##
**Multiple solutions for a classical problem in the calculus of variations.**
*(English)*
Zbl 0872.49001

In this paper, the authors consider the classical problem of calculus of variations,
\[
\int^T_0 L(t,x,\dot x)ds\tag{\(*\)}
\]
over all suitable paths satisfying \(x(0)=x_0\), \(x(T)=x_1\) in \(\mathbb{R}^N\).

This problem has been widely investigated. Whenever the Lagrangian \(L(t,x,y)\) satisfies the coercivity assumption \(L(t,x,y)\geq a+b|y|^p\) for some constant \(p>1\), the integral \((*)\) is bounded from below and the infimum is attained. This result was proved by L. Tonelli, who introduced the notion of semicontinuity for this purpose.

Little or no attention has been paid to the case when \(L\) is not coercive, so that the integral is unbounded both from below and from above. Typical of this situation is: \[ L(t,x,y)= {1\over 2} |y|^2- {1\over p} |x|^p. \] In such a case, there can be no question of minimizing the integral, so critical points have to be looked for the integral \((*)\). Such critical points satisfy the Euler-Lagrange equation \[ \begin{cases} \ddot x+|x|^{p-2} x=0,\\ x(0)= x_0,\;x(T)=x_1.\end{cases} \] In this paper, the authors look for solutions of the boundary value problem \[ \begin{cases} \ddot x+ V'(x)=0,\\ x(0)= x_0,\;x(T)=x_1,\end{cases} \] where the potential \(V(x): \mathbb{R}^N\to\mathbb{R}\) satisfies \(V(x)\sim|x|^p\) at infinity. The integral \((*)\) associated to this problem is \(I(x)= \int^T_0[\dot x^2- V(x)]ds\) on a suitable Sobolev space of curves joining \(x_0\) and \(x_1\) in \(\mathbb{R}^N\).

It is shown in the paper that if the potential \(V(x)\) is even and if \(2<p<4\), the functional \(I\) has infinitely many critical points, obtaining solutions for the variational problem \((*)\) without any coercivity assumption on the Lagrangian \(L(t,x,y)\).

The proof is based in the application to the functional \(I\) of critical point theory for unbounded functionals.

This problem has been widely investigated. Whenever the Lagrangian \(L(t,x,y)\) satisfies the coercivity assumption \(L(t,x,y)\geq a+b|y|^p\) for some constant \(p>1\), the integral \((*)\) is bounded from below and the infimum is attained. This result was proved by L. Tonelli, who introduced the notion of semicontinuity for this purpose.

Little or no attention has been paid to the case when \(L\) is not coercive, so that the integral is unbounded both from below and from above. Typical of this situation is: \[ L(t,x,y)= {1\over 2} |y|^2- {1\over p} |x|^p. \] In such a case, there can be no question of minimizing the integral, so critical points have to be looked for the integral \((*)\). Such critical points satisfy the Euler-Lagrange equation \[ \begin{cases} \ddot x+|x|^{p-2} x=0,\\ x(0)= x_0,\;x(T)=x_1.\end{cases} \] In this paper, the authors look for solutions of the boundary value problem \[ \begin{cases} \ddot x+ V'(x)=0,\\ x(0)= x_0,\;x(T)=x_1,\end{cases} \] where the potential \(V(x): \mathbb{R}^N\to\mathbb{R}\) satisfies \(V(x)\sim|x|^p\) at infinity. The integral \((*)\) associated to this problem is \(I(x)= \int^T_0[\dot x^2- V(x)]ds\) on a suitable Sobolev space of curves joining \(x_0\) and \(x_1\) in \(\mathbb{R}^N\).

It is shown in the paper that if the potential \(V(x)\) is even and if \(2<p<4\), the functional \(I\) has infinitely many critical points, obtaining solutions for the variational problem \((*)\) without any coercivity assumption on the Lagrangian \(L(t,x,y)\).

The proof is based in the application to the functional \(I\) of critical point theory for unbounded functionals.

Reviewer: A.Masiello (Bari)

### MSC:

49J05 | Existence theories for free problems in one independent variable |

49K15 | Optimality conditions for problems involving ordinary differential equations |

49J45 | Methods involving semicontinuity and convergence; relaxation |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |