p -Laplacian problems with critical Sobolev exponents

Nonlinear Analysis 66 (2007) 454–459 www.elsevier.com/locate/na

p-Laplacian problems with critical Sobolev exponents Kanishka Perera a,∗ , Elves A.B. Silva b a Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA b Departamento de Matem´atica, Universidade de Bras´ılia, 70910-900, DF, Brazil

Received 2 November 2005; accepted 22 November 2005

Abstract We study the existence and multiplicity of solutions of p-Laplacian problems with critical Sobolev exponents using variational methods. c 2005 Elsevier Ltd. All rights reserved.  MSC: primary 35J65; secondary 35B33, 47J30 Keywords: p-Laplacian problems; Critical Sobolev exponent; Existence and multiplicity; Variational methods

1. Introduction We consider the quasilinear elliptic boundary value problem  ∗ − p u = μ|u| p −2 u + λ|u| p−2 u + g(x, u) in Ω u=0 on ∂Ω

(1.1)

where Ω is a smooth bounded domain in Rn , n ≥ 3,  p u = div (|∇u| p−2 ∇u) is the pLaplacian, 1 < p < n, p∗ = np/(n − p) is the critical Sobolev exponent, μ > 0 and λ ∈ R are parameters, and g is a Carath´eodory function on Ω × R satisfying |g(x, t)| ≤ C(|t|q−1 + 1), for some 0 ≤ σ < p < q < p∗ and  G(x, t) 0 as t → 0 → ∞ as |t| → ∞ |t| p

tg(x, t) − pG(x, t) ≥ −C(1 + |t|σ )

uniformly in x.

∗ Corresponding author.

E-mail addresses: [email protected] (K. Perera), [email protected] (E.A.B. Silva). URL: http://my.fit.edu/∼kperera/ (K. Perera). c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter  doi:10.1016/j.na.2005.11.039

(1.2)

(1.3)

K. Perera, E.A.B. Silva / Nonlinear Analysis 66 (2007) 454–459

455

t Here G(x, t) = 0 g(x, s)ds and C denotes a generic positive constant. Our goal is to study the existence and multiplicity of solutions to this problem using variational methods. The first results on critical problems for the Laplacian were obtained in a celebrated paper of Br´ezis and Nirenberg [3]. This pioneering work has stimulated a vast amount of research on the subject since then (see, e.g., [1,4–8,10–14,19] and their references). The main difficulty in dealing with this class of problems is the fact that the associated functional does not satisfy the 1, p Palais–Smale compactness condition (PS) since the imbedding of the Sobolev space W0 (Ω ) ∗ p into L (Ω ) is not compact. However, using the concentration–compactness principle of Lions [15,16] it was shown in Silva and Xavier [18] that the (PS) condition holds below any fixed level if μ > 0 is sufficiently small. Our main results are Theorem 1.1. Assume λ ∈ σ (− p ), the Dirichlet spectrum of − p , and G(x, t) ≥ 0

∀(x, t).

(1.4)

Then there is a μ1 > 0 such that (1.1) has a nontrivial solution for all μ ∈ (0, μ1 ). Theorem 1.2. Assume that g is odd in t for all x. Then there is a μk > 0 such that (1.1) has k pairs of nontrivial solutions for all μ ∈ (0, μk ). Corollary 1.3. If p < q < p∗ , there is a μk > 0 such that the problem  ∗ − p u = μ|u| p −2 u + λ|u| p−2 u + |u|q−2 u in Ω u=0 on ∂Ω

(1.5)

has k pairs of nontrivial solutions for all μ ∈ (0, μk ). Our theorems are related to the results of Silva and Xavier [18] and the arguments here are adapted from Perera and Szulkin [17]. In particular, the proofs make use of a new unbounded sequence of eigenvalues of − p , constructed in [17] via a minimax scheme involving the cohomological index of Fadell and Rabinowitz [9], and the pseudo-index of Benci [2]. 2. Cohomological index and eigenvalues Let S denote the class of symmetric subsets of a Banach space W . Fadell and Rabinowitz [9] constructed an index theory i : S → N ∪ {0, ∞} with the following properties: (i) Definiteness: i (S) = 0 ⇐⇒ S = ∅. (ii) Monotonicity: If there is an odd map S → S  , then i (S) ≤ i (S  ). In particular, equality holds if S and S  are homeomorphic. (iii) Subadditivity: i (S ∪ S  ) ≤ i (S) + i (S  ). (iv) Continuity: If S is closed, then there is a closed neighborhood N ∈ S of S such that i (N) = i (S). (v) Neighborhood of zero: If U is a bounded symmetric neighborhood of 0 in W , then i (∂U ) = dim W . (vi) Stability: If S is closed and S ∗ Z2 is the join of S with Z2 = {±1}, consisting of all line segments in W ⊕ R joining ±1 to points of S, then i (S ∗ Z2 ) = i (S) + 1. (vii) Piercing property: If S, S0 , S1 are closed and ψ : S × [0, 1] → S0 ∪ S1 is an odd map such that ψ(S × [0, 1]) is closed, ψ(S × {0}) ⊂ S0 , ψ(S × {1}) ⊂ S1 , then i (ψ(S × [0, 1]) ∩ S0 ∩ S1 ) ≥ i (S).

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K. Perera, E.A.B. Silva / Nonlinear Analysis 66 (2007) 454–459 1, p

Let W = W0 (Ω ), normed by  1/ p p u = |∇u| .

(2.1)

Ω

Dirichlet eigenvalues of − p are the critical values of the C 1 functional 1 , p Ω |u|

I (u) = 

u ∈ S1 = {u ∈ W : u = 1}

(2.2)

by the Lagrange multiplier rule. Denote by A the class of compact symmetric subsets of S1 , let Fl = {A ∈ A : i (A) ≥ l} ,

(2.3)

and set λl := inf max I (u).

(2.4)

A∈Fl u∈A

It was shown in Perera and Szulkin [17] that λl  ∞ are eigenvalues of − p . 3. Proof of Theorem 1.1 Solutions of (1.1) are the critical points of  1 μ λ ∗ |∇u| p − ∗ |u| p − |u| p − G(x, u), Φμ (u) = p p Ω p

u ∈ W.

(3.1)

Since λ ∈ σ (− p ) and the case λ < λ1 has already been considered by Silva and Xavier [18], we assume that λl < λ < λl+1 for some l. Since λ > λl , there is an A0 ∈ Fl such that I ≤ λ on A0 and hence, by (1.4),   λ tp Φ0 (tu) ≤ 1− ≤ 0, u ∈ A0 , t ≥ 0. (3.2) p I (u) Take ϕ0 ∈ C(C A0 , S1 ) with ϕ0 | A0 = id, where C A0 = (A0 × [0, 1])/(A0 × {1}) is the cone over A0 . By (1.3), Φ0 (tu) → −∞

as t → ∞

(3.3)

uniformly for u on the compact set ϕ0 (C A0 ), so Φ0 ≤ 0 on Rϕ0 (C A0 ) when R > 0 is sufficiently large, and c0 :=

sup

u∈ϕ0 (C A0 ),t ≥0

Φ0 (tu) < ∞.

(3.4)

By Proposition 3.4 of Silva and Xavier [18], there is a μ1 > 0 such that Φμ satisfies the (PS) condition at all levels ≤ c0 when μ ∈ (0, μ1 ). Regarding W as a subspace of W ⊕ R and C A0 as a cone in W ⊕ R, with vertex at some point ∈ W , let A1 = {tu : u ∈ A0 , t ∈ [0, 1]} , and

 ϕ(v) =

Rv, Rϕ0 (v),

v ∈ A1 v ∈ C A0 ,

A = A1 ∪ C A0 ,

(3.5)

(3.6)

K. Perera, E.A.B. Silva / Nonlinear Analysis 66 (2007) 454–459

so that Φμ ≤ Φ0 ≤ 0 on ϕ(A). On the other hand, by (1.3),   tp λ Φμ (tu) = + o(1) as t → 0, u ∈ S1 , 1− p I (u)

457

(3.7)

and λ < λl+1 , so the infimum of Φμ on   B = u ∈ Sρ : I (u/ρ) ≥ λl+1 ,

(3.8)

where Sρ = {u = ρ}, is positive when 0 < ρ < R is sufficiently small. Lemma 3.1. (A, ϕ) links B with respect to K = {tv : v ∈ C A0 , t ∈ [0, 1]} ,

(3.9)

i.e., γ (K ) ∩ B = ∅ for every map in Γ = {γ ∈ C(K , W ) : γ | A = ϕ}. Proof. Regarding A0 ∗ Z2 as a double cone over A0 , any γ ∈ Γ can be extended to an odd map on γ = {tv : v ∈ A0 ∗ Z2 , t ∈ [0, 1]} , K

(3.10)

) ∩ B = ∅. Applying the piercing property to and it suffices to show that γ (K S = A0 ∗ Z2 ,

S0 = {u ≤ ρ} ,

S1 = {u ≥ ρ} ,

ψ(v, t) = γ (tv)

(3.11)

) ∩ Sρ ) = i (ψ(S × [0, 1]) ∩ S0 ∩ S1 ) ≥ i (S) = i (A0) + 1 ≥ l + 1, i ( γ (K

(3.12)

gives

so max

)∩Sρ u∈ γ (K

I (u/ρ) ≥ λl+1

) intersects B. and hence γ (K

(3.13) 

Set c := inf max Φμ (u).

(3.14)

γ ∈Γ u∈γ (K )

Taking γ (tv) = tϕ(v) shows that c ≤ c0 and hence Φμ satisfies (PS)c . Since max Φμ (ϕ(A)) ≤ 0 < inf Φμ (B), it follows from Lemma 3.1 that c > 0 is a critical value of Φμ . 4. Proof of Theorem 1.2 Since g is odd in t for all x, G is even in t for all x and hence the functional Φμ is even. We fix a μ0 > 0 and work with μ ≤ μ0 . Let S denote the class of compact symmetric subsets of W and Γμ the group of odd homeomorphisms γ of W such that γ = id on Φμ−1 ((−∞, 0]). Since λl  ∞, λ < λl+1 when l is sufficiently large. Using (3.7), take ρ > 0 so small that inf Φμ0 (B) > 0 where B is as in (3.8), and let i μ∗ (A) := min i (γ (A) ∩ Sρ ), γ ∈Γμ

A∈S

be the pseudo-index of Benci [2] related to i , Sρ , and Γμ .

(4.1)

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K. Perera, E.A.B. Silva / Nonlinear Analysis 66 (2007) 454–459

Lemma 4.1. For any μ ≥ 0,   Fmμ := A ∈ S : i μ∗ (A) ≥ m = ∅

∀m.

(4.2)

Proof. Take A0 ∈ Fm and, using (1.3), R > ρ so large that Φμ ≤ 0 on R A0 , and let A = {tu : u ∈ R A0 , t ∈ [0, 1]} .

(4.3)

Then for any γ ∈ Γμ , γ | R A0 = id and γ (0) = 0, and applying the piercing property to S = R A0 ,

S0 = {u ≤ ρ} ,

S1 = {u ≥ ρ} ,

ψ(u, t) = γ (tu)

(4.4)

gives i (γ (A) ∩ Sρ ) = i (ψ(S × [0, 1]) ∩ S0 ∩ S1 ) ≥ i (S) = i (A0 ) ≥ m. 

(4.5)

By Lemma 4.1, c0 :=

inf max Φ0 (u) < ∞,

(4.6)

0 u∈A A∈Fl+k

and by Proposition 3.4 of Silva and Xavier [18], there is a 0 < μk ≤ μ0 such that Φμ satisfies the (PS) condition at all levels ≤ c0 when μ ∈ (0, μk ). Set cm :=

inf

max Φμ (u),

μ A∈Fl+m u∈A

1 ≤ m ≤ k.

(4.7)

Noting that c1 ≤ · · · ≤ ck ≤ c0 , we will show that c1 > 0 and hence Φμ has k pairs of nontrivial critical points (see Benci [2]). μ Let A ∈ Fl+1 . Taking γ = id in (4.1), i (A ∩ Sρ ) ≥ l + 1, so max I (u/ρ) ≥ λl+1

(4.8)

u∈A∩Sρ

and hence A ∩ B = ∅. Therefore max Φμ (u) ≥ inf Φμ0 (u) > 0. u∈A

(4.9)

u∈B

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