[en] symmetries; [en] subcritical and supercritical exponent; [en] bifurcation; [en] boundary value problems; [en] Neumann boundary conditions; [en] Lane Emden problem; [en] Nehari manifold
Abstract :
[en] Assuming Bᵣ is a ball in Rᴺ, we analyze the positive solutions of the
problem
-Δ u+u = |u|ᵖ⁻²u in Bᵣ,
∂u/∂ν=0, on ∂Bᵣ,
that branch out from the constant solution u=1 as p grows from 2 to
+∞. The non-zero constant positive solution is the unique positive
solution for p close to 2. We show that there exist arbitrarily many
positive solutions as p → ∞ (in particular, for supercritical
exponents) or as r → ∞ for any fixed value of p > 2. We give explicit
lower bounds for p and r so that a given number of solutions exist.
The geometrical properties of those solutions are studied and
illustrated numerically. Our simulations motivate additional
conjectures. The structure of the least energy solutions (among all
or only among radial solutions) and other related problems are also
discussed.
Research center :
CREMMI - Modélisation mathématique et informatique
Disciplines :
Computer science Mathematics
Author, co-author :
Troestler, Christophe ; Université de Mons > Faculté des Sciences > Service d'Analyse numérique
