In this paper, we investigate the existence/non-existence, exact asymptotic behavior at the boundary and uniqueness of solutions to infinite boundary-value problems of the -Laplace equation with lower-order nonlinear gradient terms in bounded open subsets of . The main tool is a recently obtained comparison principle, that captures the subtlety that arises due to the presence of the nonlinear gradient term. To the best of our knowledge, the results contained herein are comprehensive and cover topics that have not been addressed in the literature, except for very special cases.
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References
1.
AlarcónS.QuaasA. (2013). Large viscosity solutions for some fully nonlinear equations. Nonlinear Differential Equations and Applications, 20, 1453–1472.
2.
BandleC.EssénM. (1994). On the solutions of quasilinear elliptic problems with boundary blow-up, partial differential equations of elliptic type (Cortona, 1992). Symposium Mathematics, XXXV, Cambridge University Press, Cambridge, 93–111.
3.
BandleC.GiarrussoE. (1996). Boundary blowup for semilinear elliptic equations with nonlinear gradient terms. Advances in Differential Equations, 1, 133–150.
4.
BhattacharyaT.MohammedA. (2021). Maximum principles for -Hessian equations with lower order terms on unbounded domains. Journal of Geometric Analysis, 31(4), 3820–3862.
5.
BieberbachL. (1916). und die automorphen funktionen. Mathematische Annalen, 77, 173–212.
6.
ChenH.FelmerP.QuaasA. (2015). Large solutions to elliptic equations involving fractional Laplacian. Annales de l’Institut Henri Poincaré C — Analyse Non Linéaire, 32(6), 1199–1228.
7.
CîrsteaF.-C.RădulescuV. (2006). Nonlinear problems with boundary blow-up: A Karamata regular variation theory approach. Asymptotic Analysis, 46(3-4), 275–298.
8.
CîrsteaF.-C.RădulescuV. (2007). Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type. Transactions of the American Mathematical Society, 359(7), 3275–3286.
9.
DíazG.LetelierR. (1993). Explosive solutions of quasilinear elliptic equations: Existence and uniqueness. Nonlinear Analysis, 20(2), 97–125.
10.
DuY. (2003). Boundary blow-up solutions and their applications(pp. 89–97). World Scientific Publishing Co., Inc. ISBN: 981-238-262-3.
11.
DuY. (2004). Asymptotic behavior and uniqueness results for boundary blow-up solutions. Differential Integral Equations, 17(7–8), 819–834.
12.
DuY.GuoZ. (2003). Boundary blow-up solutions and their applications in quasilinear elliptic equations. Journal d’Analyse Mathématique, 89, 277–302.
13.
GherguM.RadulescuV. (1997). Large solutions of quasilinear elliptic equations: Existence and qualitative properties. Bollettino dell’Unione Matematica Italiana B (7), 11, 227–252.
14.
GilbargD.TrudingerN. S. (1983). Elliptic partial differential equations of second order (2nd ed.). Springer.
15.
GladialiF.PorruG. (1998). Estimates for explosive solutions to -Laplace equations, Progress in Partial Differential Equations, Vol I (Pont-à-Mousson, 1997), (PP. 177–127). Pitman Re. Notes Math. Ser. 383, Longman, Harlow.
16.
HörmanderL. (1983). The Analysis of Partial differential operators I. Springer, Berlin, ix+391 pp.
17.
KarlsM.MohammedA. (2016). Ahmed solutions of -Laplace equations with infinite boundary values: The case of non-autonomous and non-monotone nonlinearities. Proceedings of the Edinburgh Mathematical Society, 59(4), 959–987.
18.
KellerJ. B. (1957). On solutions of . Communications on Pure and Applied Mathematics, 10, 503–510.
19.
LairA. V.MohammedA. (2023). Necessary and sufficient conditions for the existence of large solutions to semilinear elliptic equations with gradient terms. Journal of Differential Equations, 374, 593–631.
20.
LasryJ. M.LionsP. L. (1989). Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. Mathematische Annalen, 283, 583–630.
21.
LeonoriT.PorrettaA. (2016). Large solutions and gradient bounds for quasilinear elliptic equations. Communications in Partial Differential Equations, 41, 952–998.
22.
LeonoriT.PorrettaA.RieyG. (2017). Comparison principles for p-Laplace equations with lower order terms. Annali di Matematica Pura ed Applicata, 196(3), 877–903.
23.
LiebermanG. (1988). Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Analysis, 12, 1203–1219.
24.
LiebermanG. (1991). The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Communications in Partial Differential Equations, 16, 311–361.
25.
LiebermanG. (2011). Asymptotic behavior and uniqueness of blow-up solution of quasilinear elliptic equation. Journal d’Analyse Mathématique, 115, 213–249.
MohammedA. (2007). Boundary asymptotic and uniqueness of solutions to the -Laplacian with infinite boundary values. Journal of Mathematical Analysis and Applications, 325(1), 480–489.
28.
MohammedA. (2010). Existence and uniqueness of nonnegative solutions for boundary blow-up problem. Journal of Mathematical Analysis and Applications, 371, 534–545.
29.
MohammedA.RădulescuV.VitoloA. (2020). Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness. Advances in Nonlinear Analysis, 9(1), 39–64.
30.
MohammedA.VitoloA. (2019). Large solutions of fully nonlinear equations: Existence and uniqueness. NoDEA Nonlinear Differential Equations Application, 26(6), Article number 42.
31.
MohammedA.VitoloA. (2024a). The effects of nonlinear perturbation terms on comparison principles for the -Laplacian. Bulletin of Mathematical Sciences, 14(2), 2450005.
32.
MohammedA.VitoloA. (2024b). Remarks on comparison principles for p-Laplacian with extension to (p, q)-Laplacian. Bulletin of Mathematical Sciences, 14(3), 2450011.
33.
OssermanR. (1957). On the inequality . Pacific Journal of Mathematics, 7, 1641–1647.
34.
PorrettaA. (2004). Local estimates and large solutions for some elliptic equations with absorption. Advances in Differential Equations, 9(3-4), 329–351.
35.
RademacherH. (1943). Einige besondere probleme partieller Differentialgleichungen. In Die differential und integralgleichungen der mechnik und physik, I (2nd ed., pp. 838–845). Rosenberg.
36.
VéronL. (2017). Local and global aspects of quasilinear degenerate equations: Quasilinear singular elliptic problems. World Scientific.
37.
ZhangZ. (2006). Existence of large solutions for a semilinear elliptic problem via explosive sub-supersolutions. Electronic Journal of Differential Equations, 2006(2), 1–8.
38.
ZhangZ. (2014). The existence and boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms. Advances in Nonlinear Analysis, 3(3), 165–185.
39.
ZhangZ. (2017). Boundary behavior of large solutions to -Laplacian elliptic equations. Nonlinear Analysis Real World Applications, 33, 40–57.
40.
ZhangX.DuY. (2018). Sharp conditions for the existence of boundary blow-up solutions to the Monge–Ampére equation. Calculus of Variations and Partial Differential Equations, 57(2), article number 30.
41.
ZhangX.FengM. (2019). The existence and asymptotic behavior of boundary blow-up solutions to the -Hessian equation. Journal of Differential Equations, 267(8), 4626–4672.
42.
ZhangX.KanS. (2023). Sufficient and necessary conditions on the existence and estimates of boundary blow-up solutions for singular -Laplacian equations. Acta Mathematica Scientia (Series B) (Engl. Ed.), 43(3), 1175–1194.
