Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents

Previous Article
Control of attractors in nonlinear dynamical systems using external noise: Effects of noise on synchronization phenomena
PROC Home
This Issue
Next Article
Initial boundary value problem for the singularly perturbed Boussinesq-type equation

2013, 2013(special): 695-707. doi: 10.3934/proc.2013.2013.695

Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents

1.	Department of Mathematics, University of Ulsan, Ulsan 680-749, South Korea
2.	Department of Mathematics Education, Sangmyung University, Seoul 110-743, South Korea

Received August 2012 Revised March 2013 Published November 2013

Abstract
Related Papers

We study the following nonlinear problem \begin{equation*} -div(w(x)|\nabla u|^{p(x)-2}\nabla u)=\lambda f(x,u)\quad in \Omega \end{equation*} which is subject to Dirichlet boundary condition. Under suitable conditions on $w$ and $f$, employing the variational methods, we show the existence of solutions for the above problem in the weighted variable exponent Lebesgue-Sobolev spaces. Also we obtain the positivity of the infimum eigenvalue for the problem.

Keywords: weighted variable exponent Lebesgue-Sobolev spaces, fountain theorem, mountain pass theorem, $p(x)$-Laplacian, eigenvalue..

Mathematics Subject Classification: 35J20, 35J60, 35J70, 47J10, 46E3.

Citation: Inbo Sim, Yun-Ho Kim. Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents. Conference Publications, 2013, 2013 (special) : 695-707. doi: 10.3934/proc.2013.2013.695

References:

[1]	T. Bartsch, Z. Liu, On a superlinear elliptic $p$-Laplacian equation,, J. Differential Equations 198 (2004), (2004), 149. Google Scholar
[2]	N. Benouhiba, On the eigenvalues of weighted $p(x)$-Laplacian on $\mathbbR^N$,, Nonlinear Anal. 74 (2011), (2011), 235. Google Scholar
[3]	Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration,, SIAM J. Appl. Math. 66 (2006), (2006), 1383. Google Scholar
[4]	L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(\cdot)}$ and $W^{k,p(\cdot)}$,, Math. Nachr. 268 (2004), (2004), 31. Google Scholar
[5]	G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian,, Portugal. Math. 58 (2001), (2001), 339. Google Scholar
[6]	P. Drábek, A. Kufner, F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities,, de Gruyter, (1997). Google Scholar
[7]	P. De Napoli, M. Mariani, Mountain pass solutions to equations of $p$-Laplacian type,, Nonlinear Anal. 54 (2003), (2003), 1205. Google Scholar
[8]	X. Fan, D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, J. Math. Anal. Appl. 263 (2001), (2001), 424. Google Scholar
[9]	X.L. Fan, Q.H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem,, Nonlinear Anal. 52 (2003), (2003), 1843. Google Scholar
[10]	X. Fan, Q. Zhang, D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem,, J. Math. Anal. Appl. 302 (2005), (2005), 306. Google Scholar
[11]	H. Galewski, On the continuity of the Nemyskij operator between the spaces $L^{p_1(x)}$ and $L^{p_2(x)}$,, Georgian Math. Journal. 13 (2006), (2006), 261. Google Scholar
[12]	P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values,, Math. Bohem. 132 (2007), (2007), 125. Google Scholar
[13]	Y. Huang, Existence of positive solutions for a class of the $p$-Laplace equations,, J. Austral. Math. Soc. Sect. B 36 (1994), (1994), 249. Google Scholar
[14]	Y.-H. Kim, L. Wang, C. Zhang, Global bifurcation for a class of degenerate elliptic equations with variable exponents,, J. Math. Anal. Appl. 371 (2010), (2010), 624. Google Scholar
[15]	O. Kováčik, J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, Czechoslovak Math. J. 41 (1991), (1991), 592. Google Scholar
[16]	J. Musielak, Orlicz spaces and modular spaces,, Springer-Verlag, (1983). Google Scholar
[17]	K. Rajagopal, M. R.užička, Mathematical modeling of electrorheological materials,, Continuum Mech. Thermodyn. 13 (2001), (2001), 59. Google Scholar
[18]	M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, in:, Lecture Notes in Mathematics, (1748). Google Scholar
[19]	A. Szulkin, M.Willem, Eigenvalue problem with indefinite weight,, Studia Math. 135 (1995), (1995), 191. Google Scholar
[20]	M. Willem, Minimax Theorems,, Birkhauser, (1996). Google Scholar
[21]	V.V. Zhikov, On some variational problems,, Russ. J. Math. Phys. 5 (1997), (1997), 105. Google Scholar
[22]	V.V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 226 (2004), (2004), 67. Google Scholar

show all references

References:

[1]	T. Bartsch, Z. Liu, On a superlinear elliptic $p$-Laplacian equation,, J. Differential Equations 198 (2004), (2004), 149. Google Scholar
[2]	N. Benouhiba, On the eigenvalues of weighted $p(x)$-Laplacian on $\mathbbR^N$,, Nonlinear Anal. 74 (2011), (2011), 235. Google Scholar
[3]	Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration,, SIAM J. Appl. Math. 66 (2006), (2006), 1383. Google Scholar
[4]	L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(\cdot)}$ and $W^{k,p(\cdot)}$,, Math. Nachr. 268 (2004), (2004), 31. Google Scholar
[5]	G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian,, Portugal. Math. 58 (2001), (2001), 339. Google Scholar
[6]	P. Drábek, A. Kufner, F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities,, de Gruyter, (1997). Google Scholar
[7]	P. De Napoli, M. Mariani, Mountain pass solutions to equations of $p$-Laplacian type,, Nonlinear Anal. 54 (2003), (2003), 1205. Google Scholar
[8]	X. Fan, D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, J. Math. Anal. Appl. 263 (2001), (2001), 424. Google Scholar
[9]	X.L. Fan, Q.H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem,, Nonlinear Anal. 52 (2003), (2003), 1843. Google Scholar
[10]	X. Fan, Q. Zhang, D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem,, J. Math. Anal. Appl. 302 (2005), (2005), 306. Google Scholar
[11]	H. Galewski, On the continuity of the Nemyskij operator between the spaces $L^{p_1(x)}$ and $L^{p_2(x)}$,, Georgian Math. Journal. 13 (2006), (2006), 261. Google Scholar
[12]	P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values,, Math. Bohem. 132 (2007), (2007), 125. Google Scholar
[13]	Y. Huang, Existence of positive solutions for a class of the $p$-Laplace equations,, J. Austral. Math. Soc. Sect. B 36 (1994), (1994), 249. Google Scholar
[14]	Y.-H. Kim, L. Wang, C. Zhang, Global bifurcation for a class of degenerate elliptic equations with variable exponents,, J. Math. Anal. Appl. 371 (2010), (2010), 624. Google Scholar
[15]	O. Kováčik, J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, Czechoslovak Math. J. 41 (1991), (1991), 592. Google Scholar
[16]	J. Musielak, Orlicz spaces and modular spaces,, Springer-Verlag, (1983). Google Scholar
[17]	K. Rajagopal, M. R.užička, Mathematical modeling of electrorheological materials,, Continuum Mech. Thermodyn. 13 (2001), (2001), 59. Google Scholar
[18]	M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, in:, Lecture Notes in Mathematics, (1748). Google Scholar
[19]	A. Szulkin, M.Willem, Eigenvalue problem with indefinite weight,, Studia Math. 135 (1995), (1995), 191. Google Scholar
[20]	M. Willem, Minimax Theorems,, Birkhauser, (1996). Google Scholar
[21]	V.V. Zhikov, On some variational problems,, Russ. J. Math. Phys. 5 (1997), (1997), 105. Google Scholar
[22]	V.V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 226 (2004), (2004), 67. Google Scholar

[1]	Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[2]	Carla Baroncini, Julián Fernández Bonder. An extension of a Theorem of V. Šverák to variable exponent spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1987-2007. doi: 10.3934/cpaa.2015.14.1987
[3]	Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[4]	Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[5]	Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427
[6]	Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[7]	SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026
[8]	Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109
[9]	Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems & Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631
[10]	Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[11]	Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021
[12]	Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure & Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044
[13]	Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237
[14]	Mateusz Krukowski. Arzelà-Ascoli's theorem in uniform spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 283-294. doi: 10.3934/dcdsb.2018020
[15]	G. Conner, Christopher P. Grant, Mark H. Meilstrup. A Sharkovsky theorem for non-locally connected spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3485-3499. doi: 10.3934/dcds.2012.32.3485
[16]	Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 323-326. doi: 10.3934/dcds.2011.29.323
[17]	Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[18]	Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745
[19]	Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345
[20]	Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743

Impact Factor:

American Institute of Mathematical Sciences