MULTIPLE POSITIVE SOLUTIONS OF SINGULAR p ... - Hindawi

MULTIPLE POSITIVE SOLUTIONS OF SINGULAR p-LAPLACIAN PROBLEMS VIA VARIATIONAL METHODS KANISHKA PERERA AND ELVES A. B. SILVA

We obtain multiple positive solutions of singular p-Laplacian problems using variational methods. Copyright © 2006 K. Perera and E. A. B. Silva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Consider the boundary value problem −Δ p u = f (x,u)

in Ω,

u > 0 in Ω,

(1.1)

u = 0 on ∂Ω, where Ω is a bounded domain in Rn , n ≥ 1 of class C 1,α for some α ∈ (0,1), Δ p u = div(|∇u| p−2 ∇u) is the p-Laplacian of u, 1 < p < ∞, and f is a Carathéodory function on Ω × (0, ∞) satisfying ( f1 ) a0 (x) ≤ f (x,t) ≤ a1 (x)t −γ for 0 < t < t0 , ( f2 ) MT := sup(x,t)∈Ω×[t0 ,T] | f (x,t)| < ∞ for all T ≥ t0 for some nontrivial measurable functions a0 ,a1 ≥ 0 and constants γ,t0 > 0, so that it may be singular at t = 0 and changes sign. We assume that =n ( f3 ) there exists ϕ ≥ 0 in C01 (Ω) such that a1 ϕ−γ ∈ Lq (Ω), where q = (p∗ ) if p (resp., q > 1 if p = n). Here (p∗ ) = p∗ /(p∗ − 1) is the Hölder conjugate of the critical Sobolev exponent p∗ = np/(n − p) if p < n (resp., p∗ = ∞ if p ≥ n). The semilinear case p = 2 has been studied extensively in both bounded and unbounded domains (see, e.g., [2–4, 8–13, 15, 17–21, 24–26, 29, 30] and their references). The quasilinear ODE case 1 < p < ∞, n = 1, was studied using fixed point theory by Agarwal et al. [1]. The general quasilinear case 1 < p < ∞, n ≥ 1, was studied using a simple cutoff argument and variational methods by Perera and Zhang [23] for q > n and by Agarwal et al. [5], and Perera and Silva [22] for q > n/ p. We remove these restrictions on q in this paper. Hindawi Publishing Corporation Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 915–924

916

Positive solutions of singular p-Laplacian problems

Note that ( f3 ) implies a1 ∈ Lq (Ω). If a1 ∈ L∞ (Ω) and γ < 1/(p∗ ) , ( f3 ) is satisfied with any ϕ whose interior normal derivative ∂ϕ/∂ν > 0 on ∂Ω. A typical example is f (x,t) = t −γ +g(x,t), where g is a Carathéodory function on Ω × [0, ∞) that is bounded on bounded t intervals. However, ( f3 ) does not necessarily require γ < 1 as usually assumed in the literature. For example, when Ω is the unit ball, a1 (x) = (1 − |x|2 )σ , σ ≥ 0, and γ < σ + 1/(p∗ ) , we can take ϕ(x) = 1 − |x|2 . We may assume that a0 ∈ L∞ (Ω) by replacing it with min{a0 ,1} if necessary, so the problem −Δ p u = a0 (x)

in Ω,

(1.2)

u = 0 on ∂Ω

has a unique weak solution u ∈ C01 (Ω) (see, e.g., Azizieh and Clément [7, Lemma 1.1]). Since a0 ≥ 0 and is nontrivial, so is u and hence u > 0 in Ω,

∂u >0 ∂ν

on ∂Ω,

(1.3)

by the strong maximum principle of Vázquez [28]. Fix 0 < ε ≤ 1 so small that u := ε1/(p−1) u < t0 . Then −Δ p u − f (x,u) ≤ −(1 − ε)a0 (x) ≤ 0

(1.4)

by ( f1 ), so u is a subsolution of (1.1). Let ⎧ ⎪ ⎨ f (x,t),

t ≥ u(x),

fu (x,t) = ⎪ ⎩ f x,u(x),

t < u(x),

(1.5)

and consider the problem −Δ p u = fu (x,u)

u=0

in Ω, (1.6)

on ∂Ω.

A standard argument shows that weak solutions of this problem are ≥ u and hence also solutions of (1.1). We have a0 (x) ≤ fu (x,t) ≤ a1 (x)u(x)−γ ,

t < t0 ,

(1.7)

by ( f1 ), inf Ω (u/ϕ) > 0 by (1.3) and hence a1 u−γ ∈ Lq (Ω) by ( f3 ), and fu (x,t) = f (x,t) for t ≥ t0 . So the solutions of the modified problem (1.6) are the critical points of the C 1 functional:

Φ(u) = where F(x,t) =

t

0 fu (x,s)ds

Ω

|∇u| p − pF(x,u),

1,p

u ∈ W0 (Ω),

if f grows at most critically:

(1.8)

K. Perera and E. A. B. Silva 917 ( f4 ) | f (x,t)| ≤ Ct r −1 for t ≥ t0 , where r = p∗ if p < n (resp., r > p if p ≥ n). As usual, C denotes a generic positive constant. We exploit this variational framework to 1,p seek W0 (Ω) solutions to the original problem. 2. Existence Lemma 2.1. If ( f1 ) and ( f3 ) hold, and (1.1) has a supersolution u ≥ u in W 1,p (Ω), then it has a solution in the order interval [u,u] in the cases: (i) u ∈ L∞ (Ω) and ( f2 ) holds; (ii) ( f4 ) holds. Proof. Let ⎧ ⎪ ⎨ fu x,u(x) ,

t > u(x),

⎩ f (x,t), u

t ≤ u(x).

f u (x,t) = ⎪

(2.1)

If u ∈ L∞ (Ω) and ( f2 ) holds, f u ≤ a1 u−γ + M|u| ∈ Lq (Ω) ∞

(2.2)

by (1.7), where we set MT = 0 for T < t0 for convenience. If ( f4 ) holds, f u ≤ a1 u−γ + Cur −1 ∈ Lq (Ω).

(2.3)

t

So the functional Φ with F(x,t) = 0 f u (x,s)ds is bounded from below and coercive, and hence has a global minimizer by weak lower semicontinuity, in both cases. Theorem 2.2. If ( f1 )–( f3 ) hold and there is a t1 > t0 such that

f x,t1 ≤ 0,

x ∈ Ω,

(2.4)

then (1.1) has a solution ≤ t1 . Proof. It follows from Lemma 2.1, taking u ≡ t1 .

Example 2.3. Problem (1.1) with f (x,t) = t −γ − et has a solution ≤ 1 for all γ < 1/(p∗ ) . Let λ1 > 0 be the first eigenvalue of −Δ p , with the eigenfunction ϕ1 > 0. Theorem 2.4. If ( f1 ), ( f3 ), and ( f4 ) hold, and f (x,t) ≤ λt p−1 + C,

t ≥ t0 ,

(2.5)

for some 0 ≤ λ < λ1 , then (1.1) has a solution. Proof. Let p −1

f (x,t) = λ t +

+ C + a1 (x)u(x)−γ .

(2.6)

918


The functional Φ with F(x,t) = u ≥ u since

t 0

f (x,s)ds has a global minimizer u since λ < λ1 , and

−Δ p u = f (x,u) ≥ a1 (x)u(x)−γ ≥ a0 (x) ≥ εa0 (x) = −Δ p u

(2.7)

−Δ p u ≥ λu p−1 + C + a1 (x)u−γ ≥ f (x,u)

(2.8)

by ( f1 ). Then

by (2.5) and ( f1 ), so u is a supersolution of (1.1), and the conclusion follows from Lemma 2.1. Example 2.5. Problem (1.1) with f (x,t) = t −γ + λt p−1 + t s−1 has a solution for all γ < 1/(p∗ ) , λ < λ1 , and 1 ≤ s < p. Theorem 2.6. If ( f1 ), ( f3 ), and ( f4 ) hold, and f (x,t) ≤ λ1 t p−1 − g(x,t),

t ≥ t1 ,

(2.9)

for some t1 ≥ t0 and a Carathéodory function g on Ω × [t1 , ∞) satisfying g(x,t) ≤ Ct r −1 ,

(2.10)

where r is as in ( f4 ), G(x,t) := lim G(x,t) = ∞

t →∞

t t1

g(x,s)ds ≥ −C,

on a set of positive measure,

(2.11) (2.12)

then (1.1) has a solution. Proof. Take h ∈ C(R,[0,1]) such that h(t) = 1 for t ≤ t1 and h(t) = 0 for t ≥ some t2 > t1 , and let

f (x,t) = 1 − h(t) λ1 t p−1 − g(x,t) + h(t)a1 (x)u(x)−γ .

(2.13)

Following the proof of Theorem 2.4, it suffices to show that the functional Φ with F(x,t) = t t2 f (x,s)ds is bounded from below and coercive. We have ⎧ λ1 ⎪ ⎪ ⎨ t p − G(x,t) + C,

F(x,t) ≤ ⎪ p ⎪ ⎩C, and hence

Φ(u) ≥

Ω

|∇u| p − λ1 (u+ ) p + p

so Φ is bounded from below by (2.11).

t ≥ t2 ,

(2.14)

t < t2 , u≥t2

G(x,u) − C,

(2.15)

K. Perera and E. A. B. Silva 919 Suppose Φ is not coercive, say, ρ j := u j → ∞ and Φ(u j ) ≤ C. Then for a subse1,p quence, u j := u j /ρ j converges to some u weakly in W0 (Ω), strongly in L p (Ω), and a.e. u j ≡ 1,

u ≤ 1. By (2.11) and (2.15), in Ω. Since

λ1

Ω

u+j

p

p

≥ ρ j − C,

(2.16)

p

and dividing by ρ j and passing to the limit give

p

p

u p ≥ λ1 |

u| p , λ1 u + p ≥ 1 ≥

(2.17)

so u = ϕ1 . Then u j (x) = ρ j u j (x) → ∞ a.e., and hence

Φ uj ≥ p

u j ≥t 2

G x,u j − C −→ ∞

(2.18)

by (2.11), (2.12), (2.15), and Fatou’s lemma, a contradiction.

Example 2.7. Problem (1.1) with f (x,t) = t −γ + λ1 t p−1 − t s−1 has a solution for all γ < 1/(p∗ ) and 1 ≤ s < p. Remark 2.8. When p > n, the supersolutions u constructed in the proofs of Theorems 2.4 and 2.6 are in L∞ (Ω) by the Sobolev embedding and hence the weaker condition ( f2 ) can be used in place of ( f4 ). The same is true when p ≤ n by the regularity results of Guedda and Véron [16] if q > n/ p in ( f3 ), in which case solutions ≥ u of (1.1) are also in L∞ (Ω) (see Agarwal et al. [5, proof of Proposition 2.1]). 3. Multiplicity Throughout this section, we assume ( f1 ), ( f3 ), ( f4 ), and t ( f5 ) there exists t2 > t1 > t0 and λ < λ1 such that t1 f (x,s)ds ≤ (λ/ p)(t − t1 ) p for t1 ≤ t ≤ t2 . In particular, f (x,t1 ) ≤ 0 and hence (1.1) has a solution u0 ∈ [u,t1 ] by Theorem 2.2. Noting that u0 is also a subsolution of (1.1), we seek a second solution u1 ≥ u0 as another critical point of the functional Φ with

F(x,t) =

t t1

fu0 (x,s)ds,

⎧ ⎪ ⎪ ⎨ f (x,t),

fu0 (x,t) = ⎪ ⎪ ⎩ f x,u0 (x) ,

t ≥ u0 (x), (3.1) t < u0 (x).

By ( f1 ) and ( f2 ), fu (x,t) ≤ a1 (x)u(x)−γ + Mt ∈ Lq (Ω), 0 1

since u0 ≥ u.

t < t1 ,

(3.2)

920


given by (1.8) with F replaced by The functional Φ

F(x,t) =

t

f u0 (x,s)ds,

t1

⎧ ⎪ ⎪ ⎨ fu0 x,t1 ,

t > t1 ,

⎪ ⎩ fu (x,t), 0

t ≤ t1 ,

f u0 (x,t) = ⎪

(3.3)

has a global minimizer in [u0 ,t1 ] as in the proof of Lemma 2.1, which we assume is u0 itself since otherwise we are done. Lemma 3.1. u0 is a local minimizer of Φ. Proof. If u j → u0 , writing u j = v j + w j , where

v j = min u j ,t1 −→ u0 ,

w j = max u j ,t1 − t1 −→ 0,

(3.4)

we have

p

Φ u j = Φ v j + w j − p

Ω

F x,w j + t1 .

(3.5)

Since v j ,u0 ≤ t1 and u0 is a global minimizer of Φ,

vj ≥ Φ

u0 = Φ u0 . Φ vj = Φ

(3.6)

p r λ t − t1 + C t − t1 , p

(3.7)

By ( f4 ) and ( f5 ), F(x,t) ≤

t ≥ t1 ,

and hence

p

Ω

λ w j p + C w j r . λ1

F x,w j + t1 ≤

Since λ < λ1 and r > p, it follows that Φ(u j ) ≥ Φ(u0 ) for large j.

(3.8)

We recall that Φ satisfies the Cerami compactness condition (C) if every sequence (u j ), such that

Φ u j is bounded,

1 + u j Φ u j −→ 0,

(3.9)

has a convergent subsequence. For t ≥ t1 , let g(x,t) = f (x,t) − λ1 t p−1 , G(x,t) =

t

t1

g(x,s)ds = F(x,t) −

λ1 p p t − t1 . p

(3.10)

K. Perera and E. A. B. Silva 921 Lemma 3.2. If ( f1 ) and ( f3 )–( f5 ) hold, G(x,t) ≥ −C, lim G(x,t) = ∞

t ≥ t1 ,

(3.11)

on a set of positive measure,

t →∞

(3.12)

and Φ satisfies (C), then (1.1) has two solutions u1 ≥ u0 . Proof. We have

Φ tϕ1 ≤ C − p

tϕ1 ≥t1

G x,tϕ1 −→ −∞

as t −→ ∞

(3.13)

by (3.2), (3.11), (3.12), and Fatou’s lemma, and the conclusion follows from Lemma 3.1 and the mountain pass lemma. Theorem 3.3. If ( f1 ) and ( f3 )–( f5 ) hold, G(x,t) ≤ Ct p ,

t ≥ t1 ,

(3.14)

G(x,t) = 0 a.e., tp H(x,t) := pG(x,t) − tg(x,t) ≥ −C, lim

(3.15)

t →∞

t ≥ t1 ,

(3.16)

lim H(x,t) = ∞ on a set of positive measure,

(3.17)

t →∞

then (1.1) has two solutions u1 ≥ u0 . Proof. We apply Lemma 3.2. We have

∂ G(x,t) H(x,t) = − p+1 , ∂t tp t

(3.18)

and hence G(x,t) = t p

∞ t

H(x,s) 1 ds ≥ inf H(x,s) s p+1 p s ≥t

a.e.

(3.19)

by (3.15), so (3.16) and (3.17) imply (3.11) and (3.12), respectively. As usual, to verify (C) it suffices to show that every sequence (u j ) satisfying (3.9) is bounded. Suppose ρ j := u j → ∞ along a subsequence. Then for a further subsequence, 1,p u j := u j /ρ j converges to some u weakly in W0 (Ω), strongly in L p (Ω), and a.e. in Ω. By (3.2),

p

Φ u j ≥ ρ j − λ1

Ω

u+j

p

−p

u j ≥t 1

G x,u j − C ρ j + 1 .

(3.20)

p

Dividing by ρ j and noting that

u j ≥t 1

G x,u j = p ρj

u j ≥t 1

G x,u j p u j −→ 0 p uj

(3.21)

922


by (3.14), (3.15), and Lebesgue’s dominated convergence theorem, u = ϕ1 as in the proof of Theorem 2.6. Then u j (x) → ∞ a.e., and hence

p 1 + Φ u j u j + 2u−j − Φ u j ≥ u−j + p

u j ≥t 1

H x,u j − C u−j + 1 −→ ∞

by (3.2), (3.16), (3.17), and Fatou’s lemma, contradicting (3.9).

(3.22)

Example 3.4. Problem (1.1) with f (x,t) = t −γ + λ1 t p−1 + t s−1 − μ has two-ordered solutions for all γ < 1/(p∗ ) , 1 < s < p, and large μ > 0. Theorem 3.5. If ( f1 )–( f3 ) and ( f5 ) hold, and λ≤

f (x,t) ≤ C, t p −1

t ≥ t3 ,

(3.23)

for some t3 > t2 and λ > λ1 , then (1.1) has two solutions u1 ≥ u0 . Proof. Clearly, (3.23) implies (3.11) and (3.12). To verify (C), suppose (u j ) satisfies (3.9), 1,p ρ j := u j → ∞, and u j := u j /ρ j → u weakly in W0 (Ω), strongly in L p (Ω), and a.e. in Ω as in the proof of Theorem 3.3. By (3.2) and ( f4 ),

Φ u j v p −1 pρ j

=

Ω

p −2 p −1 ∇

u j ∇

u j · ∇v − α j (x)u v + o v , j

(3.24)

where ⎧ ⎪ f x,u j (x) ⎪ ⎪ ⎨ ,

α j (x) = ⎪ u j (x) ⎪ ⎪ ⎩0,

p −1

u j (x) ≥ t3 ,

(3.25)

u j (x) < t3 .

≡ 0, reBy (3.23), 0 ≤ α j ≤ C, so taking v = u − , u j and passing to the limit give u ≥ 0, spectively. Moreover, a subsequence of (α j ) converges to some 0 ≤ α ≤ C weakly in Ls (Ω) for any 1 < s < ∞, and passing to the limit in (3.24) shows that u satisfies −Δ p u

= α(x)u

p −1

u = 0

in Ω, (3.26)

on ∂Ω.

So u ∈ L∞ (Ω) ∩ C 1 (Ω) by Anane [6] and DiBenedetto [14], and hence u > 0 by the Harnack inequality of Trudinger [27]. This implies that α ≥ λ and that the first eigenvalue of −Δ p with weight α given by

inf

1,p

Ω

u∈W0 (Ω)\{0} Ω

|∇u| p

α(x)|u| p

= 1.

(3.27)

K. Perera and E. A. B. Silva 923 Then

1 ≤ Ω

∇ϕ1 p

Ω

p α(x)ϕ1

≤

λ1 < 1, λ

(3.28)

a contradiction.

Example 3.6. Problem (1.1) with f (x,t) = t −γ + λt p−1 − t s−1 − μ has two-ordered solutions for all γ < 1/(p∗ ) , λ > λ1 , 1 ≤ s < p, and large μ > 0. Theorem 3.7. If ( f1 ), ( f3 ), ( f4 ) with r < p∗ , and ( f5 ) hold, and 0 < θF(x,t) ≤ t f (x,t),

t ≥ t3 ,

(3.29)

for some t3 > t2 and θ > p, then (1.1) has two solutions u1 ≥ u0 . Proof. It follows from Lemma 3.2 since (3.29) implies that

F(x,t) ≥ F x,t3 and that Φ satisfies (C).

t θ

t3

,

t ≥ t3 ,

(3.30)

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