Some recent results on singular p-Laplacian equations

Guarnotta, Umberto; Livrea, Roberto; Marano, Salvatore A.

doi:10.1515/dema-2022-0031

Artykuł - szczegóły

Tytuł artykułu

Some recent results on singular p-Laplacian equations

Autorzy

Guarnotta Umberto , Livrea Roberto , Marano Salvatore A.

Wybrane pełne teksty z tego czasopisma

http://www.degruyter.com/view/j/dema

Identyfikatory

DOI

10.1515/dema-2022-0031

Warianty tytułu

Języki publikacji

Abstrakty

A short account of some recent existence, multiplicity, and uniqueness results for singular p-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided.

Słowa kluczowe

quasi-linear elliptic equation gradient dependence singular terms entire solutions strong solution

równanie eliptyczne gradient rozwiązanie mocne

Wydawca

De Gruyter

Czasopismo

Demonstratio Mathematica

Rocznik

2022

Tom

Vol. 55, nr 1

Strony

416--428

Opis fizyczny

Bibliogr. 86 poz.

Twórcy

autor

Guarnotta Umberto

umberto.guarnotta@unipa.it

Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy

autor

Livrea Roberto

roberto.livrea@unipa.it

Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy

autor

Marano Salvatore A.

marano@dmi.unict.it

Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy

Bibliografia

[1] J. Hernández, F. J. Mancebo, and J. M. Vega, Nonlinear singular elliptic problems: recent results and open problems, in: Nonlinear Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Applications, Vol. 64, Birkhäuser, Basel, 2005, pp. 227–242.
[2] J. Hernández and F. J. Mancebo, Singular elliptic and parabolic equations, in: M. Chipot and P. Quittner (eds), Handbook of Differential Equations, Vol. 3, Elsevier, Amsterdam, 2006, pp. 317–400.
[3] V. Radulescu, Singular phenomena in nonlinear elliptic problems: from blow-up boundary solutions to equations with singular nonlinearities, in: Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. IV, Elsevier/North-Holland, Amsterdam, 2007, pp. 485–593.
[4] M. Ghergu and V. D. Radulescu, Singular elliptic problems: bifurcation and asymptotic analysis, Oxford Lecture Ser. Math. Appl., Vol. 37, Oxford University Press, Oxford, 2008.
[5] K. Saoudi, The fibering map approach to a ( )p x -Laplacian equation with singular nonlinearities and nonlinear Neumann boundary conditions, Rocky Mountain J. Math. 48 (2018), 927–946.
[6] N. S. Papageorgiou, C. Vetro, and F. Vetro, Singular Neumann (p q, )-equations, Positivity 24 (2021), 1017–1040.
[7] N. S. Papageorgiou, V. Radulescu, and D. Repovs, Robin double-phase problems with singular and superlinear terms, Nonlinear Anal. Real World Appl. 58 (2021), 103217.
[8] N. S. Papageorgiou, V. Radulescu, and D. Repovs, Positive solutions for nonlinear Neumann problems with singular terms and convections, J. Math. Pures Appl. (9) 136 (2020), 1–21.
[9] I. De Bonis and D. Giachetti, Nonnegative solutions for a class of singular parabolic problems involving p-Laplacian, Asymptot. Anal. 91 (2015), 147–183.
[10] F. Oliva and F. Petitta, A nonlinear parabolic problem with singular terms and nonregular data, Nonlinear Anal. 194 (2020), 111472.
[11] J. Giacomoni, D. Kumar, and K. Sreenadh, A qualitative study of (p q, ) singular parabolic equations: local existence, Sobolev regularity and asymptotic behavior, Adv. Nonlinear Stud. 21 (2021), 199–227.
[12] S. Ciani and U. Guarnotta, On a non-homogeneous parabolic equation with singular and convective reaction, preprint.
[13] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011.
[14] C. A. Stuart, Existence and approximation of solutions of non-linear elliptic equations, Math. Z. 147 (1976), 53–63.
[15] M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differ. Equ. 2 (1977), 193–222.
[16] M. M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differ. Equ. 14 (1989), 1315–1327.
[17] A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991), 721–730.
[18] Y. S. Choi, A. C. Lazer, and P. J. McKenna, Some remarks on a singular elliptic boundary value problem, Nonlinear Anal. 32 (1998), 305–314.
[19] L. Orsina and F. Petitta, A Lazer-McKenna type problem with measures, Differ. Integral Equ. 29 (2016), 19–36.
[20] F. Oliva and F. Petitta, Finite and infinite energy solutions of singular elliptic problems: Existence and uniqueness, J. Differ. Equ. 264 (2018), 311–340.
[21] C. Aranda and E. LamiDozo, Multiple solutions to a singular Lane-Emden-Fowler equation with convection term, Electron J. Differ. Equ. 2007 (2007), Paper no. 5, 21 p.
[22] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM Control Optim. Calc. Var. 14 (2008), 411–426.
[23] D. Arcoya, J. Carmona, T. Leonori, P. J. Martínez-Aparicio, L. Orsina, and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differ. Equ. 246 (2009), 4006–4042.
[24] U. Guarnotta, S. A. Marano, and D. Motreanu, On a singular Robin problem with convection terms, Adv. Nonlinear Stud. 20 (2020), 895–909.
[25] N. S. Papageorgiou and Y. Zhang, Nonlinear nonhomogeneous Dirichlet problems with singular and convection terms, Bound. Value Probl. 2020 (2020), Paper no. 153, 21 pp.
[26] S. M. Gomes, On a singular nonlinear elliptic problem, SIAM J. Math. Anal. 17 (1986), 1359–1369.
[27] J. Chabrowski, Existence results for singular elliptic equations, Hokkaido Math. J. 20 (1991), 465–475.
[28] S. B. Cui, Positive solutions for Dirichlet problems associated to semilinear elliptic equations with singular nonlinearity, Nonlinear Anal. 21 (1993), 181–190.
[29] L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differ. Equ. 37 (2010), 363–380.
[30] J. Giacomoni, D. Kumar, and K. Sreenadh, Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems, Calc. Var. Partial Differ. Equ. 60 (2021), 121.
[31] N. S. Papageorgiou and P. Winkert, Singular Dirichlet (p q, )-equations, Mediterr. J. Math. 18 (2021), 141.
[32] W. Fulks and J. S. Maybee, A singular non-linear equation, Osaka Math. J. 12 (1960), 1–19.
[33] W. L. Perry, A monotone iterative technique for solution of pth order (
[34] G. Astrita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974.
[35] K. Perera and E. A. B. Silva, Existence and multiplicity of positive solutions for singular quasilinear problems, J. Math. Anal. Appl. 323 (2006), 1238–1252.
[36] K. Perera and Z. Zhang, Multiple positive solutions of singular p-Laplacian problems by variational methods, Bound. Value Probl. 2005 (2005), 377–382.
[37] C. Aranda and T. Godoy, Existence and multiplicity of positive solutions for a singular problem associated to the p-Laplacian operator, Electron. J. Differ. Equ. 2004 (2004), Paper No. 132, 15 pp.
[38] A. Canino, B. Sciunzi, and A. Trombetta, Existence and uniqueness for p-Laplace equations involving singular nonlinearities, NoDEA Nonlinear Differ. Equ. Appl. 23 (2016), 8.
[39] J. V. Gonçalves, M. L. Carvalho, and C. A. Santos, About positive ( )W Ωloc1,Φ -solutions to quasilinear elliptic problems with singular semilinear term, Topol. Methods Nonlinear Anal. 53 (2019), 491–517.
[40] L. M. De Cave, R. Durastanti, and F. Oliva, Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data, NoDEA Nonlinear Differ. Equ. Appl. 25 (2018), 18.
[41] V. De Cicco, D. Giachetti, F. Oliva, and F. Petitta, The Dirichlet problem for singular elliptic equations with general nonlinearities, Calc. Var. Partial Differ. Equ. 58 (2019), 129.
[42] S. T. Kyritsi and N. S. Papageorgiou, Pairs of positive solutions for singular p-Laplacian equations with a p-superlinear potential, Nonlinear Anal. 73 (2010), 1136–1142.
[43] N. S. Papageorgiou and G. Smyrlis, A bifurcation-type theorem for singular nonlinear elliptic equations, Methods Appl. Anal. 22 (2015), 147–170.
[44] N. S. Papageorgiou and G. Smyrlis, Nonlinear elliptic equations with singular reaction, Osaka J. Math. 53 (2016), 489–514.
[45] N. S. Papageorgiou and P. Winkert, Solutions with sign information for nonlinear nonhomogeneous problems, Math. Z. 292 (2019), 871–891.
[46] F. Faraci and G. Smyrlis, Three solutions for a singular quasilinear elliptic problem, Proc. Edinb. Math. Soc. (2) 62 (2019), 179–196.
[47] J. Giacomoni, I. Schindler, and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 117–158.
[48] J. I. Diaz, J. M. Morel, and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differ. Equ. 12 (1987), 1333–1344.
[49] P. Candito, U. Guarnotta, and K. Perera, Two solutions for a parametric singular p-Laplacian problem, J. Nonlinear Var. Anal. 4 (2020), 455–468.
[50] J. V. A. Gonçalves, M. C. Rezende, and C. A. Santos, Positive solutions for a mixed and singular quasilinear problem, Nonlinear Anal. 74 (2011), 132–140.
[51] D. D. Hai, On a class of singular p-Laplacian boundary value problems, J. Math. Anal. Appl. 383 (2011), 619–626.
[52] B. Bougherara, J. Giacomoni, and J. Hernández, Existence and regularity of weak solutions for singular elliptic problems, Electron. J. Differ. Equ. Conf. 22 (2015), 19–30.
[53] A. Canino and B. Sciunzi, A uniqueness result for some singular semilinear elliptic equations, Commun. Contemp. Math. 18 (2016), 1550084.
[54] F. Faraci and D. Puglisi, A singular semilinear problem with dependence on the gradient, J. Differ. Equ. 260 (2016), 3327–3349.
[55] 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), 28.
[56] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
[57] P. Pucci and J. Serrin, The maximum principle, Progress in Nonlinear Differential Equations and Applications, Vol. 73, Birkhäuser Verlag, Basel, 2007.
[58] M. L. Carvalho, J. V. Goncalves, E. D. Silva, and C. A. P. Santos, A type of Brézis-Oswald problem to Phi-Laplacian operator with strongly-singular and gradient terms, Calc. Var. Partial Differ. Equ. 60 (2021), 195.
[59] T. Kusano and C. A. Swanson, Entire positive solutions of singular semilinear elliptic equations, Japan. J. Math. (N.S.) 11 (1985), 145–155.
[60] R. Dalmasso, Solutions daéquations elliptiques semi-linéaires singulières, Ann. Mat. Pura Appl. (4) 153 (1988), 191–201, (in French).
[61] A. L. Edelson, Entire solutions of singular elliptic equations, J. Math. Anal. Appl. 139 (1989), 523–532.
[62] A. V. Lair and A. W. Shaker, Entire solution of a singular semilinear elliptic problem, J. Math. Anal. Appl. 200 (1996), 498–505.
[63] A. V. Lair and A. W. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl. 211 (1997), 371–385.
[64] Z. Zhang, A remark on the existence of entire solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl. 215 (1997), 579–582.
[65] F. C. S. Cîrstea and V. D. Rădulescu, Existence and uniqueness of positive solutions to a semilinear elliptic problem in RN , J. Math. Anal. Appl. 229 (1999), 417–425.
[66] J. V. Gonçalves and C. A. Santos, Existence and asymptotic behavior of non-radially symmetric ground states of semilinear singular elliptic equations, Nonlinear Anal. 65 (2006), 719–727.
[67] J. V. Gonçalves, A. L. Melo, and C. A. Santos, On existence of Linfty-ground states for singular elliptic equations in the presence of a strongly nonlinear term, Adv. Nonlinear Stud. 7 (2007), 475–490.
[68] M. Ghergu and V. D. Rădulescu, Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, J. Math. Anal. Appl. 333 (2007), 265–273.
[69] J. V. Gonçalves and F. K. Silva, Existence and nonexistence of ground state solutions for elliptic equations with a Convection term, Nonlinear Anal. 72 (2010), 904–915.
[70] J. Chabrowski and M. König, On entire solutions of elliptic equations with a singular nonlinearity, Comment. Math. Univ. Carolin. 31 (1990), 643–654.
[71] C. O. Alves, J. V. Gonçalves, and L. A. Maia, Singular nonlinear elliptic equations in RN , Abstr. Appl. Anal. 3 (1998), 411–423.
[72] A. J. Callegari and M. B. Friedman, An analytical solution of a nonlinear, singular boundary value problem in the theory of viscous fluids, J. Math. Anal. Appl. 21 (1968), 510–529.
[73] A. Callegari and A. Nachman, Some singular, nonlinear differential equations arising in boundary layer theory, J. Math. Anal. Appl. 64 (1978), 96–105.
[74] A. Callegari and A. Nachman, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), 275–281.
[75] A. C. Fowler, Mathematical Models in the Applied Sciences, Cambridge University Press, Cambridge, 1997.
[76] J. V. Gonçalves and C. A. Santos, Positive solutions for a class of quasilinear singular equations, Electron. J. Differ. Equ. 2004 (2004), 56, 15 pp.
[77] D.-P. Covei, Existence and uniqueness of positive solutions to a quasilinear elliptic problem in R N, Electron. J. Differ. Equ. 2005 (2005), 139, 15 pp.
[78] D.-P. Covei, Existence and asymptotic behavior of positive solution to a quasilinear elliptic problem in RN , Nonlinear Anal. 69 (2008), 2615–2622.
[79] X. Liu, Y. Guo, and J. Liu, Solutions for singular p-Laplacian equations in RN , J. Syst. Sci. Complex. 22 (2009), 597–613.
[80] S. Carl and K. Perera, Generalized solutions of singular p-Laplacian problems in RN , Nonlinear Stud. 18 (2011), 113–124.
[81] C. A. Santos, Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms, Nonlinear Anal. 72 (2010), 3813–3819.
[82] 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.
[83] P. Drábek and L. Sankar, Singular quasilinear elliptic problems on unbounded domains, Nonlinear Anal. 109 (2014), 148–155.
[84] J. I. Diaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 521–524.
[85] M. Chhetri, P. Drábek, and R. Shivaji, Analysis of positive solutions for classes of quasilinear singular problems on exterior domains, Adv. Nonlinear Anal. 6 (2017), 447–459.
[86] L. Gambera and U. Guarnotta, Strongly singular convective elliptic equations in RN driven by a non-homogeneous operator, Comm. Pure Appl. Anal. DOI: http://dx.doi.org/10.3934/cpaa.2022088.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).

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Bibliografia

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