locally defined near the origin with respect to the second variable, we show that $(1)$ possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.

참고문헌 (28)

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