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July  2020, 25(7): 2621-2637. doi: 10.3934/dcdsb.2020025

The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China

Received  February 2019 Revised  September 2019 Published  July 2020 Early access  April 2020

In this paper we study the asymptotic behaviour of the first eigenvalues $ \lambda^{1}_{p_{n}(\cdot)} $ and the corresponding eigenfunctions $ u_{n} $ of (1) as $ p_{n}(x)\rightarrow \infty $. Under adequate hypotheses on the sequence $ p_{n} $, we prove that $ \lambda^{1}_{p_{n}(\cdot)} $ converges to 1 and the positive first eigenfunctions $ u_{n} $, normalized by $ |u_{n}|_{L^{p_{n}(x)}(\partial \Omega)} = 1 $, converge, up to subsequences, to $ u_{\infty} $ uniformly in $ C^{\alpha}(\overline{\Omega}) $, for some $ 0<\alpha<1 $, where $ u_{\infty} $ is a nontrivial viscosity solution of a problem involving the $ \infty $-Laplacian subject to appropriate boundary conditions.

Citation: Lujuan Yu. The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2621-2637. doi: 10.3934/dcdsb.2020025
References:
[1]

F. Abdullayev and M. Bocea, The Robin eigenvalue problem for the $p(x)$-Laplacian as $p\rightarrow\infty$, Nonlinear Anal., 91 (2013), 32-45.  doi: 10.1016/j.na.2013.06.005.

[2]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140.  doi: 10.1007/s002050100117.

[3]

G. Barles, Fully non-linear Neumann type boundary conditions for second-order elliptic and parabolic equations, J. Differential Equations, 106 (1993), 90-106.  doi: 10.1006/jdeq.1993.1100.

[4]

M. Bocea and M. Mihǎilescu, The principal frequency of $\triangle_{\infty}$ as a limit of Rayleigh quotients involving Luxemburg norms, Bull. Sci.math., 138 (2014), 236-252.  doi: 10.1016/j.bulsci.2013.06.001.

[5]

M. Bocea and M. Mihǎilescu, $\Gamma$-convergence of power-law functionals with variable exponents, Nonlinear Anal., 73 (2010), 110-121.  doi: 10.1016/j.na.2010.03.004.

[6]

Y. M. ChenS. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[7]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Am. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[8]

S. G. Deng, Eigenvalues of the $p(x)$-Laplacian Steklov problem, J. Math. Anal. Appl., 339 (2008), 925-937.  doi: 10.1016/j.jmaa.2007.07.028.

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[10]

D. E. Edmunds and J. Rákosník, Sobolev embedding with variable exponent, Studia Math., 143 (2000), 267-293.  doi: 10.4064/sm-143-3-267-293.

[11]

X. L. Fan and X. Han, Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbb{R}^{N}$, Nonlinear Anal., 59 (2004), 173-188.  doi: 10.1016/j.na.2004.07.009.

[12]

X. L. FanJ. S. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.  doi: 10.1006/jmaa.2001.7618.

[13]

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.  doi: 10.1006/jmaa.2000.7617.

[14]

X. L. Fan, Y. Z. Zhao and Q. H. Zhang, A strong maximum principle for $p(x)$-Laplace equations, Chinese J. Contemp. Math., 24 (2003), 277–282. (Translation of Chinese Ann. Math. Ser. A, 24 (2003) 495–500.)

[15]

G. Franzina and P. Lindqvist, An eigenvalue problem with variable exponents, Nonlinear Anal., 85 (2013), 1-16.  doi: 10.1016/j.na.2013.02.011.

[16]

N. FukagaiM. Ito and K. Narukawa, Limit as $p\rightarrow\infty$ of $p$-Laplace eigenvalue problems and $L^{\infty}$ inequality of the Poincaré type, Differ. Integral Equations, 12 (1999), 183-206. 

[17]

P. HarjulehtoP. HästöÚ. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574.  doi: 10.1016/j.na.2010.02.033.

[18]

P. JuutinenP. Lindqvist and J. Manfredi, The $\infty$-eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89-105.  doi: 10.1007/s002050050157.

[19]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}(\Omega)$ and $W^{k, p(x)}(\Omega)$, Czechoslovak Math. J, 41 (1991), 592-618. 

[20]

A. Lê, On the first engenvalue of the Steklov eigenvalue problem for the infinity Laplacian, Electron. J. Differential Equations, 2006 (2006), 1-9. 

[21]

P. Lindqvist, Notes on the $p$-Laplace Equation, Report. University of Jyväskylä Department of Mathematics and Statistics, 102. University of Jyväskylä, Jyväskylä, 2006.

[22]

P. Lindqvist and T. Lukkari, A curious equation involving the $\infty$-Laplacian, Adv. Calc. Var., 3 (2010), 409-421. 

[23]

J. J. ManfrediJ. D. Rossi and J. M. Urbano, Limits as $p(x)\rightarrow\infty$ of $p(x)$-harmonic functions, Nonlinear Anal., 72 (2010), 309-315.  doi: 10.1016/j.na.2009.06.054.

[24]

J. J. ManfrediJ. D. Rossi and J. M. Urbano, $p(x)$-harmonic functions with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2581-2595.  doi: 10.1016/j.anihpc.2009.09.008.

[25]

J. Musielak, Orlicz Spaces and Modular Spaces, in: Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.

[26]

M. Pérez-Llanos and J. D. Rossi, Limits as $p(x)\rightarrow\infty$ of $p(x)$-harmonic functions with non-homogeneous Neumann boundary conditions, Nonlinear Elliptic Partial Differential Equations, 187–201, Contemp. Math., 540, Amer. Math. Soc., Providence, RI, 2011. doi: 10.1090/conm/540/10665.

[27]

M. Pérez-Llanos and J. D. Rossi, The behaviour of the $p(x)$-Laplacian eigenvalue problem as $p(x)\rightarrow \infty$, J. Math. Anal. Appl., 363 (2010), 502-511.  doi: 10.1016/j.jmaa.2009.09.044.

[28]

M. Råžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[29]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv., 29 (1987), 33-66. 

[30]

V. V. Zhikov, On Lavrentiev$'$s phenomenon, Russ. J. Math. Phys., 3 (1995), 249-269. 

show all references

References:
[1]

F. Abdullayev and M. Bocea, The Robin eigenvalue problem for the $p(x)$-Laplacian as $p\rightarrow\infty$, Nonlinear Anal., 91 (2013), 32-45.  doi: 10.1016/j.na.2013.06.005.

[2]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140.  doi: 10.1007/s002050100117.

[3]

G. Barles, Fully non-linear Neumann type boundary conditions for second-order elliptic and parabolic equations, J. Differential Equations, 106 (1993), 90-106.  doi: 10.1006/jdeq.1993.1100.

[4]

M. Bocea and M. Mihǎilescu, The principal frequency of $\triangle_{\infty}$ as a limit of Rayleigh quotients involving Luxemburg norms, Bull. Sci.math., 138 (2014), 236-252.  doi: 10.1016/j.bulsci.2013.06.001.

[5]

M. Bocea and M. Mihǎilescu, $\Gamma$-convergence of power-law functionals with variable exponents, Nonlinear Anal., 73 (2010), 110-121.  doi: 10.1016/j.na.2010.03.004.

[6]

Y. M. ChenS. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[7]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Am. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[8]

S. G. Deng, Eigenvalues of the $p(x)$-Laplacian Steklov problem, J. Math. Anal. Appl., 339 (2008), 925-937.  doi: 10.1016/j.jmaa.2007.07.028.

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[10]

D. E. Edmunds and J. Rákosník, Sobolev embedding with variable exponent, Studia Math., 143 (2000), 267-293.  doi: 10.4064/sm-143-3-267-293.

[11]

X. L. Fan and X. Han, Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbb{R}^{N}$, Nonlinear Anal., 59 (2004), 173-188.  doi: 10.1016/j.na.2004.07.009.

[12]

X. L. FanJ. S. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.  doi: 10.1006/jmaa.2001.7618.

[13]

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.  doi: 10.1006/jmaa.2000.7617.

[14]

X. L. Fan, Y. Z. Zhao and Q. H. Zhang, A strong maximum principle for $p(x)$-Laplace equations, Chinese J. Contemp. Math., 24 (2003), 277–282. (Translation of Chinese Ann. Math. Ser. A, 24 (2003) 495–500.)

[15]

G. Franzina and P. Lindqvist, An eigenvalue problem with variable exponents, Nonlinear Anal., 85 (2013), 1-16.  doi: 10.1016/j.na.2013.02.011.

[16]

N. FukagaiM. Ito and K. Narukawa, Limit as $p\rightarrow\infty$ of $p$-Laplace eigenvalue problems and $L^{\infty}$ inequality of the Poincaré type, Differ. Integral Equations, 12 (1999), 183-206. 

[17]

P. HarjulehtoP. HästöÚ. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574.  doi: 10.1016/j.na.2010.02.033.

[18]

P. JuutinenP. Lindqvist and J. Manfredi, The $\infty$-eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89-105.  doi: 10.1007/s002050050157.

[19]

O. Kováčik and J. Rákosník, On spaces $L^{p(x)}(\Omega)$ and $W^{k, p(x)}(\Omega)$, Czechoslovak Math. J, 41 (1991), 592-618. 

[20]

A. Lê, On the first engenvalue of the Steklov eigenvalue problem for the infinity Laplacian, Electron. J. Differential Equations, 2006 (2006), 1-9. 

[21]

P. Lindqvist, Notes on the $p$-Laplace Equation, Report. University of Jyväskylä Department of Mathematics and Statistics, 102. University of Jyväskylä, Jyväskylä, 2006.

[22]

P. Lindqvist and T. Lukkari, A curious equation involving the $\infty$-Laplacian, Adv. Calc. Var., 3 (2010), 409-421. 

[23]

J. J. ManfrediJ. D. Rossi and J. M. Urbano, Limits as $p(x)\rightarrow\infty$ of $p(x)$-harmonic functions, Nonlinear Anal., 72 (2010), 309-315.  doi: 10.1016/j.na.2009.06.054.

[24]

J. J. ManfrediJ. D. Rossi and J. M. Urbano, $p(x)$-harmonic functions with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2581-2595.  doi: 10.1016/j.anihpc.2009.09.008.

[25]

J. Musielak, Orlicz Spaces and Modular Spaces, in: Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.

[26]

M. Pérez-Llanos and J. D. Rossi, Limits as $p(x)\rightarrow\infty$ of $p(x)$-harmonic functions with non-homogeneous Neumann boundary conditions, Nonlinear Elliptic Partial Differential Equations, 187–201, Contemp. Math., 540, Amer. Math. Soc., Providence, RI, 2011. doi: 10.1090/conm/540/10665.

[27]

M. Pérez-Llanos and J. D. Rossi, The behaviour of the $p(x)$-Laplacian eigenvalue problem as $p(x)\rightarrow \infty$, J. Math. Anal. Appl., 363 (2010), 502-511.  doi: 10.1016/j.jmaa.2009.09.044.

[28]

M. Råžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[29]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv., 29 (1987), 33-66. 

[30]

V. V. Zhikov, On Lavrentiev$'$s phenomenon, Russ. J. Math. Phys., 3 (1995), 249-269. 

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