Uniqueness of positive solutions to some degenerate equations in radially symmetric domains

J. García Melián and J. Sabina de Lis Dpto. Analisis Matematico Universidad de la Laguna 38271 - La Laguna Tenerife, Spain

                           jjgarmel@ull.es

This talk is concerned with uniqueness of positive solutions as λ → +∞ of the problem

\begin{matrix}−Δ_{p}u = λmu^{p−1} − u^{q} + g(u) & in & D \\ u = 0 & on & ∂D , \end{matrix}  (P)

where Δ_{p}u = div(|∇u|^{p−2}∇u) stands for the p-Laplacian operator, p > 2, q > p − 1, m > 0 and D is a bounded rotationally invariant domain of ℝ^{ℕ}, i.e. a ball or an annulus. The perturbation term g ∈ C^{1}(ℝ) is supposed to verify

lim_{u→0+} \frac{g(u)}{u^{p−1}} = 0 ,  lim_{u→+∞} \frac{g(u)}{u^{q}} = 0 .

In addition, it is proved that every solution u to (P) in an arbitrary smooth bounded domain Ω of ℝ^{ℕ} (where uniqueness for λ large is still an open problem) exhibits a “dead core” O = {x ∈ Ω : u(x) = u_{0}(λ)}, for λ large, where u_{0}(λ) is the largest value of v verifying λmv^{p−1} − v^{q} + g(v) = 0. More importantly, an exact estimate for the distance from O to ∂Ω as λ → +∞ is also obtained.

Our main tool is the study of the Cauchy problem

\begin{matrix}−(r^{N−1}φ_{p}(u′))′ = r^{N−1}f(u, λ), r > d \\ u(d) = u_{0}, u′(d) = 0 \end{matrix}  (CP)

where φ_{p}(z) = |z|^{p−2}z, z ∈ ℝ, u_{0} ∈ ℝ, d ≥ 0, in the “critical” case f(u_{0}) = 0, where particular examples have been given where uniqueness fails. We prove the existence of a unique nontrivial solution u(r, d) in an interval of the form 0 ≤ r ≤ δ, showing also its differentiability with respect to parameters and data (by nontrivial it is meant u(r, d) < u_{0} if 0 < r ≤ δ). These results also allows us studying uniqueness of positive radial solutions to problems of the form

\begin{matrix}−Δ_{p}u = λf(u, λ) & in & D \\ u = 0 & on & ∂D , \end{matrix}

for a class of sublinear type f’s. More precisely, those solutions which are uniformly close to u_{0} in compacts of D.