# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2022088
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## Strongly singular convective elliptic equations in $\mathbb{R}^N$ driven by a non-homogeneous operator

 1 Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy 2 Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy

*Corresponding author

Received  December 2021 Early access May 2022

Fund Project: The second author is supported by the following research projects: 1) PRIN 2017 'Nonlinear Differential Problems via Variational, Topological and Set-valued Methods' (Grant No. 2017AYM8XW) of MIUR; 2) PIACERI 20-22 Linea 3 of the University of Catania

Existence of a generalized solution to a strongly singular convective elliptic equation in the whole space is established. The differential operator, patterned after the $(p,q)$-Laplacian, can be non-homogeneous. The result is obtained by solving some regularized problems through fixed point theory, variational methods and compactness results, besides exploiting nonlinear regularity theory and comparison principles.

Citation: Laura Gambera, Umberto Guarnotta. Strongly singular convective elliptic equations in $\mathbb{R}^N$ driven by a non-homogeneous operator. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022088
##### References:
 [1] L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8. [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [3] P. Candito, U. Guarnotta and and K. Perera, Two solutions for a parametric singular $p$-Laplacian problem, J. Nonlinear Var. Anal., 4 (2020), 455-468. [4] P. Candito, R. Livrea and A. Moussaoui, Singular quasilinear elliptic systems involving gradient terms, Nonlinear Anal. Real World Appl., 55 (2020), 15 pp. doi: 10.1016/j.nonrwa.2020.103142. [5] A. Canino, B. Sciunzi and A. Trombetta, Existence and uniqueness for $p$-Laplace equations involving singular nonlinearities, NoDEA Nonlinear Differ. Equ. Appl., 23 (2016), 13 pp. doi: 10.1007/s00030-016-0361-6. [6] S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and their Inequalities, Springer Monographs in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3. [7] S. Carl and K. Perera, Generalized solutions of singular $p$-Laplacian problems in $\mathbb{R}^N$, Nonlinear Stud., 18 (2011), 113-124. [8] A. Cianchi and V. G. Maz'ya, Global gradient estimates in elliptic problems under minimal data and domain regularity, Commun. Pure Appl. Anal., 14 (2015), 285-311.  doi: 10.3934/cpaa.2015.14.285. [9] D. P. Covei, Existence and asymptotic behavior of positive solution to a quasilinear elliptic problem in $\mathbb{R}^N$, Nonlinear Anal., 69 (2008), 2615-2622.  doi: 10.1016/j.na.2007.08.039. [10] L. F. O. Faria, O. H. Miyagaki and M. Tanaka, Existence of a positive solution for problems with $(p, q)$-Laplacian and convection term in $\mathbb{R}^N$, Bound. Value Probl. 2016, Paper no. 158. doi: 10.1186/s13661-016-0665-9. [11] J. Franců, Monotone operators. A survey directed to applications to differential equations, Apl. Mat., 35 (1990), 257-301. [12] M. Ghergu and V. D. Radulescu, Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, J. Math. Anal. Appl., 333 (2007), 265-273.  doi: 10.1016/j.jmaa.2006.09.074. [13] M. Ghergu and V. D. Radulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications 37, The Clarendon Press, Oxford University Press, Oxford, 2008. [14] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies 105, Princeton University Press, Princeton, 1983. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [16] U. Guarnotta, S. A. Marano and and D. Motreanu, On a singular Robin problem with convection terms, Adv. Nonlinear Stud., 20 (2020), 895-909.  doi: 10.1515/ans-2020-2093. [17] U. Guarnotta, S. A. Marano and A. Moussaoui, Singular quasilinear convective elliptic systems in $\mathbb{R}^N$, Adv. Nonlinear Anal., 11 (2022), 741-756.  doi: 10.1515/anona-2021-0208. [18] U. Guarnotta, S. A. Marano and N. S. Papageorgiou, Multiple nodal solutions to a Robin problem with sign-changing potential and locally defined reaction, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30 (2019), 269-294.  doi: 10.4171/RLM/847. [19] S. Kakutani, A proof of Schauder's theorem, J. Math. Soc. Japan, 3 (1951), 228-231.  doi: 10.2969/jmsj/00310228. [20] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Commun. Partial Differ. Equ., 16 (1991), 311-361.  doi: 10.1080/03605309108820761. [21] Z. Liu, D. Motreanu and S. Zeng, Positive solutions for nonlinear singular elliptic equations of $p$-Laplacian type with dependence on the gradient, Calc. Var. Partial Differ. Equ., 58 (2019), 22 pp. doi: 10.1007/s00526-018-1472-1. [22] S. A. Marano, G. Marino and A. Moussaoui, Singular quasilinear elliptic systems in $\mathbb{R}^N$, Ann. Mat. Pura Appl., 198 (2019), 1581-1594.  doi: 10.1007/s10231-019-00832-1. [23] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267-284.  doi: 10.1007/BF00251503. [24] N. S. Papageorgiou and P. Winkert, Singular Dirichlet $(p, q)$-equations, Mediterr. J. Math., 18 (2021), Paper no. 141. doi: 10.1007/s00009-021-01780-y. [25] P. Pucci and J. Serrin, The maximum principle, Prog. Nonlinear Differential Equations Appl. 73, Birkhäuser Verlag, Basel, 2007. [26] M. C. Rezende and C. A. Santos, Positive solutions for a quasilinear elliptic problem involving sublinear and superlinear terms, Tokyo J. Math., 38 (2015), 381-407.  doi: 10.3836/tjm/1452806047.

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##### References:
 [1] L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.  doi: 10.1016/0362-546X(92)90023-8. [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [3] P. Candito, U. Guarnotta and and K. Perera, Two solutions for a parametric singular $p$-Laplacian problem, J. Nonlinear Var. Anal., 4 (2020), 455-468. [4] P. Candito, R. Livrea and A. Moussaoui, Singular quasilinear elliptic systems involving gradient terms, Nonlinear Anal. Real World Appl., 55 (2020), 15 pp. doi: 10.1016/j.nonrwa.2020.103142. [5] A. Canino, B. Sciunzi and A. Trombetta, Existence and uniqueness for $p$-Laplace equations involving singular nonlinearities, NoDEA Nonlinear Differ. Equ. Appl., 23 (2016), 13 pp. doi: 10.1007/s00030-016-0361-6. [6] S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and their Inequalities, Springer Monographs in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3. [7] S. Carl and K. Perera, Generalized solutions of singular $p$-Laplacian problems in $\mathbb{R}^N$, Nonlinear Stud., 18 (2011), 113-124. [8] A. Cianchi and V. G. Maz'ya, Global gradient estimates in elliptic problems under minimal data and domain regularity, Commun. Pure Appl. Anal., 14 (2015), 285-311.  doi: 10.3934/cpaa.2015.14.285. [9] D. P. Covei, Existence and asymptotic behavior of positive solution to a quasilinear elliptic problem in $\mathbb{R}^N$, Nonlinear Anal., 69 (2008), 2615-2622.  doi: 10.1016/j.na.2007.08.039. [10] L. F. O. Faria, O. H. Miyagaki and M. Tanaka, Existence of a positive solution for problems with $(p, q)$-Laplacian and convection term in $\mathbb{R}^N$, Bound. Value Probl. 2016, Paper no. 158. doi: 10.1186/s13661-016-0665-9. [11] J. Franců, Monotone operators. A survey directed to applications to differential equations, Apl. Mat., 35 (1990), 257-301. [12] M. Ghergu and V. D. Radulescu, Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, J. Math. Anal. Appl., 333 (2007), 265-273.  doi: 10.1016/j.jmaa.2006.09.074. [13] M. Ghergu and V. D. Radulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications 37, The Clarendon Press, Oxford University Press, Oxford, 2008. [14] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies 105, Princeton University Press, Princeton, 1983. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [16] U. Guarnotta, S. A. Marano and and D. Motreanu, On a singular Robin problem with convection terms, Adv. Nonlinear Stud., 20 (2020), 895-909.  doi: 10.1515/ans-2020-2093. [17] U. Guarnotta, S. A. Marano and A. Moussaoui, Singular quasilinear convective elliptic systems in $\mathbb{R}^N$, Adv. Nonlinear Anal., 11 (2022), 741-756.  doi: 10.1515/anona-2021-0208. [18] U. Guarnotta, S. A. Marano and N. S. Papageorgiou, Multiple nodal solutions to a Robin problem with sign-changing potential and locally defined reaction, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30 (2019), 269-294.  doi: 10.4171/RLM/847. [19] S. Kakutani, A proof of Schauder's theorem, J. Math. Soc. Japan, 3 (1951), 228-231.  doi: 10.2969/jmsj/00310228. [20] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Commun. Partial Differ. Equ., 16 (1991), 311-361.  doi: 10.1080/03605309108820761. [21] Z. Liu, D. Motreanu and S. Zeng, Positive solutions for nonlinear singular elliptic equations of $p$-Laplacian type with dependence on the gradient, Calc. Var. Partial Differ. Equ., 58 (2019), 22 pp. doi: 10.1007/s00526-018-1472-1. [22] S. A. Marano, G. Marino and A. Moussaoui, Singular quasilinear elliptic systems in $\mathbb{R}^N$, Ann. Mat. Pura Appl., 198 (2019), 1581-1594.  doi: 10.1007/s10231-019-00832-1. [23] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal., 105 (1989), 267-284.  doi: 10.1007/BF00251503. [24] N. S. Papageorgiou and P. Winkert, Singular Dirichlet $(p, q)$-equations, Mediterr. J. Math., 18 (2021), Paper no. 141. doi: 10.1007/s00009-021-01780-y. [25] P. Pucci and J. Serrin, The maximum principle, Prog. Nonlinear Differential Equations Appl. 73, Birkhäuser Verlag, Basel, 2007. [26] M. C. Rezende and C. A. Santos, Positive solutions for a quasilinear elliptic problem involving sublinear and superlinear terms, Tokyo J. Math., 38 (2015), 381-407.  doi: 10.3836/tjm/1452806047.
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