On positive solutions for a class of Indefinite weight semilinear elliptic boundary value problems with critical Sobolev exponent

J. Korean Math. Soc. 44 (2007), No. 2, pp. 249–260

ON POSITIVE SOLUTIONS FOR A CLASS OF INDEFINITE WEIGHT SEMILINEAR ELLIPTIC BOUNDARY VALUE

PROBLEMS WITH CRITICAL SOBOLEV EXPONENT

Bongsoo Ko and Seungpil Kang

Reprinted from the

Journal of the Korean Mathematical Society Vol. 44, No. 2, March 2007

c

°2007 The Korean Mathematical Society

ON POSITIVE SOLUTIONS FOR A CLASS OF INDEFINITE WEIGHT SEMILINEAR ELLIPTIC BOUNDARY VALUE

PROBLEMS WITH CRITICAL SOBOLEV EXPONENT

Bongsoo Ko and Seungpil Kang

Abstract. By variational methods, we prove the existence of positive solutions of a class of indefinite weight semilinear elliptic boundary value problems on critical Sobolev exponent.

1. Introduction

We have known several famous results for the existence or the non-existence of positive solutions about semilinear elliptic boundary value problems in crit- ical Sobolev exponent case ([5], [2]). As some different studies from that, we discuss the existence of positive solutions of the following indefinite weight semilinear elliptic boundary value problems:

(I λα )

( −∆u = λg(x)u(1 + |u| ^p ) in Ω, (1 − α) ∂u

∂n + αu = 0 on ∂Ω,

where λ and α are real parameters, Ω is an open bounded domain in R ^N , N ≥ 3, with the smooth boundary ∂Ω. We shall consider the critical Sobolev exponent case p = _{N −2} ⁴ and the function g : Ω −→ R ¹ is smooth and changes sign.

We proved the existence of positive solutions of the case 0 < p < _{N −2} ⁴ ([4]).

Here α ∈ (0, 1) or Z

Ω

g(x)dx 6= 0 and α ∈ (α 0 , 0] for some constant α 0 < 0. We used the constrained minimization method of the functional

E λ (u) = Z

Ω

|∇u| ² − λ Z

Ω

gu ² + α (1 − α)

Z

∂Ω

u ² dS x

on the constrained set

{u ∈ W ^1,2 (Ω) : λ Z

Ω

g|u| ^p+2 = 1}

Received February 5, 2004.

2000 Mathematics Subject Classification. 35J60, 35J20.

Key words and phrases. indefinite weight semilinear elliptic problems, critical Sobolev exponent, positive solutions, variational methods.

c

°2007 The Korean Mathematical Society

249

to prove the existence if α 6= 1. The other case can be proved by the similar method on the Sobolev space W ₀ ^1,2 (Ω). In this paper, we assume that if α = 1, the considering space is W ₀ ^1,2 (Ω).

If p = _{N −2} ⁴ , the above constrained set may not be weakly closed, and so we should find a different method to get positive solutions.

In Section 2, we show that a minimizing sequence of the functional which is induced by the weighted problem (I λα ) : For α ∈ [0, 1),

J λα (u) = 1 2 Z

Ω

|∇u| ² − λ 2 Z

Ω

gu ² − λ p + 2

Z

Ω

g|u| ^p+2 + α 2(1 − α)

Z

∂Ω

u ² dS x

on the Nehari manifold:

M λα = ©

u ∈ W ^1,2 (Ω) : u 6= 0, hJ _λα ⁰ (u), ui = 0 ª , where

hJ _λα ⁰ (u), ui = Z

Ω

|∇u| ² − λ Z

Ω

gu ² (1 + |u| ^p ) + α 1 − α

Z

∂Ω

u ² dS x , converges to a positive function in W ^1,2 (Ω) which is a classical positive solution of the problem (I λα ) if λ ⁻ _α < λ < λ ⁺ _α , and λ is near to either λ ⁻ _α or λ ⁺ _α , where λ ⁻ _α and λ ⁺ _α are the principal eigenvalues of the following problem ([1]):

(L α )

( −∆u = λg(x)u in Ω, (1 − α) ∂u

∂n + αu = 0 on ∂Ω.

Furthermore, we estimate the length of the intervals about λ in which the existence is guaranteed. We also have the similar result for α = 1 using the following functional

J λ1 (u) = 1 2

Z

Ω

|∇u| ² − λ 2 Z

Ω

gu ² − λ p + 2

Z

Ω

g|u| ^p+2 .

In the end of Section 2, we can show that (I λα ) has a positive solution for all λ ∈ (λ ⁻ _α , λ ⁺ _α ), except λ 6= 0 if g(x) < 0 for all x ∈ ∂Ω. However, we note that if Ω is a ball, g = 1, N = 3 and α = 1, then (I λα ) has a positive solution if and only if ¹ ₄ λ 1 < λ < λ 1 , where λ 1 is the principal eigenvalue of −∆ with the homogeneous Dirichlet boundary condition ([2]). As the application of the result, we can prove the existence of a positive solution of the following

problem: 





−∆u = g(x)u

^{N +2N −2}

in Ω, (1 − α) ∂u

∂n + αu = 0 on ∂Ω,

if the function g satisfies the above same condition. We also note that, if Ω is

an open ball, g = 1 in ¯ Ω and α = 1, we have had the nonexistence result of

any positive solution ([5]). The above difference between the existence and the

non-existence may be appeared from the special properties of the function g in

the indefinite weight problems.

2. The main results

We first recall some facts about how the method of eigencurves can be used to define principal eigenvalues. We define µ(λ, α) by

µ(λ, α) = inf

½Z

Ω

(|∇u| ² − λgu ² )dx+ α 1 − α

Z

∂Ω

u ² dS x : u ∈ W ^1,2 (Ω), Z

Ω

u ² = 1

¾ . It can be shown in [1] that µ(0, α) > 0 on α ∈ [α 0 , 1] where α 0 ≤ 0 for some small negative, and the function λ −→ µ(λ, α) is a concave function such that µ(0, λ) → −∞ as λ → ±∞. So it follows that λ → µ(λ, α) has exactly two zeros λ ⁻ _α and λ ⁺ _α , and those are principal eigenvalues for (L α ). Furthermore, the eigencurves λ → µ(λ, α) can be used to produce an equivalent norm for W ^1,2 (Ω) if α 6= 1. In this case α = 1, we use the following function µ:

µ(λ) = inf

½Z

Ω

£ |∇| ² − λgu ² ¤

dx : u ∈ W ₀ ^1,2 (Ω), Z

Ω

u ² = 1

¾ .

Lemma 2.1 ([4]). Suppose α ∈ (0, 1) or that Z

Ω

gdx 6= 0 and α ∈ (α 0 , 0] so that (L α ) has principal eigenvalues λ ⁻ _α and λ ⁺ _α . For any λ ∈ (λ ⁻ _α , λ ⁺ _α ),

kuk λα =

½Z

Ω

[|∇u| ² − λgu ² ]dx + α 1 − α

Z

∂Ω

u ² dS x

¾

¹

2

defines a norm in W ^1,2 (Ω) which is equivalent to the usual norm for W ^1,2 (Ω).

For the simplicity, we use the following function K : R → R

K(α) =

( 0 if α = 1, α

1 − α otherwise.

Lemma 2.2. Let λ ∈ (λ ⁻ _α , λ ⁺ _α ), λ 6= 0 and let M λα = ©

u ∈ W ^1,2 (Ω) : u 6= 0, hJ _λα ⁰ (u), ui = 0 ª , Then M λα is a nonempty subset of W ^1,2 (Ω).

Proof. Since g changes sign, we can choose a nonzero function u 0 ∈ W ^1,2 (Ω)

so that Z

Ω

g|u 0 | ^p+2 > 0.

Let

t ^p = R

Ω |∇u 0 | ² − λ R

Ω gu ² ₀ + K(α) R

∂Ω u ² ₀ dS x

λ R

Ω g|u 0 | ^p+2 .

Then u = tu 0 ∈ M λα . ¤

Definition 2.3. We define the following functions:

K _λα ⁺ = inf

½Z

Ω

[|∇u| ² − λgu ² ] + K(α) Z

∂Ω

u ² dS _x : u ∈ W ^1,2 (Ω), Z

Ω

g|u| ^p+2 = 1

¾ ,

K _λα ⁻ = inf

½Z

Ω

£ |∇u| ² −λgu ² ¤ +K(α)

Z

∂Ω

u ² dS x : u ∈ W ^1,2, (Ω), Z

Ω

g|u| ^p+2 = −1

¾ ,

K _0α ⁺ = inf

½Z

Ω

|∇u| ² + K(α) Z

∂Ω

u ² dS x : u ∈ W ^1,2 (Ω), Z

Ω

g|u| ^p+2 = 1

¾

and K _0α ⁻ = inf

½Z

Ω

|∇u| ² + K(α) Z

∂Ω

u ² dS x : u ∈ W ^1,2 (Ω), Z

Ω

g|u| ^p+2 = −1

¾ . Lemma 2.4. K _0α ⁻ > 0 and K _0α ⁺ > 0 if α ∈ (0, 1].

Proof. Let α 6= 1. We show that K _0α ⁺ > 0. If not, there is a sequence u n ∈ W ^1,2 (Ω) so that

n→∞ lim

·Z

Ω

|∇u _n | ² + K(α) Z

∂Ω

u ² _n dS _x

¸

= 0 and Z

Ω

g|u _n | ^p+2 = 1.

By the Sobolev embedding : W ^1,2 (Ω) ,→ L

^{N −22N}

(Ω), it is impossible.

The proof of K _0α ⁻ > 0 is exactly the same as the above.

By the similar method, we can prove that in the case α = 1. ¤ Remark 2.5. Let α ∈ [0, 1]. We note that K _λα ⁻ and K _λα ⁺ are concave continu- ous curves on the interval [λ ⁻ _α , λ ⁺ _α ]. Hence, K _λα ⁺ ≤ K _0α ⁺ for all λ ∈ [0, λ ⁺ _α ] and K _λα ⁻ ≤ K _0α ⁻ for all λ ∈ [λ ⁻ _α , 0]. Furthermore, by the Sobolev embedding, the equivalent norm, and the relations between the principal eigenvalues and the function g: λ

Z

Ω

gφ ^p+1 dx > 0 for all p ≥ 1, where λ 6= 0 is a principal eigen- value with corresponding positive principal eigenfunction φ ([4]), the following properties hold: (i) K _λ ⁻

−

α

= K _λ ⁺

+

α

= 0, (ii) K _λα ⁻ > 0 if λ ∈ (0, λ ⁺ _α ) and K _λα ⁺ > 0 if λ ∈ (0, λ ⁺ _α ).

Definitions and Remarks. Let λ ∈ (λ ⁻ _α , λ ⁺ _α ). We define the following sets:

H λ =

½

u ∈ W ^1,2 (Ω) : λ Z

Ω

g|u| ^p+2 = 1

¾ .

Let u ∈ H λ . Then ||u|| _λα

^p2

u ∈ M λα . If u ∈ M λα , then ||u|| ⁻ _λα

^p+22

u ∈ H λ . We define the functional E λα : H λ −→ R ¹ by

E λα (u) = Z

Ω

|∇u| ² − λ Z

Ω

gu ² + K(α) Z

∂Ω

u ² dS x . Then we obtain

E λα (u) =

· 2(p + 2) p J λα

µ

||u|| _λα

^2p

u

¶¸

^p

p+2

and

J λα (u) = p 2(p + 2) E λα

µ

||u|| ⁻ _λα

^p+22

u

¶

^p+2

p

. If we let

Q λα = inf E λα (H λ ) and C λα = inf J λα (M λα ), then by the simple calculation it follows that

Q λα =

· 2(p + 2) p C λα

¸

^p

p+2

.

This implies that if {u n } is a minimizing sequence of E λα on H λ , then {||u n || _λα

^2p

u n } is also a minimizing sequence of J λα on M λα and vice versa.

Remark 2.6. Let α 6= 1. We can prove that u = 0 is not a limit point of M λα if λ ⁻ _α < λ < λ ⁺ _α . To show that, we assume there is a sequence {u n } in M λα so that ku n k λα → 0 as n → ∞. From the Sobolev embedding: W ^1,2 (Ω) ,→ L

^{N −22N}

(Ω), the sequence {w n } which is defined by w n = _ku ^u

ⁿ

n

k

λα

is a bounded sequence in L

^{N −22N}

(Ω). We hence have the following result:

0 = hJ _λα ⁰ (u n ), u n i

||u n || ² _λ

= R

Ω |∇u n | ² − λ R

Ω gu ² _n + K(α) R

∂Ω u ² _n dS x

||u n || ² _λα + (||u n || λα ) ^p+2 Z

Ω

g|w n | ^p+2 → 1 as n → ∞, which leads to a contradiction.

We can also have the same argument for the case α = 1 by the similar method.

Lemma 2.7. Let α ∈ (0, 1] or Z

Ω

gdx 6= 0 if α = 0. There are two positive numbers δ 1 and δ 2 such that for any λ ∈ (λ ⁻ _α , λ ⁻ _α + δ 1 ) ∪ (λ ⁺ _α − δ 2 , λ ⁺ _α ), if {u n } be a minimizing sequence of J λα on M λα . Then

lim inf

n→∞

¯ ¯

¯ ¯ Z

Ω

gu ² _n

¯ ¯

¯ ¯ > 0.

Proof. Let ϕ ⁻ and ϕ ⁺ be the corresponding eigenfunctions to the principal eigenvalues λ ⁻ _α and λ ⁺ _α , respectively. We can assume that

Z

Ω

g|ϕ ⁻ | ^p+2 = −1, Z

Ω

g|ϕ ⁺ | ^p+2 = 1.

(Lemma 3.1 in [4]). We also note that Z

Ω

g(ϕ ⁻ ) ² < 0, Z

Ω

g(ϕ ⁺ ) ² > 0.

Let

δ 2 = λ ⁺ _α − R

Ω |∇ϕ ⁺ | ² − K _0α ⁺ + K(α) R

∂Ω |ϕ ⁺ | ² dS x

R

Ω g|ϕ ⁺ | ² = R K _0α ⁺

Ω g|ϕ ⁺ | ² .

Then for λ ∈ (λ ⁺ _α −δ 2 , λ ⁺ _α ) and if {u n } is a minimizing sequence of J λα on M λα , it is bounded in W ^1,2 (Ω), and then u n → u weakly in W ^1,2 (Ω) and u n → u strongly in L ² (Ω). We assume that λ > 0. By the previous equality about minimums we know that {ku n k ⁻ _λα

^p+22

u n } is a minimizing sequence of E λα on H λ , and so there is a positive number q such that

n→∞ lim λ

^p+22

ku n k ⁻ _λα

^p+24

·Z

Ω

|∇u n | ² − λ Z

Ω

g(u n ) ² + K(α) Z

∂Ω

u ² _n dS x

¸

< q < K _0α ⁺ for some g > 0. Since ku n k λα 9 0 as n → ∞, if R

Ω g(u n ) ² → 0 as n → ∞, we get

K _0α ⁺ ≤ q < K _0α ⁺ , which leads to a contradiction. Therefore,

n→∞ lim Z

Ω

gu ² _n 6= 0.

Let δ 1 =

R

Ω |∇ϕ ⁻ | ² − K _0α ⁻ + K(α) R

∂Ω |ϕ ⁻ | ² dS x

R

Ω g|ϕ ⁻ | ² − λ ⁻ _α = − R K _0α ⁻

Ω g|φ ⁻ | ² . For the value λ ∈ (λ ⁻ _α , λ ⁻ _α + δ 1 ) and λ < 0, we can get the same results by the above methods.

This completes the proof. ¤

We denote by B ε (X) the ball in a Hilbert space X centered at 0 and of radius ε. We state the following:

Proposition 2.8 ([3], pp. 6). Let J be a C ¹ -functional on a Hilbert space X and let M be a closed subset of X verifying the following property:

For any u ∈ M with J ⁰ (u) 6= 0, there exists, for a small enough ε > 0, a Fr´echet differentiable function s u : B ε (X) −→ R ¹ such that, by setting t u (δ) = s u

³ δ _||J ^J

⁰0

^(u) (u)||

´

for 0 ≤ δ ≤ ε, we have

t u (0) = 1 and t u (δ) µ

u − δ J ⁰ (u)

||J ⁰ (u)||

¶

∈ M.

If J is bounded below on M , then for any minimizing sequence {v n } in M for J, there exists another minimizing sequence {u n } in M of J such that

J(u n ) ≤ J(v n ), lim

n→∞ ||u n − v n || = 0 and

||J ⁰ (u n )|| ≤ 1 n

¡ 1 + ||u n |||t ⁰ _u

_n

(0)| ¢

+ |t ⁰ _u

_n

(0)||hJ ⁰ (u n ), u n i|, where h , i is the inner product in X.

Proof. Let C = inf J(M ). Use Ekeland’s variational principle ([3]) to get a

minimizing sequence {u n } in M with the following properties:

(i): J(u n ) ≤ J(v n ) < C + 1 n , (ii): lim

n→∞ ||u n − v n || = 0, (iii): J(w) ≥ J(u n ) − 1

n ||w − u n || for all w ∈ M .

Let us assume ||J ⁰ (u n )|| > 0 for n large, since otherwise we are done. Apply the hypothesis on the set M with u = u n to find t n (δ) = s u

n

³

δ _||J ^J

⁰0

^(u (u

ⁿn

⁾ )||

´ such that w δ = t n (δ)

³

u n − δ _||J ^J

⁰0

^(u (u

ⁿn

⁾ )||

´

∈ M for all small enough δ ≥ 0.

Use now the mean value theorem to get 1

n ||w δ − u n || ≥ J(u n ) − J(w δ )

= (1 − t n (δ))hJ ⁰ (w δ ), u n i + δt n (δ)hJ ⁰ (w δ ), J ⁰ (u n )

||J ⁰ (u n )|| i + o(δ),

where ^o(δ) _δ → 0 as δ → 0. Dividing by δ > 0 and passing to the limit as δ → 0 we derive

1

n (1 + |t ⁰ _n (0)|||u n ||) ≥ −t ⁰ _n (0)hJ ⁰ (u n ), u n i + ||J ⁰ (u n )||,

which is our claim. ¤

Lemma 2.9. Let α ∈ (0, 1] or that Z

Ω

gdx 6= 0 for α = 0. Given λ ∈ (λ ⁻ _α , λ ⁺ _α ), λ 6= 0, J λα is bounded below on M λα and there exists a minimiz- ing sequence {u n } of J λα on M λα so that

n→∞ lim ||J _λα ⁰ (u n )|| λα = 0 and

n→∞ lim J λα (u n ) = inf J λα (M λα ).

Proof. Let λ ∈ (λ ⁻ _α , λ ⁺ _α ) and let λ 6= 0. We show that J λα is bounded below on M _λα . In fact, the following can be checked easily: if u ∈ M _λα , then

λ Z

Ω

g|u| ^p+2 > 0 and

J λα (u) = pλ 2(p + 2)

Z

Ω

g|u| ^p+2 .

Let u ∈ M λα . Define G : R ¹ × W ^1,2 (Ω) −→ R ¹ by G(s, w) = Φ λα (s(u − w)), where Φ λα : W ^1,2 (Ω) −→ R ¹ is a functional defined by

Φ λα (u) = Z

Ω

|∇u| ² − λ Z

Ω

gu ² − λ Z

Ω

g|u| ^p+2 + K(α) Z

∂Ω

u ² dS x .

Then G(1, 0) = 0 and d

ds G(1, 0)

= 2 Z

Ω

|∇u| ² − 2λ Z

Ω

gu ² − λ(p + 2) Z

Ω

g|u| ^p+2 + 2K(α) Z

∂Ω

u ² dS x

= −p µZ

Ω

£ |∇u| ² − λgu ² ¤

+ K(α) Z

∂Ω

u ² dS x

¶ 6= 0.

Hence, we can apply the Implicit Function Theorem at (1, 0) and get that for δ > 0 small enough, there exists a differentiable function

s u : B δ (W ^1,2 (Ω)) −→ R ¹ such that s u (0) = 1, s u (w)(u − w) ∈ M λα , and

hs ⁰ _u (0), wi = hΦ ⁰ _λα (u), wi hΦ ⁰ _λα (u), ui

for all w ∈ B _δ (W ^1,2 (Ω)). From the identification of duality to the Hilbert space W ^1,2 (Ω), we let

w u = J _λα ⁰ (u)

||J _λα ⁰ (u)|| λα and t u (ρ) = s u (ρw u ) for all 0 ≤ ρ ≤ δ. Then t u (0) = 1 and

t u (ρ)(u − ρw u ) = s u (ρw u )(u − ρw u ) ∈ M λα .

From Proposition 2.8, there is a minimizing sequence {u n } of J λα on M λα so that

J λα (u n ) ≤ J λα (v n ) < inf J λα (M λα ) + 1 n , lim

n→∞ ||u n − v n || λα = 0, and

||J _λα ⁰ (u n )|| λα ≤ 1 n

¡ 1 + |t ⁰ _u

_n

(0)|||u n || λα

¢ + |t ⁰ _u

_n

(0)||hJ _λα ⁰ (u n ), u n i|.

Since J λα (u n ) = _2(p+2) ^λp ||u n || ² _λα , so the sequence {u n } is bounded in W ^1,2 (Ω).

Let ||u n || λα ≤ C 1 for all n. Then

||J _λα ⁰ (u n )|| λα ≤ 1 n

¡ 1 + |t ⁰ _u

_n

(0)|C 1

¢ .

Since

|t ⁰ _u

_n

(0)| = |hΦ ⁰ _λα (u n ), w n i|

p||u n || ² _λα ,

where w n = w u

n

, and lim n→∞ inf ||u n || λα > 0, if we show that |t ⁰ _u

_n

(0)| is uniformly bounded on n, we are done. In fact, we have the following inequality

|hΦ ⁰ _λα (u n ), w n i|

≤ 2 Z

Ω

|∇u n · ∇w n |+2λ Z

Ω

|u n w n |+λ(p+2) Z

Ω

|g||u n | ^p+1 |w n | +2K(α)

Z

∂Ω

u n w n dS x .

From the well-known Sobolev embedding theorem, ||w n || λα = 1 for all n, and H¨older inequality, we have two positive constants C 2 and C 3 so that

|hΦ ⁰ _λα (u n ), w n i| ≤ C 2 ||u n || λα + C 3 .

Since {u n } is a bounded sequence in W ^1,2 (Ω), so is hΦ ⁰ _λα (u n ), w n i on n. There- fore, we can conclude that

n→∞ lim ||J _λα ⁰ (u n )|| λα = 0.

Clearly, we note that

n→∞ lim J λα (u n ) = inf J λα (M λα ).

¤ Theorem 2.10. Let α ∈ (0, 1] or that

Z

Ω

gdx 6= 0 for α = 0. For any λ ∈ (λ ⁻ _α , λ ⁻ _α + δ ₁ ) ∪ (λ ⁺ _α − δ ₂ , λ ⁺ _α ), λ 6= 0, the problem (I _λα ) has a positive solution.

Proof. Let

c = inf J λα (M λα ) and let {u n } be a sequence in M λα such that

n→∞ lim J λα (u n ) = c.

By Lemma 2.9, we can assume that

n→∞ lim ||J _λα ⁰ (u n )|| λα = 0.

Then {u n } is bounded and we can find a weak limit point u of the sequence in W ^1,2 (Ω). We can also assume that {u n } converges weakly to u and, by the Rellich-Kondrakov Theorem ([3]), that u n → u strongly in L ^q (Ω) for all q < _{N −2} ^2N . In particular, for any v ∈ W ^1,2 (Ω),

hJ _λα ⁰ (u n ), vi = Z

Ω

∇u n · ∇v − λ Z

Ω

gu n v − λ Z

Ω

gu n |u n | ^p v + K(α) Z

∂Ω

u n vdS x , which converges as n → ∞ to

Z

Ω

(∇u · ∇v − λguv − λgu|u| ^p v)dx + K(α) Z

∂Ω

uvdS x = hJ _λα ⁰ (u), vi.

Hence, hJ _λα ⁰ (u), vi = 0 for all v ∈ W ^1,2 (Ω) which means that u is a weak solution for (I λα ). In particular, hJ _λα ⁰ (u), ui = 0. Since lim inf

n→∞

¯ ¯

¯ ¯ Z

Ω

gu ² _n

¯ ¯

¯ ¯ > 0 by Lemma 2.7, we have that u 6= 0. Therefore, u ∈ M λα .

Since J λα is weakly lower semi-continuous, we get c ≤ J λα (u) ≤ lim

n→∞ J λα (u n ) = c.

It follows that J λα (u) = c and that ||u n || λα → ||u|| λα which implies that u _n → u strongly in W ^1,2 (Ω). Since J _λα ⁰ is continuous at u, we get J _λα ⁰ (u) = 0.

The positivity of u is clear from the equality J λα (u) = J λα (|u|).

This completes the proof. ¤

Theorem 2.11. Let α ∈ (0, 1] or that Z

Ω

gdx 6= 0 for α = 0. If g(x) < 0 for all x ∈ ∂Ω, for any λ ∈ (0, λ ⁺ _α ), the problem (I λα ) has a positive solution.

Proof. By Theorem 2.10, we have a positive solution u λ of the problem:

( −∆u = λg(x)u + g(x)u|u| ^p in Ω, (1 − α) ∂u

∂n + αu = 0 on ∂Ω, for λ ∈ (λ ⁻ _α + δ 2 , λ ⁻ _α ) ∪ (λ ⁺ _α − δ 1 , λ ⁺ _α ).

Let 0 < λ < λ ⁺ _α and let {u n } be a minimizing sequence of J λα in M λα . If K _λα ⁺ < K _0α ⁺ on (0, λ ⁺ _α ), we can have the same inequality about the limit lim inf

n→∞

¯ ¯

¯ ¯ Z

Ω

gu ² _n

¯ ¯

¯ ¯ > 0 by the same method in Lemma 2.7, and so using the same method in Theorem 2.10 we have the existence of a positive solution of (I λα ).

We note that K _λα ⁺ ≤ K _0α ⁺ for all λ ∈ (0, λ ⁺ _α ).

Suppose that there is λ 0 so that 0 < λ 0 < λ ⁺ _α and K _λ ⁺

0

α = K _0α ⁺ and K _λα ⁺ <

K _0α ⁺ on (λ 0 , λ ⁺ _α ). Let u λ be the positive minimizer of the functional J λα on M λα for λ ∈ (λ 0 , λ ⁺ _α ). Let

t ^p _λ = λ R

Ω gu ^p+2 _λ + (λ − λ 0 ) R

Ω gu ² _λ + K(α) R

∂Ω u ² dS x

λ 0

R

Ω gu ^p+2 _λ .

Then t λ u λ ∈ M λ

0

α , and J λ

0

α (t λ u λ ) = t ² _λ

·

J λα (u λ ) + p

2(p + 2) (λ − λ 0 ) Z

Ω

gu ² _λ

¸ . As the previous calculation in Remark, we note that

λ→λ inf

0

Z

Ω

gu ^p+2 _λ > 0, and we also note that

lim inf

λ→λ

0

J λα (u λ ) < ∞

implies that

lim inf

λ→λ

0

¯ ¯

¯ ¯ Z

Ω

gu ² _λ

¯ ¯

¯ ¯ 6= ∞,

and hence, t λ → 1 as λ → λ 0 . Since {t λ u λ } is a minimizing sequence of J λ

0

α

as λ → λ 0 , we get the weak limit u λ

0

of u λ so that

λ→λ lim

0

t λ u λ = u λ

0

in L ² (Ω).

If u λ

0

6= 0, we know that it is the minimizer of J λ

0

α and is the positive solution of the above boundary value problem with respect to λ ₀ . Let

v λ = λ

^p+21

ku λ k ⁻ _λα

^p+22

u λ . Then

K _λα ⁺ = Z

Ω

|∇v λ | ² − λ Z

Ω

g(v λ ) ² + K(α) Z

∂Ω

v _λ ² dS x ,

K _λ ⁺

₀

_α = Z

Ω

|∇v λ

0

| ² − λ 0

Z

Ω

g(v λ

0

) ² + K(α) Z

∂Ω

v _λ ²

₀

dS x , and

− Z

Ω

g(v λ ) ² ≤ K _λα ⁺ − K _λ ⁺

₀

_α λ − λ 0 ≤ −

Z

Ω

g(v λ

0

) ² . Taking the limit on the both sides as λ → λ 0 , we get

dK _λα ⁺

dλ (λ 0 ) = − Z

Ω

g(v λ

0

) ² . Hence, K _λα ⁺ is differentiable at λ = λ 0 , and so R

Ω g(v λ

0

) ² = 0. Since K _λ ⁺

₀

_α = K _0α ⁺ =

Z

Ω

|∇v λ

0

| ² + K(α) Z

∂Ω

v _λ ²

₀

dS x , so v λ

0

is also a positive solution of the problem:

( −∆u = g(x)u|u| ^p in Ω, (1 − α) ∂u

∂n + αu = 0 on ∂Ω, which leads to a contradiction.

Let u λ

0

= 0. The u λ → 0 a.e. in Ω as λ → λ 0 . By Harnack Inequality we can argue that u λ → 0 uniformly on any compact subset of Ω. Also by the Maximum Principle {u λ } is uniformly bounded on any neighborhood of

∂Ω. Therefore, u λ → 0 uniformly bounded in Ω, and then by the Lebesgue dominated convergence Theorem

1 = lim

λ→λ

0

ku λ k ⁻² _λα Z

Ω

g|u λ | ^p+2 = 0

since ku λ k λα 9 0 as λ → λ 0 , which also leads to a contradiction.

This completes the proof. ¤

Corollary 2.12. Let α ∈ (0, 1] or that Z

Ω

gdx 6= 0 for α = 0. If g satisfies the condition in Theorem 2.11, then the following problem:

 



−∆u = g(x)u|u|

^{N −24}

in Ω, (1 − α) ∂u

∂n + αu = 0 on ∂Ω, has a positive solution.

Proof. With the result of Theorem 2.11 if we let λ 0 = 0, the proof for the convergence of a minimizing sequence of the functional:

J(u) = 1 2

Z

Ω

|∇u| ² − 1 p + 2

Z

Ω

g|u| ^p+2 dx + 2K(α) Z

∂Ω

u ² dS x

on the Nehari manifold {u ∈ W ^1,2 (Ω) : u 6= 0,

Z

Ω

|∇u| ² − Z

Ω

g|u| ^p+2 + K(α) Z

∂Ω

u ² dS x = 0}

can be produced by the one of Theorem 2.11. ¤

Acknowledgement. This work was supported by grant No. R05-2002-000- 00605-0 from the Basic Research Program of the Korea Science and Engineering Foundation.

References

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[2] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477.

[3] N. Ghoussoub, Duality and perturbation methods in critical point theory, With appendices by David Robinson. Cambridge Tracts in Mathematics, 107. Cambridge University Press, Cambridge, 1993.

[4] B. Ko and K. Brown, The existence of positive solutions for a class of indefinite weight semilinear elliptic boundary value problems, Nonlinear Anal. 39 (2000), no. 5, Ser. A:

Theory Methods, 587–597.

[5] R. Poho˜ zaev, Eigenfunctions on the equation ∆u + λf (u) = 0, Soviet Math. Dokl. 6 (1965), 1408–1411.

Bongsoo Ko

Department of Mathematics Education Educational Research Institute Cheju National University Cheju 690-756, Korea

E-mail address: [email protected] Seungpil Kang

Department of Mathematics Education

Educational Research Institute

Cheju National University

Cheju 690-756, Korea