# zbMATH — the first resource for mathematics

A gradient bound for entire solutions of quasi-linear equations and its consequences. (English) Zbl 0819.35016
The paper deals with a maximum principle for entire solutions to quasilinear elliptic equations of the form $$\text{div}(\Phi' (| \nabla u|^ 2) \nabla u)= F'(u)$$ and various consequences which can be derived from it. It is assumed that $$F\geq 0$$ and $$\Phi$$ satisfies hypotheses modeled on one of the following two typical cases: (A) the $$p$$-Laplacian, i.e. $$\Phi(s)= {2\over p} [(a+ s)^{p/2}- a^{p/2}]$$, $$a\geq 0$$, $$p>1$$, (B) the mean curvature operator, i.e. $$\Phi(s)= 2(\sqrt{1 +s}-1)$$. The main result is the inequality $P(u; x)\equiv 2\Phi' (| \nabla u(x)|^ 2) | \nabla u(x)|^ 2- \Phi(| \nabla u(x)|^ 2)- 2F(u (x))\leq 0$ for bounded entire solutions in case (A) and for bounded entire solutions which also have bounded derivatives in case (B). The proof is based on an elliptic differential inequality for $$P(u; x)$$ and a compactness argument on the space of solutions to the above equation. As a consequence the authors obtain the following Liouville type theorem: If, under the above assumptions on $$u$$, there exists $$x_ 0\in \mathbb{R}^ n$$ with $$u(x_ 0) =0$$ and if, moreover, in case (A), $$p\geq 2$$, $$F$$ behaves like $$F(u)= O(| u-u_ 0 |^ p)$$ $$(u\to u_ 0)$$ at any zero $$u_ 0$$ of $$F$$, then it follows that $$u= \text{const}$$ in $$\mathbb{R}^ n$$. Another consequence is the montonicity of the scaled energy $$E(r)= {1\over {r^{n-1}}} \int_{B_ r} (\Phi (| \nabla u|^ 2)+ 2F(u)) dx$$ with $$B_ r= \{x\in \mathbb{R}^ n \mid | x|\leq r\}$$. From this it can be concluded that any of the above solutions with $$F(u)\in L_ 1 (\mathbb{R}^ n)$$ must be constant. The final result states that if $$P(u; x_ 0)=0$$ for some $$x_ 0\in \mathbb{R}^ n$$ then $$u$$ must be of the form $$u(x)= g(\alpha+ bx)$$ with $$\alpha\in \mathbb{R}$$ and $$b\in \mathbb{R}^ n$$.

##### MSC:
 35B50 Maximum principles in context of PDEs 35J60 Nonlinear elliptic equations
Full Text:
##### References:
 [1] di Benedetto, Nonlinear Anal. 7 pp 827– (1983) [2] Bombieri, Arch. Rational Mech. Anal. 32 pp 255– (1965) [3] Garofalo, Amer. J. Math. 11 pp 9– (1989) [4] , and , Symmetry of positive solutions of nonlinear elliptic equations in Rn, pp. 369–402 in: Mathematical Analysis and Applications, ed., Advances in Mathematics Vol. 7A, Academic Press, New York, 1981. [5] Convergence problems for functionals and operators, pp. 131–188 in: Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, , and , eds., Pitagora, Bologna, 1979. [6] and , Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag. Berlin, 1983. · Zbl 0361.35003 [7] and , Linear and Quasilinear Equations of Elliptic Type, Academic Press, New York, 1968. [8] Ladyzhenskaya, Comm. Pure Appl. Math. 23 pp 677– (1970) [9] Lewis, Indiana Univ. Math. J. 32 pp 849– (1983) [10] Lieberman, Nonlinear Anal. 12 pp 1203– (1988) [11] Boundary C1.{$$\beta$$} regularity of p-harmonic functions, preprint. [12] Modica, Comm. Pure Appl. Math. 38 pp 679– (1985) [13] Monotonicity of the energy for entire solutions of semilinear elliptic equations, pp. 843–850 in: Partial Differential Equations and the Calculus of Variations: Essays in Honor of E. de Giorgi, Vol. 2, et al., eds., BirkhĂ¤user, Boston, 1989. [14] Modica, Boll. Un. Mat. Ital. 5 pp 614– (1980) [15] Maximum Principles and Their Applications, Academic Press, New York, 1981. · Zbl 0454.35001 [16] Tolksdorf, J. Differential Equations 51 pp 126– (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.