## An application of variational methods to Dirichlet boundary value problem with impulses.(English)Zbl 1191.34039

The authors consider the impulsive Dirichlet boundary value problem
\begin{aligned} &-u''(t) + \lambda u(t) = f(t,u(t)) + p(t), \quad t \in [0,T], \\ &\triangle u'(t_j) = I_j(u(t_j)),\;j = 1,\dots,k, \\ &u(0) = u(T) = 0, \end{aligned}
where $$0 < t_1 < \dots < t_k < T$$ are impulse instants, the impulsive functions $$I_j : {\mathbb R} \to {\mathbb R}$$ and the right-hand side $$f$$ are continuous, $$p \in L^2[0,T]$$, $$\lambda > -\pi^2/T^2$$. Sufficient conditions for the existence of at least one and infinitely many weak solutions are found. The proofs are based on the critical points theory.

### MSC:

 34B37 Boundary value problems with impulses for ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 47J30 Variational methods involving nonlinear operators
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