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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), (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]

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[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

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P. Drábek, A. Kufner, F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities,, de Gruyter, (1997).   Google Scholar

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

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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

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