For several classes of functions including the special case , , we obtain Liouville type, boundedness and symmetry results for solutions of the non-linear -Laplacian problem defined on the whole space . Suppose is a solution. We have that either
(1) if does not change sign, then is a constant (hence, or or ); or
(2) if changes sign, then , moreover on ; or
(3) if on and the level set lies on one side of a hyperplane and touches that hyperplane, i.e., there exists and such that for all , then depends on one variable only (in the direction of ).
Submitted May 29, 2003. Published September 25, 2003.
Math Subject Classifications: 35J15, 35J25, 35J60.
Key Words: Quasi-linear elliptic equations, comparison Principle, boundary blow-up solutions, moving plane method, sliding method, symmetry of solution.
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| Zhenyi Zhao |
Department of Mathematical Sciences
Beijing, 100084, China
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