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Keywords:
$p(x)$-Laplace operator; generalized Lebesgue-Sobolev space; critical point; weak solution; electrorheological fluid
Summary:
We study the boundary value problem $-{\mathrm div}((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in ${\mathbb{R}} ^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x)=\max \lbrace p_1(x),p_2(x)\rbrace $ for any $x\in \overline{\Omega }$ or $m(x)<q(x)< \frac{N\cdot m(x)}{(N-m(x))}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda $ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ${\mathbb{Z}} _2$-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.
References:
[1] E. Acerbi and G. Mingione: Regularity results for a class of functionals with nonstandard growth. Arch. Rational Mech. Anal. 156 (2001), 121–140. MR 1814973
[2] C. O. Alves and M. A. S. Souto: Existence of solutions for a class of problems involving the $p(x)$-Laplacian. Progress in Nonlinear Differential Equations and Their Applications 66 (2005), 17–32. MR 2187792
[3] A. Ambrosetti and P. H. Rabinowitz: Dual variational methods in critical point theory and applications. J. Functional Analysis 14 (1973), 349–381. MR 0370183
[4] H. Brezis: Analyse fonctionnelle: théorie et applications. Masson, Paris, 1992. MR 0697382
[5] L. Diening: Theorical and numerical results for electrorheological fluids. Ph.D. thesis, University of Freiburg, Germany, 2002.
[6] D. E. Edmunds, J. Lang and A. Nekvinda: On $L^{p(x)}$ norms. Proc. Roy. Soc. London Ser. A 455 (1999), 219–225. MR 1700499
[7] D. E. Edmunds and J. Rákosník: Density of smooth functions in $W^{k,p(x)}(\Omega )$. Proc. Roy. Soc. London Ser. A 437 (1992), 229–236. MR 1177754
[8] D. E. Edmunds and J. Rákosník: Sobolev embedding with variable exponent. Studia Math. 143 (2000), 267–293. MR 1815935
[9] X. Fan, J. Shen and D. Zhao: Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega )$. J. Math. Anal. Appl. 262 (2001), 749–760. MR 1859337
[10] X. L. Fan and Q. H. Zhang: Existence of solutions for $p(x)$-Laplacian Dirichlet problem. Nonlinear Anal. 52 (2003), 1843–1852. MR 1954585
[11] X. L. Fan, Q. H. Zhang and D. Zhao: Eigenvalues of $p(x)$-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302 (2005), 306–317. MR 2107835
[12] X. L. Fan and D. Zhao: On the spaces $L^{p(x)}(\Omega )$ and $W^{m,p(x)}(\Omega )$. J. Math. Anal. Appl. 263 (2001), 424–446. MR 1866056
[13] T. C. Halsey: Electrorheological fluids. Science 258 (1992), 761–766.
[14] O. Kováčik and J. Rákosník: On spaces $L^{p(x)}$ and $W^{1,p(x)}$. Czech. Math. J. 41 (1991), 592–618. MR 1134951
[15] H. G. Leopold: Embedding on function spaces of variable order of differentiation in function spaces of variable order of integration. Czech. Math. J. 49 (1999), 633–644. MR 1708338
[16] P. Marcellini: Regularity and existence of solutions of elliptic equations with $p,q$ growth conditions. J. Differential Equations 90 (1991), 1–30. MR 1094446 | Zbl 0724.35043
[17] M. Mihăilescu: Elliptic problems in variable exponent spaces. Bull. Austral. Math. Soc. 74 (2006), 197–206.
[18] M. Mihăilescu and V. Rădulescu: A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc. Roy. Soc. London Ser. A 462 (2006), 2625–2641. MR 2253555
[19] M. Mihăilescu and V. Rădulescu: On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent. Proceedings of the American Mathematical Society 135 (2007), no. 9, 2929–2937. MR 2317971
[20] J. Musielak: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, Vol. 1034, Springer, Berlin, 1983. MR 0724434 | Zbl 0557.46020
[21] J. Musielak and W. Orlicz: On modular spaces. Studia Math. 18 (1959), 49–65. MR 0101487
[22] H. Nakano: Modulared Semi-Ordered Linear Spaces. Maruzen Co., Ltd., Tokyo, 1950. MR 0038565 | Zbl 0041.23401
[23] W. Orlicz: Über konjugierte Exponentenfolgen. Studia Math. 3 (1931), 200–211. Zbl 0003.25203
[24] C. Pfeiffer, C. Mavroidis, Y. Bar-Cohen and B. Dolgin: Electrorheological fluid based force feedback device, in Proceedings of the 1999 SPIE Telemanipulator and Telepresence Technologies VI Conference (Boston, MA), Vol. 3840. 1999, pp. 88–99.
[25] P. Rabinowitz: Minimax methods in critical point theory with applications to differential equations, Expository Lectures from the CBMS Regional Conference held at the University of Miami, American Mathematical Society, Providence, RI. 1984. MR 0845785
[26] M. Ruzicka: Electrorheological Fluids Modeling and Mathematical Theory. Springer-Verlag, Berlin, 2002. MR 1810360
[27] I. Sharapudinov: On the topology of the space $L^{p(t)}([0;1])$. Matem. Zametki 26 (1978), 613–632. MR 0552723
[28] M. Struwe: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Heidelberg, 1996. MR 1411681 | Zbl 0864.49001
[29] I. Tsenov: Generalization of the problem of best approximation of a function in the space $L^s$. Uch. Zap. Dagestan Gos. Univ. 7 (1961), 25–37.
[30] W. M. Winslow: Induced fibration of suspensions. J. Appl. Phys. 20 (1949), 1137–1140.
[31] Q. Zhang: A strong maximum principle for differential equations with nonstandard $p(x)$-growth conditions. J. Math. Anal. Appl. 312 (2005), 24–32. MR 2175201 | Zbl 1162.35374
[32] V. Zhikov: Averaging of functionals in the calculus of variations and elasticity. Math. USSR Izv. 29 (1987), 33–66.
[33] V. Zhikov: On passing to the limit in nonlinear variational problem. Math. Sb. 183 (1992), 47–84. MR 1187249
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