Geodesic
Existence of weak solutions for elliptic Dirichlet problems with variable exponent
Mathematica Bohemica, Tome 148 (2023) no. 3, pp. 283-302
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This paper presents several sufficient conditions for the existence of weak solutions to general nonlinear elliptic problems of the type $$ \begin{cases} -{\rm div} a(x, u, \nabla u)+b(x, u, \nabla u)=0 \text {in} \ \Omega ,\\ u=0 \text {on} \ \partial \Omega , \end{cases} $$ where $\Omega $ is a bounded domain of $\mathbb R^n$, $n\ge 2$. In particular, we do not require strict monotonicity of the principal part $a(x,z,\cdot )$, while the approach is based on the variational method and results of the variable exponent function spaces.
This paper presents several sufficient conditions for the existence of weak solutions to general nonlinear elliptic problems of the type $$ \begin{cases} -{\rm div} a(x, u, \nabla u)+b(x, u, \nabla u)=0 \text {in} \ \Omega ,\\ u=0 \text {on} \ \partial \Omega , \end{cases} $$ where $\Omega $ is a bounded domain of $\mathbb R^n$, $n\ge 2$. In particular, we do not require strict monotonicity of the principal part $a(x,z,\cdot )$, while the approach is based on the variational method and results of the variable exponent function spaces.
DOI : 10.21136/MB.2022.0069-21
Classification : 35J20, 35J25, 35J70
Keywords: variable exponent; existence; variational methods; Dirichlet problem 
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Kim, Sungchol; Ri, Dukman. Existence of weak solutions for elliptic Dirichlet problems with variable exponent. Mathematica Bohemica, Tome 148 (2023) no. 3, pp. 283-302. doi: 10.21136/MB.2022.0069-21

[1] Boccardo, L., Dacorogna, B.: A characterization of pseudo-monotone differential operators in divergence form. Commun. Partial Differ. Equations 9 (1984), 1107-1117. | DOI | MR | JFM

[2] Bogachev, V. I.: Measure Theory. Volume I. Springer, Berlin (2007). | DOI | MR | JFM

[3] Bonanno, G., Chinn{\`ı, A.: Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent. J. Math. Anal. Appl. 418 (2014), 812-827. | DOI | MR | JFM

[4] Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66 (2006), 1383-1406. | DOI | MR | JFM

[5] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, New York (2013). | DOI | MR | JFM

[6] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017. Springer, Berlin (2011). | DOI | MR | JFM

[7] Fan, X.: On the sub-supersolution method for $p(x)$-Laplacian equations. J. Math. Anal. Appl. 330 (2007), 665-682. | DOI | MR | JFM

[8] Fan, X.: Remarks on eigenvalue problems involving the $p(x)$-Laplacian. J. Math. Anal. Appl. 352 (2009), 85-98. | DOI | MR | JFM

[9] Fan, X.: Existence and uniqueness for the $p(x)$-Laplacian-Dirichlet problems. Math. Nachr. 284 (2011), 1435-1445. | DOI | MR | JFM

[10] Fan, X., Shen, J., Zhao, D.: Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$. J. Math. Anal. Appl. 262 (2001), 749-760. | DOI | MR | JFM

[11] Fan, X., Zhang, Q.: Existence of solutions for $p(x)$-Laplacian Dirichlet problem. Nonlinear Anal., Theory Methods Appl., Ser. A 52 (2003), 1843-1852. | DOI | MR | JFM

[12] Fan, X., Zhang, Q., Zhao, D.: Eigenvalues of $p(x)$-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302 (2005), 306-317. | DOI | MR | JFM

[13] Fu, Y., Yang, M.: Existence of solutions for quasilinear elliptic systems in divergence form with variable growth. J. Inequal. Appl. 2014 (2014), Article ID 23, 16 pages. | DOI | MR | JFM

[14] Fu, Y., Yu, M.: The Dirichlet problem of higher order quasilinear elliptic equation. J. Math. Anal. Appl. 363 (2010), 679-689. | DOI | MR | JFM

[15] Galewski, M.: On the existence and stability of solutions for Dirichlet problem with $p(x)$-Laplacian. J. Math. Anal. Appl. 326 (2007), 352-362. | DOI | MR | JFM

[16] Gossez, J.-P., Mustonen, V.: Pseudo-monotonicity and the Leray-Lions condition. Differ. Integral Equ. 6 (1993), 37-45. | MR | JFM

[17] Harjulehto, P., Hästö, P., Lê, Ú. V., Nuortio, M.: Overview of differential equations with non-standard growth. Nonlinear Anal., Theory Methods Appl., Ser. A 72 (2010), 4551-4574. | DOI | MR | JFM

[18] Ji, C.: Remarks on the existence of three solutions for the $p(x)$-Laplacian equations. Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 2908-2915. | DOI | MR | JFM

[19] Kim, S., Ri, D.: Global boundedness and Hölder continuity of quasiminimizers with the general nonstandard growth conditions. Nonlinear Anal., Theory Methods Appl., Ser. A 185 (2019), 170-192. | DOI | MR | JFM

[20] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. | DOI | MR | JFM

[21] Lions, J. L.: Quelques méthodes de résolution des problémes aux limites nonlinéaires. Etudes mathematiques. Dunod, Paris (1969), French. | MR | JFM

[22] Mashiyev, R. A., Cekic, B., Buhrii, O. M.: Existence of solutions for $p(x)$-Laplacian equations. Electron. J. Qual. Theory Differ. Equ. 2010 (2010), Article ID 65, 13 pages. | DOI | MR | JFM

[23] Mihăilescu, M., Repovš, D.: On a PDE involving the $\mathcal{A}_{p(\cdot)}$-Laplace operator. Nonlinear Anal., Theory Methods Appl., Ser. A 75 (2012), 975-981. | DOI | MR | JFM

[24] Pucci, P., Zhang, Q.: Existence of entire solutions for a class of variable exponent elliptic equations. J. Differ. Equations 257 (2014), 1529-1566. | DOI | MR | JFM

[25] Rădulescu, V. D.: Nonlinear elliptic equations with variable exponent: Old and new. Nonlinear Anal., Theory Methods Appl., Ser. A 121 (2015), 336-369. | DOI | MR | JFM

[26] Roubíček, T.: Nonlinear Partial Differential Equations with Applications. ISNM. International Series of Numerical Mathematics 153. Birkhäuser, Basel (2005). | DOI | MR | JFM

[27] Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748. Springer, Berlin (2000). | DOI | MR | JFM

[28] Yu, C., Ri, D.: Global $L^{\infty}$-estimates and Hölder continuity of weak solutions to elliptic equations with the general nonstandard growth conditions. Nonlinear Anal., Theory Methods Appl., Ser. A 156 (2017), 144-166. | DOI | MR | JFM

[29] Zeidler, E.: Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization. Springer, New York (1985). | DOI | MR | JFM

[30] Zhang, A.: $p(x)$-Laplacian equations satisfying Cerami condition. J. Math. Anal. Appl. 337 (2008), 547-555. | DOI | MR | JFM

[31] Zhikov, V. V.: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 29 (1987), 33-66 translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50 1986 675-710. | DOI | MR | JFM

[32] Zhou, Q.-M.: On the superlinear problems involving $p(x)$-Laplacian-like operators without AR-condition. Nonlinear Anal., Real World Appl. 21 (2015), 161-196. | DOI | MR | JFM

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