Variational method to the impulsive equation with Neumann boundary conditions
The authors consider the impulsive boundary value problem \[ \begin{aligned} -(p(t)u'(t))' + r(t)u'(t) + q(t)u(t) &= g(t,u(t)),\quad \text{a.e. } t \in [0,1],\;t \neq t_k, \\ -\triangle(p(t_k)u'(t_k)) &= I_k(u(t_k)),\quad k = 1,\ldots,p-1,\\ u'(0+) &= u'(1-) = 0, \end{aligned} \] where \(0 < t_1 < \ldots < t_{p-1} < 1\); \(p \in C^1([0,1])\), \(r\), \(q \in C([0,1])\), \(g\), \(I_k\) are continuous functions. Sufficient conditions for the existence of at least one (two, infinitely many) solutions are obtained by using a variational approach and critical point theorems.
scientific article; zbMATH DE number 5626150
scientific article