A note on the stability of solutions to quasilinear elliptic equations

In this note, we prove a stability theorem for a class of quasilinear elliptic equations -δ<inf>p</inf> u = a(x)u-f(x,u) in Ω, u=0 on δΩ, where δ<inf>p</inf> u= div(/∇<inf>u</inf>u/ ^p-2∇<inf>u</inf>) 2 ≤ p < p < ∞ ,Ω ⊂ &Rdbl<inf>N</inf> is an open, smooth and bounded subset. We show that if u is an unstable solution of the above problem, then u vanishes at some point of Ω. In this work, a and f may change sign. © de Gruyter 2013.