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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

Inbo Sim 1, and  Yun-Ho Kim 2,

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

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), 149-175.

[2]

N. Benouhiba, On the eigenvalues of weighted $p(x)$-Laplacian on $\mathbb{R}^N2$, Nonlinear Anal. 74 (2011), 235-243.

[3]

Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406.

[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), 31-43.

[5]

G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Portugal. Math. 58 (2001), 339-378.

[6]

P. Drábek, A. Kufner, F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, de Gruyter, Berlin, 1997.

[7]

P. De Napoli, M. Mariani, Mountain pass solutions to equations of $p$-Laplacian type, Nonlinear Anal. 54 (2003), 1205-1219.

[8]

X. Fan, D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263 (2001), 424-446.

[9]

X.L. Fan, Q.H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843-1852.

[10]

X. Fan, Q. Zhang, D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306-317.

[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), 261-265.

[12]

P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values, Math. Bohem. 132 (2007), 125-136.

[13]

Y. Huang, Existence of positive solutions for a class of the $p$-Laplace equations, J. Austral. Math. Soc. Sect. B 36 (1994), 249-264.

[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), 624-637.

[15]

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

[16]

J. Musielak, Orlicz spaces and modular spaces, Springer-Verlag, Berlin, 1983.

[17]

K. Rajagopal, M. R.užička, Mathematical modeling of electrorheological materials, Continuum Mech. Thermodyn. 13 (2001), 59-78.

[18]

M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000.

[19]

A. Szulkin, M.Willem, Eigenvalue problem with indefinite weight, Studia Math. 135 (1995), 191-201.

[20]

M. Willem, Minimax Theorems, Birkhauser, Basel, 1996.

[21]

V.V. Zhikov, On some variational problems, Russ. J. Math. Phys. 5 (1997), 105-116.

[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), 67-81.

show all references

References:
[1]

T. Bartsch, Z. Liu, On a superlinear elliptic $p$-Laplacian equation, J. Differential Equations 198 (2004), 149-175.

[2]

N. Benouhiba, On the eigenvalues of weighted $p(x)$-Laplacian on $\mathbb{R}^N2$, Nonlinear Anal. 74 (2011), 235-243.

[3]

Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406.

[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), 31-43.

[5]

G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian, Portugal. Math. 58 (2001), 339-378.

[6]

P. Drábek, A. Kufner, F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, de Gruyter, Berlin, 1997.

[7]

P. De Napoli, M. Mariani, Mountain pass solutions to equations of $p$-Laplacian type, Nonlinear Anal. 54 (2003), 1205-1219.

[8]

X. Fan, D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263 (2001), 424-446.

[9]

X.L. Fan, Q.H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843-1852.

[10]

X. Fan, Q. Zhang, D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306-317.

[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), 261-265.

[12]

P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values, Math. Bohem. 132 (2007), 125-136.

[13]

Y. Huang, Existence of positive solutions for a class of the $p$-Laplace equations, J. Austral. Math. Soc. Sect. B 36 (1994), 249-264.

[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), 624-637.

[15]

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

[16]

J. Musielak, Orlicz spaces and modular spaces, Springer-Verlag, Berlin, 1983.

[17]

K. Rajagopal, M. R.užička, Mathematical modeling of electrorheological materials, Continuum Mech. Thermodyn. 13 (2001), 59-78.

[18]

M. R.užička, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000.

[19]

A. Szulkin, M.Willem, Eigenvalue problem with indefinite weight, Studia Math. 135 (1995), 191-201.

[20]

M. Willem, Minimax Theorems, Birkhauser, Basel, 1996.

[21]

V.V. Zhikov, On some variational problems, Russ. J. Math. Phys. 5 (1997), 105-116.

[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), 67-81.

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