Critical quasilinear elliptic problems using concave-convex nonlinearities Claudiney Goulart Universidade Federal de Goiás - Regional Jataí
[email protected] Join work with Carvalho, M. L. M., Silva. E. D., Goncalves, J. V.
Introduction In this work we deal with existence, multiplicity and asymptotic behaviour of nonnegative solutions of the problem ∗
− ∆Φu = λa(x)|u|q−2u + b(x)|u|` −2u in Ω, u = 0 on ∂Ω,
(1)
where Ω ⊂ RN is a bounded smooth domain, λ > 0 is a parameter, `∗ := N `/(N − `) with 1 < ` < N and a, b : Ω → R are two indefinite functions in sign. The operator ∆Φ is named Φ-Laplacian which is given by ∆Φu = div(φ(|∇u|)∇u) where φ : (0, ∞) → (0, ∞) is a C 2-function satisfying (φ1) lim sφ(s) = 0, lim sφ(s) = ∞;
It is important to mention that, we shall prove that u 7→ hJλ0 (u), ui is 1,Φ in C 1 class. So that Nλ is a C 1-submanifold of W0 (Ω). At this moment we shall define the fibering map γu : [0, +∞) → R Z ∗ ` q ∗ t λt q ` a(x)|u| − ∗ b(x)|u| . Fiby: γu(t) := Jλ(tu). = Φ(t|∇u|) − q ` Ω bering maps have been considered together the Nehari manifold in order to ensure the existence of critical points for Jλ. In particular, for concave-convex nonlinearities.It is easy to see that u ∈ Nλ if and only if γu0 (1) = 0. Notice also that, using (φ3), we deduce that γu is of class C 2. As was pointed in for example [4] it is natural to divide Nλ into three sets: Nλ± := {u ∈ Nλ : γu00(1) ≷ 0}; Nλ0 := {u ∈ Nλ : γu00(1) = 0}.
s→∞
s→0
(φ2) s 7→ sφ(s) is strictly increasing. (tφ(t))00t (tφ(t))00t ≤ sup =: m − 2 < N − 2. (φ3) −1 < ` − 2 := inf 0 0 t>0 (tφ(t)) t>0 (tφ(t)) We extend s 7→ sφ(s) to R as an odd function. The function Φ is giRt ven by Φ(t) = 0 sφ(s)ds, t ≥ 0. The conditions (φ1) − (φ3) imply that those spaces are Banach, reflexive and saparable. For further results on Orlicz and Orlicz-Sobolev framework we refer the reader to [1]. Quasilinear elliptic problems such as (1) have been considered in order to explain many physical problems which arise from Nonlinear Elasticity, Plasticity and both Newtonian and non-Newtonian fluids. We refer the reader to [6, 7, 9]. When φ := 2, a = b := 1 we notice that ` = 2. Then problem (1) reads as ∗ q−2 2 − ∆u = λ|u| u + |u| −2u in Ω, u = 0 on ∂Ω.
(2)
In the pioneering paper [3] the authors proved results on existence of positive solutions of (2). A new variational technique was developed to overcome difficulties due to the presence of the critical Sobolev exponent 2∗ = N2N −2 . Problem (2) was later considered in [2] where among other results it was shown that there is some Λ > 0 such that (2) has a positive minimal solution for all λ ∈ (0, Λ). It is important to mention that ∗ 1,Φ ` we deal with to the lack of compactness in W0 (Ω) ,→ L (Ω). In order to overcome the difficulty with compactness we apply the concentration compactness principle, see [8], together with variational methods as in [3]. Furthermore, the Brezis-Lieb Lemma for convex functions plays a crucial role.
Main Results
Analysis of the Fibering Maps We give a complete description on the geometry for the fibering maps associated to the problem (1). We consider the auxiliary function of C 1 class given by Z Z ∗ ∗ 1,Φ 2−q 2 ` −q ` mu(t) = t φ(t|∇u|)|∇u| − t b(x)|u| , t ≥ 0, u ∈ W0 (Ω). Ω
Ω
Now if (φ1) − (φ3) hold, then tu ∈ Nλ if and only if t is a solution of Z mu(t) = λ a(x)|u|q . Ω
We shall consider the following results; Lemma 1. Suppose that (φ1) − (φ3) hold. Z ∗ ` 1. Suppose that b(x)|u| ≤ 0 holds. Then we obtain mu(0) := Ω
lim mu(t) = 0, mu(∞) := lim mu(t) = ∞ and m0u(t) > 0 for any t→∞ t→0 t > 0. Z ∗ ` b(x)|u| > 0. Then there exists an only critical point for
2. Suppose Ω
mu, i.e, there is an only point t˜ > 0 in such way that m0u(t˜) = 0. Furthermore, we know that t˜ > 0 is a global maximum point for mu and mu(∞) = −∞. 1,Φ Lemma 2. Suppose that (φ1) − (φ3) and (H) hold. Let u ∈ W0 (Ω)/{0}
be a fixed function. Then we shall consider the following assertions: Z ∗ ` 1. Assume that b(x)|u| ≤ 0. Then γu0 (t) 6= 0 for any t > 0 and Ω Z λ > 0 whenever a(x)|u|q ≤ 0. Furthermore, there exist an unique Ω
In this paper we shall assume the following set of technical conditions: `(`∗ − m) ∗, a, b ∈ L∞(Ω), a+, b+ 6≡ 0. (H) 1 < q < ≤ ` ≤ m < ` `∗ − ` The hypothesis (H) is trivially satisfied for Laplacian operator, pLaplacian operator for each p > 1 and many others quasilinear operators in divergent form. Recall that under (φ1) − (φ3) the functional 1,Φ Jλ : W0 (Ω) → R given by Z Z Z ∗ λ 1 1,Φ q ` a(x)|u| − ∗ b(x)|u| , u ∈ W0 (Ω), Jλ(u) = Φ(|∇u|) − q Ω ` Ω Ω is well-defined. Furthermore, the functional Jλ is in C 1 class. The derivative of Jλ is given by
0 Jλ(u), v =
Z
Z φ(|∇u|)∇u∇v − λ
Ω
Ω
a(x)|u|q−2uv −
Z
∗ ` b(x)|u| −2uv
Ω
1,Φ
for any u, v ∈ W0 (Ω). Hence finding weak solutions for the problem (1) is equivalent to find critical points for the functional Jλ. The main feature in this work is to use the Nehari method in order to achieve our main results. The hypothesis (H) is essential for the minimization procedure which shows that the critical value on the Nehari manifold is negative. Our main result is the following Theorem 1. Suppose (φ1) − (φ3) and (H). Set Λ = min{Λ1, Λ2}. Then for each λ ∈ (0, Λ), problem (1) admits at least two nonnegative weak solutions 1,Φ u = uλ, v = vλ ∈ W0 (Ω) satisfying Jλ(u) < 0 < Jλ(v). Furthermore, the function u is a ground state solution for each λ ∈ (0, Λ).
The Nehari manifold The Nehari manifold associated to the functional Jλ is given by
0 1,Φ (3) Nλ = {u ∈ W0 (Ω) \ {0} : Jλ(u), u = 0}.
tZ1 = t1(u, λ) in such way that γu0 (t1) = 0 and t1u ∈ Nλ+ whenever a(x)|u|q > 0. Ω
Z 2. Assume that
∗ ` b(x)|u| > 0 holds.
Then there exists an unique
Ω
tZ1 = t1(u, λ) > t˜ such that γu0 (t1) = 0 and t1u ∈ Nλ− whenever a(x)|u|q ≤ 0. Ω
3. For each λ > 0 small enough there exists unique 0 < t1 = t1(u, λ) < t˜ < t2 = t2(u,Zλ) such that γu0 (tZ1) = γu0 (t2) = 0, t1u ∈ Nλ+ and t2u ∈ Nλ− ∗ ` b(x)|u| > 0 holds.
a(x)|u|q > 0,
whenever Ω
Ω
The Palais-Smale condition Given any Banach space X endowed with the norm k.k and taking I : X → R a functional of C 1 class we recall that a sequence (un) ∈ X is said to be a Palais-Smale sequence at level c ∈ R, in short (P S)c, when I(un) → c and I 0(un) → 0 as n → ∞. Recall that I satisfies the Palais-Smale condition at the level c, in short (P S)c condition, when any (P S)c sequence admits a convergent subsequence. Now we follow same ideas discussed in [10]. At this stage we shall prove that any minimizer sequences on the Nehari manifold in Nλ+ or Nλ− provides us a Palais-Smale sequence. Proposition 1. Suppose (φ1) − (φ3) and (H). Then we have the following assertions 1. there exists a sequence (un) ⊂ Nλ such that Jλ(un) = αλ+ + e 0 −1, Φ on(1), Jλ(un) = on(1) in W (Ω).
2. there exists a sequence (un) ⊂ Nλ− such that Jλ(un) = αλ− + e 0 −1, Φ on(1), Jλ(un) = on(1) in W (Ω),
where, αλ± = inf u∈N ± Jλ(u). λ
We are going to apply the following result, whose proof is made by using the concentration compactness principle due to Lions for Orlicz -Sobolev framework, see [11] or else in [5, 7]. Q Lemma 3. (i) φ(|∇un|)∇un * φ(|∇u|)∇u in LΦ e (Ω); ∗
∗
(ii) |un|` −2un * |u|` −2u in L
`∗ `∗ −1
(Ω).
Idea of Proof of Theorem 1 • Jλ is coercive on Nλ. ˙ − is a C 1• There is a λ1 > 0 such that Nλ0 = ∅ and Nλ = Nλ+∪N λ manifold for all λ ∈ (0, λ1). • Under these conditions we observe that any critical point for Jλ on 1,Φ Nλ is a critical point for Jλ in the whole space W0 (Ω). First Solution: ¯ 1}, λ ¯ 1 > 0 is given by Lemma 2. • Let λ < Λ1 = min{λ1, λ • We prove that αλ+ := inf Jλ(u) < 0. u∈Nλ+
• We find a function u = uλ ∈ Nλ+ in such way that Jλ(u) = min Jλ(w) =: αλ+ and J 0(u) ≡ 0. Indeed, w∈Nλ+
– By Proposition 1, there exists a minimizer sequence (un) ⊂ W 1,Φ(Ω) such that Jλ(un) = αλ+ + on(1) and Jλ0 (un) = on(1). – Since the functional Jλ is coercive in Nλ+, taking a subsequence if necessary, considering the Sobolev embedding, Lemma 3 Z φ(|∇u|)∇u∇v −
and Lebesgue convergence theorem we get, ∗
1,Φ
λa(x)|u|q−2uv − b(x)|u|` −2v = 0, v ∈ W0 (Ω). – u is a weak solution to the elliptic problem (1). Z
Ω
∗ q` > 0 and – Additionally, we obtain λ a(x)|u|q ≥ −αλ+ ∗ ` −q Ω u 6≡ 0. • Using Brezis-Lieb Lemma we prove αλ+ = Jλ(u) and un → u.
• Finally, in view of Lemma 2 we show that u ∈ Nλ+.
• Since Jλ(u) = Jλ(|u|) and Jλ0 (u) = Jλ0 (|u|), we can assume u ≥ 0.
Second Solution: ˜ 1, δ1 > 0 with α− := inf Jλ(w) ≥ δ1 > 0, ∀λ ∈ (0, λ ˜ 1). • There is λ λ − w∈Nλ
¯ 1, λ ˜ 1} • We take Λ2 = min{λ • Arguing in an analogous way to what was done to obtain the first solution we obtain a second solution v such that v ≥ 0 and J(v) > δ1.
Conclusion: • Considering Λ = min{Λ1, Λ2}, we conclude the proof of the theorem. • For more details on the proofs see [5].
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