Qualitative properties of solutions for quasi-linear elliptic equations
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For several classes of functions including the special case ƒ(u) = up-1 - um, m > p - 1 > 0, we obtain Liouville type, boundedness and symmetry results for solutions of the non-linear p-Laplacian problem -∆pu = ƒ(u) defined on the whole space ℝn. Suppose u ∈ C2 (ℝn) is a solution. We have that either (1) if u doesn't change sign, then u is a constant (hence, u ≡ 1 or u ≡ 0 or u ≡ -1); or (2) if u changes sign, then u ∈ L∞ (ℝn), moreover |u| < 1 on ℝn; or (3) if |Du| > 0 on ℝn and the level set u-1 (0) lies on one side of a hyperplane and touches that hyperplane, i.e., there exists v ∈ Sn-1 and x0 ∈ u-1(0) such that v · (x - x0) ≥ 0 for all x ∈ u-1 (0), then u depends on one variable only (in the direction of v).
CitationZhao, Z. (2003). Qualitative properties of solutions for quasi-linear elliptic equations. Electronic Journal of Differential Equations, 2003(99), pp. 1-18.
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