## 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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### References:

 [1] M. Benchohra, J. Henderson, S. Ntouyas: Impulsive Differential Equations and Inclusions. Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006. · Zbl 1130.34003 [2] Y. sChen, S. Levine, M. Rao: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66 (2006), 1383–1406. · Zbl 1102.49010 [3] X. L. Fan, Q.H. Zhang: Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear Anal., Theory Methods Appl. 52 (2003), 1843–1852. · Zbl 1146.35353 [4] X. L. Fan, D. Zhao: On the spaces L p(x)({$$\Omega$$}) and W m,p(x)({$$\Omega$$}). J. Math. Anal. Appl. 263 (2001), 424–446. · Zbl 1028.46041 [5] M. Feng, D. Xie: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J. Comput. Appl. Math. 223 (2009), 438–448. · Zbl 1159.34022 [6] W. Ge, Y. Tian: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc., II. Ser. 51 (2008), 509–527. · Zbl 1163.34015 [7] P. Harjulehto, P. Hästö, U.V. Lê, M. Nuortio: Overview of differential equations with non-standard growth. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 4551–4574. · Zbl 1188.35072 [8] T. Jankowski: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Appl. Math. Comput. 202 (2008), 550–561. · Zbl 1255.34025 [9] J. Mawhin: Problèmes de Dirichlet Variationnels non Lineaires. Les Presses de l’Universite de Montreal, Montreal, 1987. (In French.) [10] J. J. Nieto, D. O’Regan: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10 (2009), 680–690. · Zbl 1167.34318 [11] M. Ružička: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000. · Zbl 0962.76001 [12] K. Teng, Ch. Zhang: Existence of solution to boundary value problem for impulsive differential equations. Nonlinear Anal., Real World Appl. 11 (2010), 4431–4441. · Zbl 1207.34034 [13] J. L. Troutman: Variational Calculus with Elementary Convexity. With the assistence of W.Hrusa. Undergraduate Texts in Mathematics. Springer, New York, 1983. · Zbl 0523.49001 [14] H. Zhang, Z. Li: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal., Real World Appl. 11 (2010), 67–78. · Zbl 1186.34089 [15] Z. Zhang, R. Yuan: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 11 (2010), 155–162. · Zbl 1191.34039 [16] V.V. Zhikov: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 29 (1987), 33–66; translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50 (1986), 675–710. · Zbl 0599.49031
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