EXISTENCE OF WEAK SOLUTIONS FOR $p$-LAPLACIAN PROBLEM WITH IMPULSIVE EFFECTS

Abstract

In this paper, we shall adopt topological degree theory and critical point theory to study the existence of weak solutions for the $p$-Laplacian Dirichlet boundary value problem \[\begin{cases} -(|u'|^{p-2} u')' = f(t,u), \; \textrm{in } \Omega, \\ u(0) = u(1) = 0, \end{cases}\] with impulsive conditions $u(t_j^+) - u(t_j^-) = 0$, $\Delta |u'(t_j)|^{p-2} u'(t_j) = I_j(u(t_j))$, $j=1,2,\ldots,n$, where $p \in (1,+\infty)$, $\Omega = (0,1) \backslash \{t_1,\ldots,t_n\}$, $f \in C([0,1] \times \mathbb{R}, \mathbb{R})$ and $I_j \in C(\mathbb{R}, \mathbb{R})$ ($j=1,2,\ldots,n$).