1D logistic reaction and p -Laplacian diffusion as p goes to one
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1D logistic reaction and p-Laplacian diffusion as p goes to one José Sabina Lis1
· Sergio Segura de León2
Received: 12 September 2020 / Revised: 3 November 2020 / Accepted: 6 November 2020 © Università degli Studi di Napoli "Federico II" 2020
Abstract This work discusses the existence of the limit as p goes to 1 of the nontrivial solutions to the one-dimensional problem: − |u x | p−2 u x x = λ|u| p−2 u − |u|q−2 u 0 λ1 (−Δ p ), the first eigenvalue of −Δ p , while the best understood issues have to do with positive solutions. In fact, there exists a unique positive solution u λ , bifurcating from zero at λ = λ1 , whose asymptotic profile as λ → ∞ has been studied in full detail (see [14,17,18,20] for references dealing with the ‘genuine’ nonlinear diffusion case p = 2). As a characteristic feature, 1
u λ ∞ ≤ λ q− p and λ
− q−1 p
u λ → 1 as λ → ∞,
the last convergence being uniform in compact sets of Ω. Moreover, while first estimate is strict in the case 1 < p ≤ 2, the complementary range p > 2 enjoys especial 1
phenomena. In fact, the region {u λ (x) = λ q− p } becomes nonempty and converges to Ω as λ → ∞ ([17,20] and Remark 2 below). On the other hand, by means of variational arguments it can be shown the existence of an arbitrarily large number of further nontrivial (two-signed) solutions to (1.1) when λ → ∞ (see [15] for this kind of results in a closely related problem). In the present work, we are only concerned with the one-dimensional case:
−(|u x | p−2 u x )x = λ|u| p−2 u − |u|q−2 u u(0) = u(1) = 0.
0 0,
(2.7)
The existence and uniqueness of a maximal solution for this and a slightly larger class of problems have been considered in the literature (see [18,25]). However, we can proceed here in a direct way. In fact, the function E(v, vt ) defined by E(v, vt ) =
1 1 1 |vt | p + V (v), V (v) = |v| p − |v|q , p p q
(2.8)
is conserved through the solutions to (2.7). To ascertain the response of problem (2.7) it is enough to assume that α ≥ 0 since the equation is invariant with respect to the change v → −v. According to the values of α ≥ 0 and employing the fact that E(v, vt ) = V (α),
(2.9)
three cases are possible. (a) α = 1 which implies v = 1. In this regard, the restriction 1 < p ≤ 2 is crucial (see Remark 2 below). (b) α > 1. A unique solution v exists, it is increasing, satisfies V (v) < V (α) and blows-up at t = ω(α), ω(α) := { p }
− 1p
∞
α
ds 1
(V (α) − V (s)) p
< ∞.
c) 0 < α < 1. Again, a unique solution v exists which decreases from α to −α T when 0 ≤ t ≤ T , vanishes at t = , is symmetric with respect to t = T and 2 becomes periodic with period 2T where T = T (α) = 2{ p }
− 1p
0
α
ds 1
(V (α) − V (s)) p
.
(2.10)
123
J. Sabina, S. Segura
Coming back to (2.6), let v be any of its nontrivial solutions. It can be assumed without loss of generality that it verifies vt (0) > 0. Such solution must necessarily exhibit a first maximum at t = tm > 0 with value v(tm ) = α. Since v˜ = v(t − tm ) solves (2.7) then α must satisfy: 0 < α < 1. 1
This assertion
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