Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction
Annales de l'Institut Fourier, Volume 73 (2023) no. 6, pp. 2543-2566.

We provide a simple method for obtaining new Liouville-type theorems for positive supersolutions of the elliptic problem -Δ p u+b(x)|u| pq q+1 =c(x)u q in Ω, where Ω is an exterior domain in N with Np>1 and qp-1. In the case qp-1, we mainly deal with potentials of the type b(x)=|x| a , c(x)=λ|x| σ , where λ>0 and a,σ. We show that positive supersolutions do not exist in some ranges of the parameters p,q,a,σ, which turn out to be optimal. When q=p-1, we consider the above problem with general weights b(x)0, c(x)>0 and we assume that c(x)-b p (x) p p >0 for large |x|, but we also allow the case lim |x| [c(x)-b p (x) p p ]=0. The weights b and c are allowed to be unbounded. We prove that if this equation has a positive supersolution, then the potentials must satisfy a related differential inequality not depending on the supersolution. We also establish sufficient conditions for the nonexistence of positive supersolutions in relationship with the values of τ:=lim sup |x| |x|b(x). A key ingredient in the proofs is a generalized Hardy-type inequality associated to the p-Laplace operator.

Nous proposons une méthode simple pour obtenir de nouveaux théorèmes du type Liouville pour les supersolutions positives du problème elliptique -Δ p u+b(x)|u| pq q+1 =c(x)u q dans Ω, où Ω est un domaine extérieur dans N avec Np>1 et qp-1. Dans le cas qp-1, on traite principalement des potentiels du type b(x)=|x| a , c(x)=λ|x| σ , où λ>0 et a,σ. Nous montrons que les supersolutions positives n’existent pas dans certaines gammes de paramètres p,q,a,σ, qui s’avèrent optimales. Si q=p-1, on considère le problème ci-dessus avec des poids généraux b(x)0, c(x)>0 et on suppose que c(x)-b p (x) p p >0 si |x| est assez large, mais on admet aussi le cas lim |x| [c(x)-b p (x) p p ]=0. Les potentiels b et c sont autorisés à être non bornés. Nous prouvons que si cette équation a une supersolution positive, alors les potentiels doivent satisfaire une certaine inégalité différentielle ne dépendant pas de la supersolution. Nous établissons également des conditions suffisantes pour la non-existence de supersolutions positives en relation avec les valeurs de τ:=lim sup |x| |x|b(x). Un ingrédient clé des preuves est une inégalité généralisée de type Hardy associée à l’opérateur p-Laplace.

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DOI: 10.5802/aif.3576
Classification: 35J60, 35B53
Keywords: Nonlinear elliptic equation, Liouville theorem, supersolution, convection term.
Mot clés : Équation elliptique non linéaire, théorème de Liouville, supersolution, terme de convection.
Aghajani, Asadollah 1; Rădulescu, Vicenţiu D. 2, 3, 4, 5

1 School of Mathematics Iran University of Science and Technology Narmak, Tehran (Iran)
2 Faculty of Applied Mathematics AGH University of Science and Technology 30-059 Krakow (Poland)
3 Department of Mathematics University of Craiova 200585 Craiova (Romania)
4 Institute of Mathematics “Simion Stoilow” of the Romanian Academy PO Box 1-764, 014700 Bucharest, (Romania)
5 Brno University of Technology Faculty of Electrical Engineering and Communication Technická 3058/10 Brno 61600, (Czech Republic)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Aghajani, Asadollah; Rădulescu, Vicenţiu D. Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction. Annales de l'Institut Fourier, Volume 73 (2023) no. 6, pp. 2543-2566. doi : 10.5802/aif.3576. https://aif.centre-mersenne.org/articles/10.5802/aif.3576/

[1] Aghajani, A.; Cowan, C. A note on the nonexistence of positive supersolutions to elliptic equations with gradient terms, Ann. Mat. Pura Appl. (4), Volume 200 (2021) no. 1, pp. 125-135 | DOI | MR | Zbl

[2] Alarcón, S.; García-Melián, J.; Quaas, A. Nonexistence of positive supersolutions to some nonlinear elliptic problems, J. Math. Pures Appl. (9), Volume 99 (2013) no. 5, pp. 618-634 | DOI | MR | Zbl

[3] Alarcón, Salomón; García-Melián, Jorge; Quaas, Alexander Keller–Osserman type conditions for some elliptic problems with gradient terms, J. Differential Equations, Volume 252 (2012) no. 2, pp. 886-914 | DOI | MR | Zbl

[4] Alarcón, Salomón; García-Melián, Jorge; Quaas, Alexander Liouville type theorems for elliptic equations with gradient terms, Milan J. Math., Volume 81 (2013) no. 1, pp. 171-185 | DOI | MR | Zbl

[5] Alarcón, Salomón; García-Melián, Jorge; Quaas, Alexander Existence and non-existence of solutions to elliptic equations with a general convection term, Proc. Roy. Soc. Edinburgh Sect. A, Volume 144 (2014) no. 2, pp. 225-239 | DOI | MR | Zbl

[6] Arcoya, David; De Coster, Colette; Jeanjean, Louis; Tanaka, Kazunaga Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., Volume 268 (2015) no. 8, pp. 2298-2335 | DOI | MR | Zbl

[7] Armstrong, Scott N.; Sirakov, Boyan Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations, Volume 36 (2011) no. 11, pp. 2011-2047 | DOI | MR | Zbl

[8] Armstrong, Scott N.; Sirakov, Boyan Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 10 (2011) no. 3, pp. 711-728 | DOI | MR | Zbl

[9] Bandle, Catherine; Levine, Howard A. On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., Volume 316 (1989) no. 2, pp. 595-622 | DOI | MR | Zbl

[10] Berestycki, H.; Capuzzo-Dolcetta, I.; Nirenberg, L. Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., Volume 4 (1994) no. 1, pp. 59-78 | DOI | MR | Zbl

[11] Berestycki, Henri; Hamel, François; Nadirashvili, Nikolai The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc. (JEMS), Volume 7 (2005) no. 2, pp. 173-213 | DOI | MR | Zbl

[12] Berestycki, Henri; Hamel, François; Rossi, Luca Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl. (4), Volume 186 (2007) no. 3, pp. 469-507 | DOI | MR | Zbl

[13] Bidaut-Véron, Marie-Françoise Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., Volume 107 (1989) no. 4, pp. 293-324 | DOI | MR | Zbl

[14] Bidaut-Véron, Marie-Françoise; García-Huidobro, Marta; Véron, Laurent Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient, Duke Math. J., Volume 168 (2019) no. 8, pp. 1487-1537 | DOI | MR | Zbl

[15] Bidaut-Véron, Marie-Françoise; Garcia-Huidobro, Marta; Véron, Laurent A priori estimates for elliptic equations with reaction terms involving the function and its gradient, Math. Ann., Volume 378 (2020) no. 1-2, pp. 13-56 | DOI | MR | Zbl

[16] Bidaut-Véron, Marie-Françoise; Garcia-Huidobro, Marta; Véron, Laurent Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms, Discrete Contin. Dyn. Syst., Volume 40 (2020) no. 2, pp. 933-982 | DOI | MR | Zbl

[17] Bidaut-Véron, Marie-Françoise; Pohozaev, Stanislav Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., Volume 84 (2001), pp. 1-49 | DOI | MR | Zbl

[18] Birindelli, Isabeau; Demengel, Françoise Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math. (6), Volume 13 (2004) no. 2, pp. 261-287 | DOI | MR | Zbl

[19] Burgos-Pérez, M. Á.; García-Melián, J.; Quaas, A. Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 9, pp. 4703-4721 | DOI | MR | Zbl

[20] Capuzzo Dolcetta, I.; Cutrì, A. Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Commun. Contemp. Math., Volume 5 (2003) no. 3, pp. 435-448 | DOI | MR | Zbl

[21] Chen, Huyuan; Felmer, Patricio On Liouville type theorems for fully nonlinear elliptic equations with gradient term, J. Differential Equations, Volume 255 (2013) no. 8, pp. 2167-2195 | DOI | MR | Zbl

[22] Chen, Huyuan; Peng, Rui; Zhou, Feng Nonexistence of positive supersolutions to a class of semilinear elliptic equations and systems in an exterior domain, Sci. China Math., Volume 63 (2020) no. 7, pp. 1307-1322 | DOI | MR | Zbl

[23] Chen, Huyuan; Quaas, Alexander; Zhou, Feng On nonhomogeneous elliptic equations with the Hardy–Leray potentials, J. Anal. Math., Volume 144 (2021) no. 1, pp. 305-334 | DOI | MR | Zbl

[24] Cowan, Craig Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal., Volume 9 (2010) no. 1, pp. 109-140 | DOI | MR | Zbl

[25] Felmer, Patricio; Quaas, Alexander; Sirakov, Boyan Solvability of nonlinear elliptic equations with gradient terms, J. Differential Equations, Volume 254 (2013) no. 11, pp. 4327-4346 | DOI | MR | Zbl

[26] Galakhov, E. I. Positive solutions of a quasilinear elliptic equation, Mat. Zametki, Volume 78 (2005) no. 2, pp. 202-211 | DOI | MR

[27] Galakhov, E. I. Solvability of an elliptic equation with a gradient nonlinearity, Differ. Uravn., Volume 41 (2005) no. 5, p. 661-669, 718 | DOI | MR | Zbl

[28] Ghoussoub, Nassif; Moradifam, Amir Functional inequalities: new perspectives and new applications, Mathematical Surveys and Monographs, 187, American Mathematical Society, Providence, RI, 2013, xxiv+299 pages | DOI | MR | Zbl

[29] Ghoussoub, Nassif; Robert, Frédéric Hardy-singular boundary mass and Sobolev-critical variational problems, Anal. PDE, Volume 10 (2017) no. 5, pp. 1017-1079 | DOI | MR | Zbl

[30] Gidas, B.; Spruck, J. A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, Volume 6 (1981) no. 8, pp. 883-901 | DOI | MR | Zbl

[31] Gidas, Basilis Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, Nonlinear partial differential equations in engineering and applied science (Proc. Conf., Univ. Rhode Island, Kingston, R.I., 1979) (Lecture Notes in Pure and Appl. Math.), Volume 54, Dekker, New York (1980), pp. 255-273 | MR | Zbl

[32] Jeanjean, Louis; Sirakov, Boyan Existence and multiplicity for elliptic problems with quadratic growth in the gradient, Comm. Partial Differential Equations, Volume 38 (2013) no. 2, pp. 244-264 | DOI | MR | Zbl

[33] Kichenassamy, Satyanad; Véron, Laurent Singular solutions of the p-Laplace equation, Math. Ann., Volume 275 (1986) no. 4, pp. 599-615 | DOI | MR | Zbl

[34] Kondratiev, Vladimir; Liskevich, Vitali; Moroz, Vitaly Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains, Ann. Inst. H. Poincaré C Anal. Non Linéaire, Volume 22 (2005) no. 1, pp. 25-43 | DOI | MR | Zbl

[35] Kondratiev, Vladimir; Liskevich, Vitali; Moroz, Vitaly; Sobol, Zeev A critical phenomenon for sublinear elliptic equations in cone-like domains, Bull. London Math. Soc., Volume 37 (2005) no. 4, pp. 585-591 | DOI | MR | Zbl

[36] Quittner, Pavol; Souplet, Philippe Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007, xii+584 pages | MR | Zbl

[37] Rossi, Luca Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains, Commun. Pure Appl. Anal., Volume 7 (2008) no. 1, pp. 125-141 | DOI | MR | Zbl

[38] Serrin, James; Zou, Henghui Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., Volume 189 (2002) no. 1, pp. 79-142 | DOI | MR | Zbl

[39] Véron, Laurent Local and global aspects of quasilinear degenerate elliptic equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, xv+457 pages (Quasilinear elliptic singular problems) | DOI | MR | Zbl

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