Mathematica Bohemica, Vol. 148, No. 3, pp. 283-302, 2023
Existence of weak solutions for elliptic Dirichlet problems with variable exponent
Sungchol Kim, Dukman Ri
Received May 11, 2021. Published online June 27, 2022.
Abstract: This paper presents several sufficient conditions for the existence of weak solutions to general nonlinear elliptic problems of the type
$\cases-{\rm div} a(x, u, \nn u)+b(x, u, \nn u)=0 \text{in} \Omega$,
$u=0 \text{on} \partial\Omega$,
where $\Omega$ is a bounded domain of $\mathbb R^n$, $n\ge2$. 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.
Keywords: variable exponent; existence; variational methods; Dirichlet problem
References: [1] L. Boccardo, B. Dacorogna: A characterization of pseudo-monotone differential operators in divergence form. Commun. Partial Differ. Equations 9 (1984), 1107-1117. DOI 10.1080/03605308408820358 | MR 0759239 | Zbl 0562.47041
[2] V. I. Bogachev: Measure Theory. Volume I. Springer, Berlin (2007). DOI 10.1007/978-3-540-34514-5 | MR 2267655 | Zbl 1120.28001
[3] G. Bonanno, A. Chinni: Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent. J. Math. Anal. Appl. 418 (2014), 812-827. DOI 10.1016/j.jmaa.2014.04.016 | MR 3206681 | Zbl 1312.35111
[4] Y. Chen, S. Levine, M. Rao: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66 (2006), 1383-1406. DOI 10.1137/050624522 | MR 2246061 | Zbl 1102.49010
[5] D. V. Cruz-Uribe, A. Fiorenza: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, New York (2013). DOI 10.1007/978-3-0348-0548-3 | MR 3026953 | Zbl 1268.46002
[6] L. Diening, P. Harjulehto, P. Hästö, M. Růžička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017. Springer, Berlin (2011). DOI 10.1007/978-3-642-18363-8 | MR 2790542 | Zbl 1222.46002
[7] X. Fan: On the sub-supersolution method for $p(x)$-Laplacian equations. J. Math. Anal. Appl. 330 (2007), 665-682. DOI 10.1016/j.jmaa.2006.07.093 | MR 2302951 | Zbl 1206.35103
[8] X. Fan: Remarks on eigenvalue problems involving the $p(x)$-Laplacian. J. Math. Anal. Appl. 352 (2009), 85-98. DOI 10.1016/j.jmaa.2008.05.086 | MR 2499888 | Zbl 1163.35026
[9] X. Fan: Existence and uniqueness for the $p(x)$-Laplacian-Dirichlet problems. Math. Nachr. 284 (2011), 1435-1445. DOI 10.1002/mana.200810203 | MR 2832655 | Zbl 1234.35111
[10] X. Fan, J. Shen, D. Zhao: Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$. J. Math. Anal. Appl. 262 (2001), 749-760. DOI 10.1006/jmaa.2001.7618 | MR 1859337 | Zbl 0995.46023
[11] X. Fan, Q. Zhang: Existence of solutions for $p(x)$-Laplacian Dirichlet problem. Nonlinear Anal., Theory Methods Appl., Ser. A 52 (2003), 1843-1852. DOI 10.1016/S0362-546X(02)00150-5 | MR 1954585 | Zbl 1146.35353
[12] X. Fan, Q. Zhang, D. Zhao: Eigenvalues of $p(x)$-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302 (2005), 306-317. DOI 10.1016/j.jmaa.2003.11.020 | MR 2107835 | Zbl 1072.35138
[13] Y. Fu, M. Yang: Existence of solutions for quasilinear elliptic systems in divergence form with variable growth. J. Inequal. Appl. 2014 (2014), Article ID 23, 16 pages. DOI 10.1186/1029-242X-2014-23 | MR 3213021 | Zbl 1310.35112
[14] Y. Fu, M. Yu: The Dirichlet problem of higher order quasilinear elliptic equation. J. Math. Anal. Appl. 363 (2010), 679-689. DOI 10.1016/j.jmaa.2009.10.003 | MR 2564887 | Zbl 1182.35115
[15] M. Galewski: On the existence and stability of solutions for Dirichlet problem with $p(x)$-Laplacian. J. Math. Anal. Appl. 326 (2007), 352-362. DOI 10.1016/j.jmaa.2006.03.006 | MR 2277787 | Zbl 1159.35365
[16] J.-P. Gossez, V. Mustonen: Pseudo-monotonicity and the Leray-Lions condition. Differ. Integral Equ. 6 (1993), 37-45. MR 1190164 | Zbl 0786.35055
[17] P. Harjulehto, P. Hästö, Ú. V. Lê, M. Nuortio: Overview of differential equations with non-standard growth. Nonlinear Anal., Theory Methods Appl., Ser. A 72 (2010), 4551-4574. DOI 10.1016/j.na.2010.02.033 | MR 2639204 | Zbl 1188.35072
[18] C. Ji: 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 10.1016/j.na.2010.12.013 | MR 2785386 | Zbl 1210.35132
[19] S. Kim, D. Ri: 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 10.1016/j.na.2019.02.016 | MR 3926581 | Zbl 1419.49045
[20] O. Kováčik, J. Rákosník: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. DOI 10.21136/CMJ.1991.102493 | MR 1134951 | Zbl 0784.46029
[21] J. L. Lions: Quelques méthodes de résolution des problémes aux limites nonlinéaires. Etudes mathematiques. Dunod, Paris (1969). (In French.) MR 0259693 | Zbl 0189.40603
[22] R. A. Mashiyev, B. Cekic, O. M. Buhrii: Existence of solutions for $p(x)$-Laplacian equations. Electron. J. Qual. Theory Differ. Equ. 2010 (2010), Article ID 65, 13 pages. DOI 10.14232/ejqtde.2010.1.65 | MR 2735026 | Zbl 1207.35142
[23] M. Mihăilescu, D. Repovš: On a PDE involving the $\mathcal{A}_{p(\cdot)}$-Laplace operator. Nonlinear Anal., Theory Methods Appl., Ser. A 75 (2012), 975-981. DOI 10.1016/j.na.2011.09.034 | MR 2847471 | Zbl 1269.35009
[24] P. Pucci, Q. Zhang: Existence of entire solutions for a class of variable exponent elliptic equations. J. Differ. Equations 257 (2014), 1529-1566. DOI 10.1016/j.jde.2014.05.023 | MR 3217048 | Zbl 1292.35135
[25] V. D. Rădulescu: Nonlinear elliptic equations with variable exponent: Old and new. Nonlinear Anal., Theory Methods Appl., Ser. A 121 (2015), 336-369. DOI 10.1016/j.na.2014.11.007 | MR 3348928 | Zbl 1321.35030
[26] T. Roubíček: Nonlinear Partial Differential Equations with Applications. ISNM. International Series of Numerical Mathematics 153. Birkhäuser, Basel (2005). DOI 10.1007/978-3-0348-0513-1 | MR 2176645 | Zbl 1087.35002
[27] M. Růžička: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748. Springer, Berlin (2000). DOI 10.1007/BFb0104029 | MR 1810360 | Zbl 0962.76001
[28] C. Yu, D. Ri: 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 10.1016/j.na.2017.02.019 | MR 3634773 | Zbl 1375.35127
[29] E. Zeidler: Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization. Springer, New York (1985). DOI 10.1007/978-1-4612-5020-3 | MR 0768749 | Zbl 0583.47051
[30] A. Zhang: $p(x)$-Laplacian equations satisfying Cerami condition. J. Math. Anal. Appl. 337 (2008), 547-555. DOI 10.1016/j.jmaa.2007.04.007 | MR 2356093 | Zbl 1216.35065
[31] V. V. Zhikov: 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 10.1070/IM1987v029n01ABEH000958 | MR 0864171 | Zbl 0599.49031
[32] Q.-M. Zhou: On the superlinear problems involving $p(x)$-Laplacian-like operators without AR-condition. Nonlinear Anal., Real World Appl. 21 (2015), 161-196. DOI 10.1016/j.nonrwa.2014.07.003 | MR 3261587 | Zbl 1304.35471
Affiliations: Sungchol Kim, Dukman Ri (corresponding author), Department of Mathematics, University of Science, Pyongyang, Democratic People's Republic of Korea, e-mail: ksc@star-co.net.kp, ridukman@star-co.net.kp