Variational approach to some damped Dirichlet problems with impulses (Q2868885)

The paper investigates the solvability of Dirichlet boundary value problems on the real interval \([0,T]\) with impulses at fixed time instants \(0<t_1<t_2<\dots <t_k<T\), \(k\in \mathbb{N}\). The studied equation NEWLINE\[NEWLINE -u''(t)+p(t)u'(t)+q(t)u(t)=f(t,u(t)),\;t\in [0,T]\setminus \{t_1,\dots, t_k\}, \eqno(1) NEWLINE\]NEWLINE is subject to the Dirichlet conditions NEWLINE\[NEWLINE u(0)=0,\quad u(T)=0, \eqno (2) NEWLINE\]NEWLINE and to the impulse conditions NEWLINE\[NEWLINE u'(t_j-)-u'(t_j+)=I_j(u(t_j)),\;j=1,\dots, k.\eqno(3) NEWLINE\]NEWLINE Here, \(p\in C[0,T]\), \(q\in L^\infty[0,T]\), \(I_j\in C(\mathbb R,\mathbb R)\), \(j=1,\dots,k\), and \(f\in C([0,T]\times\mathbb R,\mathbb R)\).NEWLINENEWLINEThe authors provide sufficient conditions for the existence of at least one weak solution of problem (1)--(3). The main results are contained in three theorems (Theorem 1.3, Theorem 1.4 and Theorem 1.5) and cover also differential equations with nonlinear terms having locally (with respect to \(t\)) superlinear behaviour for \(|u|\to \infty\). The proofs are based on variational methods and the critical point theory. Some examples illustrating the results are given in the last section.