## Duke Mathematical Journal

### Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient

#### Abstract

We study local and global properties of positive solutions of $-\Delta u=u^{p}|{\nabla u}|^{q}$ in a domain $\Omega$ of ${\mathbb{R}}^{N}$, in the range $p+q\gt 1$, $p\geq 0$, $0\leq q\lt 2$. We first prove a local Harnack inequality and nonexistence of positive solutions in ${\mathbb{R}}^{N}$ when $p(N-2)+q(N-1)\lt N$. Using a direct Bernstein method, we obtain a first range of values of $p$ and $q$ in which $u(x)\leq c(\operatorname{dist}(x,\partial \Omega ))^{\frac{q-2}{p+q-1}}$. This holds in particular if $p+q\lt 1+\frac{4}{N-1}$. Using an integral Bernstein method, we obtain a wider range of values of $p$ and $q$ in which all the global solutions are constants. Our result contains Gidas and Spruck’s nonexistence result as a particular case. We also study solutions under the form $u(x)=r^{\frac{q-2}{p+q-1}}\omega (\sigma )$. We prove existence, nonexistence, and rigidity of the spherical component $\omega$ in some range of values of $N$, $p$, and $q$.

#### Article information

Source
Duke Math. J., Volume 168, Number 8 (2019), 1487-1537.

Dates
Revised: 6 December 2018
First available in Project Euclid: 18 May 2019

https://projecteuclid.org/euclid.dmj/1558145273

Digital Object Identifier
doi:10.1215/00127094-2018-0067

Mathematical Reviews number (MathSciNet)
MR3959864

Zentralblatt MATH identifier
07080117

Subjects
Primary: 35J62: Quasilinear elliptic equations
Secondary: 35B08: Entire solutions

#### Citation

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. 168 (2019), no. 8, 1487--1537. doi:10.1215/00127094-2018-0067. https://projecteuclid.org/euclid.dmj/1558145273

#### References

• [1] M. F. Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Ration. Mech. Anal. 107 (1989), no. 4, 293–324.
• [2] M. F. Bidaut-Véron, M. Garcia-Huidobro, and L. Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3471–3515.
• [3] M. F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math. 84 (2001), 1–49.
• [4] M. F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), no. 3, 489–539.
• [5] H. Brézis and P. L. Lions, “A note on isolated singularities for linear elliptc equations” in Mathematical Analysis and Applications, Part A, Academic Press, New York, 1981, 263–266.
• [6] M. Burgos-Pérez, J. García-Mellían, and A. Quass, Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems, Discrete Contin. Dyn. Syst. 36 (2016), no. 9, 4703–4721.
• [7] G. Caristi and E. Mitidieri, Nonexistence of positive solutions of quasilinear equations, Adv. Differential Equations 2 (1997), no. 3, 319–359.
• [8] R. Fillipucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal. 70 (2009), no. 8, 2903–2916.
• [9] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598.
• [10] G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2–3, 311–361.
• [11] J. R. Licois and L. Véron, A class of nonlinear conservative elliptic equations in cylinders. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 26 (1998), no. 2, 249–283.
• [12] P. L. Lions, Résolution de problèmes elliptiques quasilinéaires, Arch. Rational Mech. Anal. 74 (1980), no. 4, 335–353.
• [13] E. Mitidieri and S. Pohozaev, A priori estimates and the absence of solutions to nonlinear partial differential equations and inequalities (in Russian) Tr. Mat. Inst. Steklova 234 (2001), 1–384; English translation in Proc. Steklov Inst. Math. 2001, no. 3, 1–362.
• [14] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302.
• [15] J. Serrin, Singularities of solutions of quasi-linear equations. Acta Math. 113 (1965), 219-240.
• [16] J. Smoller, Schock Waves and Reaction-diffusion Equations, 2nd ed., Grundlehren Math. Wiss. 258, Springer, New York, 1994.
• [17] L. Véron, Local and Global Aspects of Quasilinear Degenerate Elliptic Equations, World Scientific, Hackensack, 2017.