Multiplicity results for a class of boundary value problems with impulsive effects (Q2805381)

The authors consider the impulsive boundary value problem of the form NEWLINENEWLINE\[NEWLINE-u'' = f(x, u), \quad x \in (0, T)\backslash \{x_1, x_2, \cdots , x_m\},NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINE\Delta u'(x_j) = J_j(u(x_j)), \quad j = 1, 2, \cdots , m.NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINEu(0) = u(T) = 0, NEWLINE\]NEWLINE where \(m\) is a positive integer, \(0 = x_0 < x_1 < \cdots < x_m < x_{m+1} =T, \;f \in C((0, T) \times \mathbb{R}, \mathbb{R}), \Delta u'(x_j) = u'(x_j^+) - u'(x_j^-)\) where \(u'(x_j^-)\) and \(u'(x_j^+)\) denote the left and right limit of \(u'(t)\) at \(t = t_j\) respectively, and \(J_j(u) \) are continuous functions on \(\mathbb{R}\).NEWLINENEWLINEThe authors study the existence of multiple nontrivial solutions of the above problem with sublinear impulsive effects. That means, there exist \(a_j, b_j > 0\) and \(0 \leq \gamma_j < 1\) such that \(|J_j(u) | \leq a_j |u|^{\gamma_j} + b_j, \quad j = 1,2, \cdots, m.\)NEWLINENEWLINEUnder some natural assumptions, by using Morse theory in combination with the minimax arguments, the authors obtain the existence of at least three nontrivial solutions. An illustrative example is provided.