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Impulsive boundary value problems for \(p(t)\)-Laplacian’s via critical point theory. (English) Zbl 1274.34083

The paper provides conditions ensuring the existence of a solution to the boundary value problem \[ \frac {\text{d}}{{\text{d}}\,t}\Big (\big | u'(t)\big | ^{p(t)-2}u'(t)\Big )=f(t,u), \quad u(0)=u(\pi )=0, \]
\[ \lim _{\tau \to t_j+}\Big (\big | u'(\tau )\big | ^{p(\tau )-2}u'(\tau )\Big )- \lim _{\tau \to t_j-}\Big (\big | u'(\tau )\big | ^{p(\tau )-2}u'(\tau )\Big ) =I_j(u(t_j)),\;j=1,2,\dots ,m, \] where \(p\in C([0,\pi ],(1,\infty )),\,f\: [0,\pi ]\times \mathbb {R}\to \mathbb {R}\) is a Carathéodory function, \(I_j\: \mathbb {R}\to \mathbb {R}\) are continuous and bounded and \(t_j,\) \(j=1,2,\dots ,m,\) are fixed and such that \(0<t_1<t_2<\dots <t_m<\pi .\)
The main idea is to consider weak formulation of the problem, then to use the Mountain Pass Principle to show the existence of a weak solution which, due to the fundamental lemma of the calculus of variations, coincides with the classical solution.
Reviewer: M. Tvrdý (Praha)

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
47J30 Variational methods involving nonlinear operators
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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