Abstract
References
- [1] H. Aoyama, Lebesgue spaces with variable on a probability space, Hiroshima Math. J. 39, 207–216, 2009.
- [2] F.E. Browder, Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Natl. Acad. Sci. USA, 74 (7), 2659–2661, 1977.
- [3] H. Cartan, Differential Calculus, Herman, Paris-France, 1971.
- [4] B. Cekic, R. Mashiyev and G.T. Alisoy, On the Sobolev-type inequality for Lebesgue spaces with a variable exponent, Int. Math. Forum, 1 (27), 1313–1323, 2006.
- [5] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, 2011.
- [6] X.L. Fan, Solutions for $p\left( x\right) $-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl. 312, 464–477, 2005.
- [7] P. Harjulehto, P. Hästö, M. Koskenoja and S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal. 25, 205–222, 2006.
- [8] P. Harjulehto, P. Hästö, U.V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72, 4551–4574, 2010.
- [9] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41(116) (4), 592–618, 1991.
- [10] B. Lahmi, E. Azroul and K. El Haitin, Nonlinear degenerated elliptic problems with dual data and nonstandard growth, Math. Reports 20(70) (1), 81–91, 2018.
- [11] R.A. Mashiyev, S. Oğraş, Z. Yucedag and M. Avci, The Nehari manifold approach for Dirichlet problem involving the $p\left(x\right) $-Laplacian equation, J. Korean Math. Soc. 47 (4), 845–860, 2010.
- [12] M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations (Second Edition), Springer, 2004.
- [13] M. Růžička, Electrorheological fluids: modeling and mathematical theory, Springer, Berlin Heidelberg New York, 2000.
- [14] B. Tian, Y. Fu and B. Xu, Function spaces with a random variable exponent, Abstr. Appl. Anal. 2011, Article ID 179068, 2011.
- [15] B. Tian, B. Xu and Y. Fu, Stochastic field exponent function spaces with applications, Complex Var. Elliptic Equ. 59 (1), 133–148, 2014.
- [16] C. Unal and I. Aydın, Weighted variable exponent Sobolev spaces with zero boundary values and capacity estimates, Sigma J. Eng. & Nat. Sci. 36 (2), 371–386, 2018.
Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth
Abstract
In this study, we consider weighted stochastic field exponent function spaces $L_{\vartheta }^{p(.,.)}\left( D\times \Omega \right) $ and $W_{\vartheta }^{k,p(.,.)}\left( D\times \Omega \right) $. Also, we study some basic properties and embeddings of these spaces. Finally, we present an application for defined spaces to the stochastic partial differential equations with stochastic field growth.
*********************************************************************************************************************************************************************
Keywords
Quasilinear elliptic equation Weighted stochastic field exponent Sobolev spaces Pseudo-monotone operator Compact embedding theorem
References
- [1] H. Aoyama, Lebesgue spaces with variable on a probability space, Hiroshima Math. J. 39, 207–216, 2009.
- [2] F.E. Browder, Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Natl. Acad. Sci. USA, 74 (7), 2659–2661, 1977.
- [3] H. Cartan, Differential Calculus, Herman, Paris-France, 1971.
- [4] B. Cekic, R. Mashiyev and G.T. Alisoy, On the Sobolev-type inequality for Lebesgue spaces with a variable exponent, Int. Math. Forum, 1 (27), 1313–1323, 2006.
- [5] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, 2011.
- [6] X.L. Fan, Solutions for $p\left( x\right) $-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl. 312, 464–477, 2005.
- [7] P. Harjulehto, P. Hästö, M. Koskenoja and S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal. 25, 205–222, 2006.
- [8] P. Harjulehto, P. Hästö, U.V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72, 4551–4574, 2010.
- [9] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41(116) (4), 592–618, 1991.
- [10] B. Lahmi, E. Azroul and K. El Haitin, Nonlinear degenerated elliptic problems with dual data and nonstandard growth, Math. Reports 20(70) (1), 81–91, 2018.
- [11] R.A. Mashiyev, S. Oğraş, Z. Yucedag and M. Avci, The Nehari manifold approach for Dirichlet problem involving the $p\left(x\right) $-Laplacian equation, J. Korean Math. Soc. 47 (4), 845–860, 2010.
- [12] M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations (Second Edition), Springer, 2004.
- [13] M. Růžička, Electrorheological fluids: modeling and mathematical theory, Springer, Berlin Heidelberg New York, 2000.
- [14] B. Tian, Y. Fu and B. Xu, Function spaces with a random variable exponent, Abstr. Appl. Anal. 2011, Article ID 179068, 2011.
- [15] B. Tian, B. Xu and Y. Fu, Stochastic field exponent function spaces with applications, Complex Var. Elliptic Equ. 59 (1), 133–148, 2014.
- [16] C. Unal and I. Aydın, Weighted variable exponent Sobolev spaces with zero boundary values and capacity estimates, Sigma J. Eng. & Nat. Sci. 36 (2), 371–386, 2018.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Publication Date | August 6, 2020 |
| Published in Issue | Year 2020 Volume: 49 Issue: 4 |