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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
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