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Keywords:
Orlicz--Sobolev space; $\varphi$-Laplacian; eigenvalue; Rayleigh quotient
Orlicz--Sobolev space; $\varphi$-Laplacian; eigenvalue; Rayleigh quotient
Summary:
In this paper we consider the $\varphi\,$-Laplacian problem with Dirichlet boundary condition, $$ -{\rm div}\Big(\varphi(|\nabla u|) \frac{\nabla u}{|\nabla u |}\Big)=\lambda g(\cdot) \varphi(u) \qquad\text{in } \Omega, \lambda\in{\mathbb{R}} \text{ and } u\vert_{\partial\Omega}=0. $$ The term $\varphi$ is a real odd and increasing homeomorphism, $g$ is a nonnegative function in $L^{\infty}(\Omega)$ and $\Omega\subseteq\mathbb{R}^N$ is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both a minimizer and an eigenfunction of the $\varphi$-Laplacian. It turns out that if, in addition, a suitable $\Delta_2$-condition holds then any number greater than or equal to the minimum of the Rayleigh quotient is an eigenvalue of the differential equation.
References:
[1] Adams R. A., Fournier J. J. F.: Sobolev Spaces. Pure and Applied Mathematics (Amsterdam), 140, Elsevier Academic Press, Amsterdam, 2003. MR 2424078 | Zbl 1098.46001
[2] Arriagada W., Huentutripay J.: Blow-up rates of large solutions for a $\phi$-Laplacian problem with gradient term. Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 4, 669–689. MR 3233749
[3] Arriagada W., Huentutripay J.: Characterization of a homogeneous Orlicz space. Electron. J. Differential Equations 2017 (2017), Paper No. 49, 17 pages. MR 3625929
[4] Arriagada W., Huentutripay J.: Regularity, positivity and asymptotic vanishing of solutions of a $\phi$-Laplacian. An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 25 (2017), no. 3, 59–72. MR 3747154
[5] Brezis H.: Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983 (French). MR 0697382
[6] Diaz G., Letelier R.: Unbounded solutions of one-dimensional quasilinear elliptic equations. Appl. Anal. 48 (1993), no. 1–4, 173–203. DOI 10.1080/00036819308840157 | MR 1278131
[7] Donaldson T. K., Trudinger N. S.: Orlicz–Sobolev spaces and imbedding theorems. J. Functional Analysis 8 (1971), 52–75. DOI 10.1016/0022-1236(71)90018-8 | MR 0301500
[8] Drábek P., Manásevich R.: On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian. Differential Integral Equations 12 (1999), no. 6, 773–788. MR 1728030
[9] Drábek P., Robinson S. B.: Resonance problems for the $p$-Laplacian. J. Funct. Anal. 169 (1999), no. 1, 189–200. DOI 10.1006/jfan.1999.3501 | MR 1726752
[10] Drábek P., Rother W.: Nonlinear eigenvalue problem for $p$-Laplacian in $\mathbb{R}^N$. Mathematische Nachrichten 173 (1995), no. 1, 131–139. DOI 10.1002/mana.19951730109 | MR 1336957
[11] Fan X., Zhang Q., Zhao D.: Eigenvalues of $p(x)$-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302 (2005), no. 2, 306–317. DOI 10.1016/j.jmaa.2003.11.020 | MR 2107835
[12] Fukagai N., Ito M., Narukawa K.: Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on ${\mathbb{R}}^N$. Funkcial. Ekvac. 49 (2006), no. 2, 235–267. DOI 10.1619/fesi.49.235 | MR 2271234
[13] Garcia Azorero J. P., Peral Alonso I.: Existence and nonuniqueness for the $p$-Laplacian: nonlinear eigenvalues. Comm. Partial Differential Equations 12 (1987), no. 12, 1389–1403. MR 0912211
[14] García-Huidobro M., Le V. K., Manásevich R., Schmitt K.: On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting. NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 2, 207–225. DOI 10.1007/s000300050073 | MR 1694787
[15] Gossez J.-P.: Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients. Trans. Amer. Math. Soc. 190 (1974), 163–205. DOI 10.1090/S0002-9947-1974-0342854-2 | MR 0342854
[16] Gossez J.-P.: Orlicz–Sobolev spaces and nonlinear elliptic boundary value problems. Nonlinear Analysis, Function Spaces and Applications, Proc. Spring School, Horni Bradlo, 1978, Teubner, Leipzig, 1979, pages 59–94. MR 0578910
[17] Gossez J.-P., Manásevich R.: On a nonlinear eigenvalue problem in Orlicz–Sobolev spaces. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 4, 891–909. MR 1926921
[18] Huentutripay J., Manásevich R.: Nonlinear eigenvalues for a quasilinear elliptic system in Orlicz–Sobolev spaces. J. Dynam. Differential Equations 18 (2006), no. 4, 901–921. DOI 10.1007/s10884-006-9049-7 | MR 2263407
[19] Krasnosel'skiĭ M. A., Rutic'kiĭ Ja. B.: Convex Functions and Orlicz Spaces. Noordhoff, Groningen, 1961.
[20] Lang S.: Real and Functional Analysis. Graduate Texts in Mathematics, 142, Springer, New York, 1993. MR 1216137 | Zbl 0831.46001
[21] Lê A.: Eigenvalue problems for the $p$-Laplacian. Nonlinear Anal. 64 (2006), no. 5, 1057–1099. MR 2196811
[22] Lieberman G. M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations. Comm. Partial Differential Equations 16 (1991), no. 2–3, 311–361. DOI 10.1080/03605309108820761 | MR 1104103
[23] Lindqvist P.: On the equation $ div(|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u=0$. Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164. DOI 10.1090/S0002-9939-1990-1007505-7 | MR 1007505
[24] Lindqvist P.: Note on a nonlinear eigenvalue problem. Rocky Mountain J. Math. 23 (1993), no. 1, 281–288. DOI 10.1216/rmjm/1181072623 | MR 1212743
[25] Mihăilescu M., Rădulescu V.: On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent. Proc. Amer. Math. Soc. 135 (2007), no. 9, 2929–2937. DOI 10.1090/S0002-9939-07-08815-6 | MR 2317971
[26] Mihăilescu M., Rădulescu V.: Eigenvalue problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces. Anal. Appl. (Singap.) 6 (2008), no. 1, 83–98. DOI 10.1142/S0219530508001067 | MR 2380887
[27] Mihăilescu M., Rădulescu V., Repovš D.: On a non-homogeneous eigenvalue problem involving a potential: an Orlicz–Sobolev space setting. J. Math. Pures Appl. (9) 93 (2010), no. 2, 132–148. DOI 10.1016/j.matpur.2009.06.004 | MR 2584738
[28] Mustonen V., Tienari M.: An eigenvalue problem for generalized Laplacian in Orlicz–Sobolev spaces. Proc. Roy. Soc. Edinburgh A 129 (1999), no. 1, 153–163. MR 1669197
[29] Pick L., Kufner A., John O., Fučík S.: Function Spaces. Vol. 1, De Gruyter Series in Nonlinear Analysis and Applications, 14, Walter de Gruyter, Berlin, 2013. MR 3024912
[30] Rădulescu V. D.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal. 121 (2015), 336–369. MR 3348928
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