Anisotropic elliptic equations with gradient-dependent lower order terms and $ L^1 $ data

Barbara Brandolini; Florica C. Cîrstea; Barbara Brandolini; Florica C. Cîrstea

doi:10.3934/mine.2023073

Mathematics in Engineering

2023, Volume 5, Issue 4: 1-33. doi: 10.3934/mine.2023073

Previous Article Next Article

Research article Special Issues

Anisotropic elliptic equations with gradient-dependent lower order terms and $ L^1 $ data

Barbara Brandolini ¹,
Florica C. Cîrstea ^{2
,
,}

1.
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, via Archirafi 34, 90123 Palermo, Italy
2.
School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

Received: 14 March 2022 Revised: 29 November 2022 Accepted: 04 January 2023 Published: 31 January 2023

We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $ \mathcal Au+\Phi(x, u, \nabla u) = \mathfrak{B}u+f $ in $ \Omega $, where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ and $ f\in L^1(\Omega) $ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $ \mathcal A $, the prototype of which is $ \mathcal A u = -\sum_{j = 1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u) $ with $ p_j > 1 $ for all $ 1\leq j\leq N $ and $ \sum_{j = 1}^N (1/p_j) > 1 $. As a novelty in this paper, our lower order terms involve a new class of operators $ \mathfrak B $ such that $ \mathcal{A}-\mathfrak{B} $ is bounded, coercive and pseudo-monotone from $ W_0^{1, \overrightarrow{p}}(\Omega) $ into its dual, as well as a gradient-dependent nonlinearity $ \Phi $ with an "anisotropic natural growth" in the gradient and a good sign condition.
- nonlinear anisotropic elliptic equations,
- Leray–Lions operators,
- pseudo-monotone operators,
- lower order terms,
- summable data
Citation: Barbara Brandolini, Florica C. Cîrstea. Anisotropic elliptic equations with gradient-dependent lower order terms and $ L^1 $ data[J]. Mathematics in Engineering, 2023, 5(4): 1-33. doi: 10.3934/mine.2023073

Related Papers:

Abstract

We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $ \mathcal Au+\Phi(x, u, \nabla u) = \mathfrak{B}u+f $ in $ \Omega $, where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ and $ f\in L^1(\Omega) $ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $ \mathcal A $, the prototype of which is $ \mathcal A u = -\sum_{j = 1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u) $ with $ p_j > 1 $ for all $ 1\leq j\leq N $ and $ \sum_{j = 1}^N (1/p_j) > 1 $. As a novelty in this paper, our lower order terms involve a new class of operators $ \mathfrak B $ such that $ \mathcal{A}-\mathfrak{B} $ is bounded, coercive and pseudo-monotone from $ W_0^{1, \overrightarrow{p}}(\Omega) $ into its dual, as well as a gradient-dependent nonlinearity $ \Phi $ with an "anisotropic natural growth" in the gradient and a good sign condition.

References

[1]	A. Alberico, I. Chlebicka, A. Cianchi, A. Zatorska-Goldstein, Fully anisotropic elliptic problems with minimally integrable data, Calc. Var., 58 (2019), 186. https://doi.org/10.1007/s00526-019-1627-8 doi: 10.1007/s00526-019-1627-8
[2]	A. Alberico, G. di Blasio, F. Feo, Comparison results for nonlinear anisotropic parabolic problems, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., 28 (2017), 305–322. https://doi.org/10.4171/RLM/764 doi: 10.4171/RLM/764
[3]	A. Alvino, V. Ferone, A. Mercaldo, Sharp a priori estimates for a class of nonlinear elliptic equations with lower order terms, Annali di Matematica, 194 (2015), 1169–1201. https://doi.org/10.1007/s10231-014-0416-4 doi: 10.1007/s10231-014-0416-4
[4]	A. Alvino, A. Mercaldo, Nonlinear elliptic equations with lower order terms and symmetrization methods, Boll. Unione Mat. Ital., 1 (2008), 645–661.
[5]	S. N. Antontsev, M. Chipot, Anisotropic equations: uniqueness and existence results, Differential Integral Equations, 21 (2008), 401–419. https://doi.org/10.57262/die/1356038624 doi: 10.57262/die/1356038624
[6]	S. N. Antontsev, J. I. Díaz, S. Shmarev, Energy methods for free boundary problems, Boston, MA: Birkhäuser Boston, Inc., 2002. https://doi.org/10.1007/978-1-4612-0091-8
[7]	S. Antontsev, S. Shmarev, Chapter 1 Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, In: Handbook of differential equations: stationary partial differential equations, North-Holland, 2006. https://doi.org/10.1016/S1874-5733(06)80005-7
[8]	P. Baroni, A. Di Castro, G. Palatucci, Intrinsic geometry and De Giorgi classes for certain anisotropic problems, Discrete Contin. Dyn. Syst. S, 10 (2017), 647–659. https://doi.org/10.3934/dcdss.2017032 doi: 10.3934/dcdss.2017032
[9]	M. Bendahmane, K. H. Karlsen, Nonlinear anisotropic elliptic and parabolic equations in $\mathbb R^N$ with advection and lower order terms and locally integrable data, Potential Anal., 22 (2005), 207–227. https://doi.org/10.1007/s11118-004-6117-7 doi: 10.1007/s11118-004-6117-7
[10]	A. Bensoussan, L. Boccardo, Nonlinear systems of elliptic equations with natural growth conditions and sign conditions, Appl. Math. Optim., 46 (2002), 143–166. https://doi.org/10.1007/s00245-002-0753-3 doi: 10.1007/s00245-002-0753-3
[11]	A. Bensoussan, L. Boccardo, F. Murat, On a nonlinear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 347–364. https://doi.org/10.1016/S0294-1449(16)30342-0 doi: 10.1016/S0294-1449(16)30342-0
[12]	M. F. Betta, A. Mercaldo, F. Murat, M. M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower-order term and right-hand side a measure, J. Math. Pure. Appl., 82 (2003), 90–124. https://doi.org/10.1016/S0021-7824(03)00006-0 doi: 10.1016/S0021-7824(03)00006-0
[13]	L. Boccardo, T. Gallouët, F. Murat, A unified presentation of two existence results for problems with natural growth, In: Progress in partial differential equations: the Metz surveys, 2 (1992), Harlow: Longman Sci. Tech., 1993,127–137.
[14]	L. Boccardo, P. Marcellini, C. Sbordone, $L^\infty$-regularity for variational problems with sharp nonstandard growth conditions, Bollettino U. M. I., 4 (1990), 219–225.
[15]	L. Boccardo, F. Murat, J.-P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Annali di Matematica, 152 (1988), 183–196. https://doi.org/10.1007/BF01766148 doi: 10.1007/BF01766148
[16]	L. Boccardo, T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal., 19 (1992), 573–579. https://doi.org/10.1016/0362-546X(92)90022-7 doi: 10.1016/0362-546X(92)90022-7
[17]	M.-M. Boureanu, A new class of nonhomogeneous differential operators and applications to anisotropic systems, Complex Var. Elliptic Equ., 61 (2016), 712–730. https://doi.org/10.1080/17476933.2015.1114614 doi: 10.1080/17476933.2015.1114614
[18]	M.-M. Boureanu, A. Vélez-Santiago, Fine regularity for elliptic and parabolic anisotropic Robin problems with variable exponents, J. Differ. Equations, 266 (2019), 8164–8232. https://doi.org/10.1016/j.jde.2018.12.026 doi: 10.1016/j.jde.2018.12.026
[19]	B. Brandolini, F. C. Cîrstea, Singular anisotropic elliptic equations with gradient-dependent lower order terms, arXiv: 2001.02887.
[20]	H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier, 18 (1968), 115–175.
[21]	F. E. Browder, Existence theorems for nonlinear partial differential equations, In: Global analysis, Providence, R.I.: Amer. Math. Soc., 1970, 1–60.
[22]	A. Cianchi, Symmetrization in anisotropic elliptic problems, Commun. Part. Diff. Eq., 32 (2007), 693–717. https://doi.org/10.1080/03605300600634973 doi: 10.1080/03605300600634973
[23]	F. C. Cîrstea, J. Vétois, Fundamental solutions for anisotropic elliptic equations: existence and a priori estimates, Commun. Part. Diff. Eq., 40 (2015), 727–765. https://doi.org/10.1080/03605302.2014.969374 doi: 10.1080/03605302.2014.969374
[24]	G. di Blasio, F. Feo, G. Zecca, Regularity results for local solutions to some anisotropic elliptic equations, Isr. J. Math., in press.
[25]	A. Di Castro, E. Montefusco, Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations, Nonlinear Anal., 70 (2009), 4093–4105. https://doi.org/10.1016/j.na.2008.06.001 doi: 10.1016/j.na.2008.06.001
[26]	X. Fan, Anisotropic variable exponent Sobolev spaces and $\overrightarrow{p}(\cdot)-$Laplacian equations, Complex Var. Elliptic Equ., 56 (2011), 623–642. https://doi.org/10.1080/17476931003728412 doi: 10.1080/17476931003728412
[27]	X. Fan, Local boundedness of quasi-minimizers of integral functions with variable exponent anisotropic growth and applications, Nonlinear Differ. Equ. Appl., 17 (2010), 619–637. https://doi.org/10.1007/s00030-010-0072-3 doi: 10.1007/s00030-010-0072-3
[28]	F. Feo, J. L. Vazquez, B. Volzone, Anisotropic $p$-Laplacian evolution of fast diffusion type, Adv. Nonlinear Stud., 21 (2021), 523–555. https://doi.org/10.1515/ans-2021-2136 doi: 10.1515/ans-2021-2136
[29]	V. Ferone, B. Messano, Comparison and existence results for classes of nonlinear elliptic equations with general growth in the gradient, Adv. Nonlinear Stud., 7 (2007), 31–46. https://doi.org/10.1515/ans-2007-0102 doi: 10.1515/ans-2007-0102
[30]	V. Ferone, F. Murat, Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces, J. Differ. Equations, 256 (2014), 577–608. https://doi.org/10.1016/j.jde.2013.09.013 doi: 10.1016/j.jde.2013.09.013
[31]	I. Fragalà, F. Gazzola, B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 715–734. https://doi.org/10.1016/j.anihpc.2003.12.001 doi: 10.1016/j.anihpc.2003.12.001
[32]	I. Fragalà, F. Gazzola, G. Lieberman, Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains, Discrete Contin. Dyn. Syst., 2005 (2005), 280–286. https://doi.org/10.3934/proc.2005.2005.280 doi: 10.3934/proc.2005.2005.280
[33]	H. Gao, F. Leonetti, W. Ren, Regularity for anisotropic elliptic equations with degenerate coercivity, Nonlinear Anal., 187 (2019), 493–505. https://doi.org/10.1016/j.na.2019.06.017 doi: 10.1016/j.na.2019.06.017
[34]	D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Berlin: Springer, 2001. https://doi.org/10.1007/978-3-642-61798-0
[35]	N. Grenon, F. Murat, A. Porretta, A priori estimates and existence for elliptic equations with gradient dependent terms, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 137–205. ttps://doi.org/10.2422/2036-2145.201106_012 doi: 10.2422/2036-2145.201106_012
[36]	H. Le Dret, Nonlinear elliptic partial differential equations. An introduction, Cham: Springer, 2018. https://doi.org/10.1007/978-3-319-78390-1
[37]	J. Leray, J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97–107.
[38]	P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267–284. https://doi.org/10.1007/BF00251503 doi: 10.1007/BF00251503
[39]	R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Providence, RI: American Mathematical Society, 1997. http://doi.org/10.1090/surv/049
[40]	M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3–24.
[41]	N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge–Ampère type, In: International Congress of Mathematicians. Vol. III, Zürich: Eur. Math. Soc., 2006,291–301. https://doi.org/10.4171/022-3/15
[42]	N. S. Trudinger, From optimal transportation to conformal geometry, In: Geometric analysis, Cham: Birkhäuser, 2020,511–520. https://doi.org/10.1007/978-3-030-34953-0_20
[43]	J. Vétois, Strong maximum principles for anisotropic elliptic and parabolic equations, Adv. Nonlinear Stud., 12 (2012), 101–114. https://doi.org/10.1515/ans-2012-0106 doi: 10.1515/ans-2012-0106
[44]	E. Zeidler, Nonlinear functional analysis and its applications, New York: Springer, 1990. https://doi.org/10.1007/978-1-4612-0981-2

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)