| | | |

## Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth

#### İ̇smail AYDIN [1] , Cihan UNAL [2]

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.

*********************************************************************************************************************************************************************

Quasilinear elliptic equation, Weighted stochastic field exponent Sobolev spaces, Pseudo-monotone operator, Compact embedding theorem
• [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.
Primary Language en Mathematics Mathematics Orcid: 0000-0001-8371-3185Author: İ̇smail AYDIN Institution: SINOP UNIVERSITYCountry: Turkey Orcid: 0000-0002-7242-393XAuthor: Cihan UNAL (Primary Author)Institution: SINOP UNIVERSITYCountry: Turkey Publication Date : August 6, 2020
 Bibtex @research article { hujms561682, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1383 - 1396}, doi = {10.15672/hujms.561682}, title = {Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth}, key = {cite}, author = {Aydın, İ̇smail and Unal, Cihan} } APA Aydın, İ , Unal, C . (2020). Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth . Hacettepe Journal of Mathematics and Statistics , 49 (4) , 1383-1396 . DOI: 10.15672/hujms.561682 MLA Aydın, İ , Unal, C . "Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1383-1396 Chicago Aydın, İ , Unal, C . "Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1383-1396 RIS TY - JOUR T1 - Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth AU - İ̇smail Aydın , Cihan Unal Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.561682 DO - 10.15672/hujms.561682 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1383 EP - 1396 VL - 49 IS - 4 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.561682 UR - https://doi.org/10.15672/hujms.561682 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth %A İ̇smail Aydın , Cihan Unal %T Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 4 %R doi: 10.15672/hujms.561682 %U 10.15672/hujms.561682 ISNAD Aydın, İ̇smail , Unal, Cihan . "Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth". Hacettepe Journal of Mathematics and Statistics 49 / 4 (August 2020): 1383-1396 . https://doi.org/10.15672/hujms.561682 AMA Aydın İ , Unal C . Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1383-1396. Vancouver Aydın İ , Unal C . Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1383-1396.

Authors of the Article