Opuscula Math. 40, no. 1 (2020), 21-36
https://doi.org/10.7494/OpMath.2020.40.1.21
Opuscula Mathematica
Some multiplicity results of homoclinic solutions for second order Hamiltonian systems
Sara Barile
Addolorata Salvatore
Abstract. We look for homoclinic solutions \(q:\mathbb{R} \rightarrow \mathbb{R}^N\) to the class of second order Hamiltonian systems \[-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}\] where \(L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}\) and \(a,b: \mathbb{R}\rightarrow \mathbb{R}\) are positive bounded functions, \(G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}\) are positive homogeneous functions and \(f:\mathbb{R}\rightarrow\mathbb{R}^N\). Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if \(f\equiv 0\) and the existence of at least three solutions if \(f\) is not trivial but small enough.
Keywords: second order Hamiltonian systems, homoclinic solutions, variational methods, compact embeddings.
Mathematics Subject Classification: 34C37, 58E05, 70H05.
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- Sara Barile (corresponding author)
https://orcid.org/0000-0001-8921-7844
- Università degli Studi di Bari Aldo Moro, Dipartimento di Matematica, Via E. Orabona 4, 70125 Bari, Italy
- Addolorata Salvatore
https://orcid.org/0000-0001-9132-6468
- Università degli Studi di Bari Aldo Moro, Dipartimento di Matematica, Via E. Orabona 4, 70125 Bari, Italy
- Communicated by Vicentiu D. Radulescu.
- Received: 2019-01-05.
- Revised: 2019-03-26.
- Accepted: 201903-26.
- Published online: 2020-02-17.