INTRODUCTION 5
(0.9) such that
(0.22)
σ(λ, d) = and limt→∞ v(t, λ, d) = 0, or
σ(λ, d) and v (σ(λ, d), λ, d) = 0.
In chapter 4 we prove the following result concerning ground states of (0.9).
Theorem 5. The equation ( 0.9) has a ground state if and only if −1 q
p
2∗
1. Moreover, such a ground state is unique.
Ground states for (0.9) have been studied by several authors in the last two
decades (see [16, 17, 18, 20, 23]). In fact it is pointed out in [16] the interest of
this problem in plasma physics. Our proof of Theorem 5 borrows ideas developed
in the latter papers.
In Chapter 4 we study the stability of stationary solutions for the constant
density case (see (0.6)) under either (0.4) or (0.5). Our main result are:
Theorem 6. Let (λ, d) Σ with λ = ξ1(d). If
i) d 1 and ξ1 is strictly decreasing in a neighborhood of d, or
ii) d 1 and ξ1 is strictly increasing in a neighborhood of d,
then u(r, λ, d) = v
1
k+1
(r, λ, d), with r = |x|, is a radial steady state solution of
(0.9) subject to (0.7) stable under non-necessarily radial uniform perturbations of
the initial data.
Theorem 7. The steady state solution u 1 of (0.3) subject to (0.5) is unsta-
ble. Moreover, any solution u(x, t) of (0.3) subject to (0.5) such that 0 u(x, 0) 1
for all x
¯(0,
B 1) (respectively u(x, 0) 1 for all x
¯(0,
B 1)) satisfies u(x, t) 0
as t uniformly (respectively u(x, t) as t uniformly.)
Theorem 8. Let v(r, d) be a solution of (0.9) subject to (0.7), respectively
(0.9) subject to (0.8), such that vd(r, d) := ∂dv(r, d) 0 for some r (0, 1]. Then
u(r, d) = v
1
k+1
(r, d), with r = |x|, is a stationary solution of (0.3) subject to (0.4)
or to (0.5) respectively, that is linearly unstable. In particular, if (λ, d) Σ and
i) d 1, λ = ξ1(d), and ξ1 is strictly increasing in a neighborhood of d, or
ii) d 1, λ = ξ1(d), and ξ1 is strictly decreasing in a neighborhood of d, or
iii) λ = ξi(d) for i = 2, 3, . . .,
then u(r, λ, d) = v
1
k+1
(r, λ, d), with r = |x|, is a radial steady state solution of (0.9)
subject to (0.7) linearly unstable, respectively a radial steady state solution of (0.9)
subject to (0.8) linearly unstable.
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