Solvability of nonlinear elliptic type equation with two unrelated non standard growths
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In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths \begin{equation*} -div\left(\left\vert\nabla u\right\vert^{p_{1}\left( x\right)-2}\nabla u\right)-\sum\limits_{i=1}^{n}D_{i}\left( \left\vert u\right\vert ^{p_{0}\left( x\right)-2}D_{i}u\right)+c\left( x,u\right) \!=\!h\left( x\right) ,~x\!\in\! \Omega \end{equation*} in a bounded domain $\Omega \subset \mathbb{R}^{n}$. Here, one of the operators in the sum is monotone and the other is weakly compact. We obtain sufficient conditions and show the existence of weak solutions of the considered problem by using monotonicity and compactness methods together.
Abstract
In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths \begin{equation*} -div\left(\left\vert\nabla u\right\vert^{p_{1}\left( x\right)-2}\nabla u\right)-\sum\limits_{i=1}^{n}D_{i}\left( \left\vert u\right\vert ^{p_{0}\left( x\right)-2}D_{i}u\right)+c\left( x,u\right) \!=\!h\left( x\right) ,~x\!\in\! \Omega \end{equation*} in a bounded domain $\Omega \subset \mathbb{R}^{n}$. Here, one of the operators in the sum is monotone and the other is weakly compact. We obtain sufficient conditions and show the existence of weak solutions of the considered problem by using monotonicity and compactness methods together.
참고문헌 (28)
- E. Acerbi, "Regularity results for stationary electro-rheological uids", Arch. Ration. Mech. Anal, 164(3), pp.213~259 (2002)
- R. A. Adams, "Sobolev Spaces" (1975), Academic Press
- S. N. Antontsev, "On stationary thermo-rheological viscous ows", Ann. Univ. Ferrara Sez. VII Sci. Mat, 52(1), pp.19~36 (2006)
- S. N. Antontsev, "A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions", Nonlinear Anal, 60(3), pp.515~545 (2005)
- S. N. Antontsev, "On the localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneration", Siberian Math. J, 46(5), pp.765~782 (2005)
- S. N. Antontsev, "Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions", Nonlinear Anal, 65(4), pp.728~761 (2006)
- Y. Chen, "Variable exponent, linear growth functionals in image restoration", SIAM J. Appl. Math, 66(4), pp.1383~1406 (2006)
- L. Diening, "Theoretical and numerical results for electrorheological uids", Ph.D., pp.2002
- L. Diening, "Lebesgue and Sobolev spaces with variable exponents" (2017), Springer
- X. Fan, "Sobolev embedding theorems for spaces Wk,p(x)(Ω)", J. Math. Anal. Appl, 262(2), pp.749~760 (2001)
- X. Fan, "On the spaces Lp(x)(Ω) and Wm,p(x)(Ω)", J. Math. Anal. Appl, 263(2), pp.424~446 (2001)
- O. Kovacik, "On spaces Lp(x) and Wk,p(x)", Czechoslovak Math. J, 41, pp.592~618 (1991)
- J. -L. Lions, "Quelques methodes de resolution des problemes aux limites non lineaires" (1969), Dunod
- G. de Marsily, "Quantitative Hydrogeology. Groundwater Hydrology for Engineers" (1986), Academic Press
- J. Musielak, "Orlicz spaces and modular spaces" (1983), Springer-Verlag
- I. P. Natanson, "Theory of Functions of a Real Variable" (1950), Moscow-Leningrad
- V. Radulescu, "Partial differential equations with variable exponents, Monographs and Research Notes in Mathematics" (2015), CRC Press
- K. Rajagopal, "Mathematical modeling of electro-rheological uids", Contin. Mech. Thermodyn, 13, pp.59~78 (2001)
- M. Ruzicka, "Electrorheological uids: modeling and mathematical theory" (2000), Springer-Verlag
- U. Sert, "On solvability of a class of nonlinear elliptic type equation with variable exponent", J. Appl. Anal. Comput, 7(3), pp.1139~1160 (2017)
- K. N. Soltanov, "Solvability of nonlinear equations with operators in the form of the sum of a pseudomonotone and a weakly compact operator", Russian Acad. Sci. Dokl. Math, 45(3), pp.676~681 (1992)
- K. N. Soltanov, "Some imbedding theorems and nonlinear differential equations", Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci, 19(5) (1999)
- K. N. Soltanov, "Some nonlinear equations of the nonstable filtration type and embedding theo-rems", Nonlinear Anal, 65(11), pp.2103~2134 (2006)
- K. N. Soltanov, "Nonlinear equations in non-re exive Banach spaces and strongly nonlinear differential equations", Adv. Math. Sci. Appl, 9(2), pp.939~972 (1999)
- B, "Nonlinear functional analysis and its applications. II", pp.1990, E. Zeidler
- V. V. Zhikov, "Averaging of functionals of the calculus of variations and elasticity theory", Math. USSR. Izv, 29, pp.33~36 (1987)
- V. V. Zhikov, "On some variational problems", Russian J. Math. Phys, 5(1), pp.105~116 (1997)
- V. V. Zhikov, "On the technique for passing to the limit in nonlinear elliptic equations", Funct. Anal. Appl, 43(2), pp.96~112 (2009)