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
quasilinear elliptic equation —Δ_{p}u = f(x)F(u) in ℜ^{n} where p ≥ 2,
0 ≤ f ∈ C (ℜ^{n}) and F ∈ C^{1}(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 wellknown critical dimensions.
For a general nonlinearity F ∈ C^{1}(ℜ) and f(x) = x^{α},
we prove Liouville theorems in dimensions n≤4(p+α)/(p1)+p,
for bounded radial stable solutions. For specific nonlinearities
F(u) = e^{u}, u^{q} where q
> p  1 and u^{q} where q < 0, known as
the Gelfand, the LaneEmden and the negative exponent non
linearities, respectively, we prove Liouville theorems for both
radialfinite Morse index (not necessarily bounded) and stable
(not necessarily radial nor bounded) solutions.
(Subscribers Only)
