Multibump solutions for a class of Lagrangian systems slowly oscillating at infinity
The authors study the existence of infinitely many solutions homoclinic to the origin \(u=0\), \(\dot u=0\), to a class of second-order Lagrangian systems of the form \[ -\ddot u+u =\alpha (t)W(u), \qquad u\in \mathbb{R}^n,\quad t\in \mathbb{R}. \tag{1} \] It is assumed that (1) \(\alpha\in C^1(\mathbb{R},\mathbb{R})\), \(W\in C^2(\mathbb{R}^n,\mathbb{R})\), (2) there exists \(\theta >2\) such that \(0<\theta W(x)\leq \nabla W(x)x\) for any \(x\in \mathbb{R}^n\setminus \{0\}\), (3) \(\nabla W(x)x<{\nabla}^2 W(x)xx\) for any \( x\in \mathbb{R}^n\setminus \{0\}\), (4) there exist \(\overline{a}\) and \(\underline{a}>0\) such that \(\overline{a}\geq \alpha (t) \geq \underline{a}>0\) for any \(t\in \mathbb{R}\), (5) \(\underline{\alpha}=\liminf _{t\to +\infty}\alpha (t)< \limsup _{t\to +\infty}\overline{\alpha} (t)=\overline{\alpha}\) and \(\lim _{t\to +\infty}\dot{\alpha} (t)=0\). The main result is the following theorem: If hypotheses (1)--(5) hold then equation (1) admits infinitely many so-called multibump solutions. More precisely, there exist \(\bar\delta >0\), a sequence of disjoint intervals \(Q_{j}\) in \(\mathbb{R}^+\) with \(| Q_{j}| \to +\infty\), and an increasing sequence of indices \((\hat \jmath _n)\) such that given any increasing sequence of indices \((j_n)\) with \(j_i\geq \hat\jmath _i\) \((i\in \mathbb{N})\) and \(\sigma\in\{ 0,1\}^\mathbb{N}\) there exists a solution \(u_{j,\sigma}\in C^2(\mathbb{R},\mathbb{R}^n)\) to equation (1) verifying: \[ | u_{j,\sigma}| < {\overline{\delta}\over 2} \text{ for all }t\in \mathbb{R}\setminus\bigcup _{\{i| \sigma _{i}=1\}}Q_{j_{i}}, \qquad\| u_{j,\sigma}\| _{L^{\infty}(Q_{j_{i}})}\geq \overline\delta \text{ if }\sigma _i=1. \] \(u_{j,\sigma}\) is a homoclinic solution to equation (1) whenever \(\sigma _i=0\) definitely. The proof uses variation techniques and is based on a localization procedure related to the time dependence of the Lagrangian.
scientific article; zbMATH DE number 1271564