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Volume 20,     Number 4,     Winter 2012

 

EFFECT OF WEIGHTS ON STABLE SOLUTIONS
OF A QUASILINEAR ELLIPTIC EQUATION
MOSTAFA FAZLY

Abstract. In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasi-linear elliptic equation —Δpu = f(x)F(u) in ℜn where p ≥ 2, 0 ≤ fC (ℜn) and FC1(ℜ). We refer to f(x) as weight and to F(u) as nonlinearity. The remarkable fact is that if the weight function is bounded from below by a strict positive constant that is 0 <C≤ f then it does not have much impact on the stable solutions, however, a nonnegative weight that is 0 ≤ f will push certain critical dimensions. This analytical observation has potential to be applied in various models to push certain well-known critical dimensions.
For a general nonlinearity FC1(ℜ) and f(x) = |x|α, we prove Liouville theorems in dimensions n≤4(p+α)/(p-1)+p, for bounded radial stable solutions. For specific nonlinearities F(u) = eu, uq where q > p - 1 and -uq where q < 0, known as the Gelfand, the Lane-Emden and the negative exponent non- linearities, respectively, we prove Liouville theorems for both radial finite Morse index (not necessarily bounded) and stable (not necessarily radial nor bounded) solutions.

 

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