On the Existence of Infinite Sequences of Ordered Positive Solutions of Nonlinear Elliptic Eigenvalue Problems

Assume that f satisfies (f1)–(f3) and let ϕ satisfy (ϕ1)–(ϕ2). There exist Γ1,Γ2>0 such that

1. ∑ i , j = 1 N ∂ ⁡ α i ∂ ⁡ η j ⁢ ( η ) ⁢ ξ i ⁢ ξ j ≥ Γ 1 ⁢ ϕ ⁢ ( | η | ) ⁢ | ξ | 2

and

1. |∑i,j=1N∂⁡αi∂⁡ηj⁢(η)|≤Γ2⁢ϕ⁢(|η|),

where ξ=(ξ1,…,ξN), η=(η1,…,ηN), η≠0, αj⁢(η)=ϕ⁢(|η|)⁢ηj, j=1,…,N. Assume, in addition, one of the conditions:

1. there is L > 0 such that f ⁢ ( x , s ) + L ⁢ s is nondecreasing in [ 0 , ∞ ) and either

   (1.4) ϕ ⁢ ( s ) → ∞   as ⁢ s → 0

   or

   (1.5) ϕ ⁢ is constant ;

2. ϕ is bounded near the origin, H is convex near the origin, and

   1. for each integer k ≥ 1 , there exist functions δ k , d k : Ω ¯ → ( 0 , ∞ ) such that

      d k ⁢ ( x ) ⁢ ( c k - s ) ⁢ ϕ ⁢ ( c k - s ) ≥ λ ⁢ f ⁢ ( x , s )   for ⁢ x ∈ Ω ¯ , s ∈ ( c k - δ k ⁢ ( x ) , c k ) .

Then, there is λ¯>0 such that, for each λ>λ¯, (1.1) admits two infinite sequences of C1,α⁢(Ω¯) positive solutions {u^m}m=1∞ and {uk}k=2∞, 0<α<1, satisfying

and

Let g∈C⁢(Ω¯×ℝ,ℝ). Assume that

1. there is s 0 > 0 such that g ⁢ ( x , s 0 ) ≤ 0 for each x ∈ Ω ¯ ;

2. there is L>0 such that g⁢(x,s)+L⁢s is increasing in [0,s0] for each x∈Ω¯.

Assume one of the conditions:

1. ϕ⁢(s)→∞ as s→0 or ϕ is constant;

2. ϕ is bounded near the origin, H is convex near the origin, and there is c>0 such that

   c ⁢ ( s 0 - s ) ⁢ ϕ ⁢ ( s 0 - s ) ≥ g ⁢ ( x , s )   for ⁢ x ∈ Ω ¯ , s ∈ [ 0 , s 0 ] .

Let u∈C1⁢(Ω¯) be a positive solution of

{ - Δ Φ ⁢ u = g ⁢ ( x , u ) in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω .

Then, ∥u∥∞≠s0.

Proof.

Assume, on the contrary, that ∥u∥∞=s0. We consider two cases.

Case 1. ϕ⁢(s)→∞ as s→0. Set h⁢(x,t):=g⁢(x,t)+L⁢t and let v∈W01,Φ⁢(Ω) with v≥0. Using conditions (i)–(ii), we have

We will use [18, Theorem 1.1]. To verify condition (1.6) in [18, Theorem 1.1], let F⁢(s)=s22 for s>0 (in the notation of [18]). Note that

F ⁢ ( s ) H ⁢ ( s ) = s 2 2 ⁢ [ s 2 ⁢ ϕ ⁢ ( s ) - Φ ⁢ ( s ) ] = 1 2 ⁢ [ ϕ ⁢ ( s ) - Φ ⁢ ( s ) s 2 ] ,

where the function H was introduced in Section 1. It follows from (2.3) that

As a consequence, there is δ>0 such that

F ⁢ ( s ) H ⁢ ( s ) ≤ 1   for ⁢ s ∈ ( 0 , δ ) .

Once H-1 is increasing, we conclude from the last inequality that condition (1.6) in [18, Theorem 1.1] is verified, which implies that s0>u⁢(x) for x∈Ω, a contradiction. The case where ϕ is a constant is simpler and is proved in a similar way.

Case 2. ϕ is bounded near the origin. Set h⁢(x,s):=-g⁢(x,s)+c⁢(s0-s)⁢ϕ⁢(s0-s), where the constant c>0 is chosen such that

h ⁢ ( x , s ) ≥ 0   for ⁢ ( x , s ) ∈ Ω ¯ × [ 0 , s 0 ] .

Such a constant c>0 exists due to condition (b). For each v∈W1,Φ⁢(Ω) with v≥0, we have

∫ Ω ϕ ⁢ ( | ∇ ⁡ ( s 0 - u ) | ) ⁢ ∇ ⁡ ( s 0 - u ) ⁢ ∇ ⁡ v + ∫ Ω c ⁢ ( s 0 - u ) ⁢ ϕ ⁢ ( s 0 - u ) ⁢ v = - ∫ Ω g ⁢ ( x , u ) ⁢ v + c ⁢ ∫ Ω ( s 0 - u ) ⁢ ϕ ⁢ ( s 0 - u ) ⁢ v ≥ 0 .

We claim that condition (1.6) in [18, Theorem 1.1] holds with F⁢(s)=c⁢Φ⁢(s). Indeed, using condition (Δ2), we have

where C>1 is a constant. Since H is convex near the origin and H⁢(0)=0, we have that H-1⁢(a⁢s)≤a⁢H-1⁢(s) for a>1 and small s. Thus, by (3.2) we find that

H - 1 ⁢ ( Φ ⁢ ( s ) ) ≤ C ⁢ s   for small ⁢ s .

The rest of the proof follows as in Case 1. ∎

Assume that the hypotheses of Theorem 1.1 are satisfied. Then, v∈Qk⁢(λ) if and only if v is a positive C1,α⁢(Ω¯)-solution of (3.3) with ∥v∥∞<ck.

Proof.

Let v∈Qk⁢(λ). Then, v is a weak solution of (3.3). By the regularity result in [14, Theorem 1.7] and the remark following it we infer that v∈C1,α⁢(Ω¯) for some α∈(0,1). Let

U 0 = { x ∈ Ω | v ⁢ ( x ) < 0 }   and   U c k = { x ∈ Ω | v ⁢ ( x ) > c k } ,

and assume that U0 and Uck are both nonempty sets. Note that

{ - Δ Φ ⁢ v = λ ⁢ f k ⁢ ( x , v ) ≥ 0 in U 0 , v = 0 on ∂ ⁡ U 0 .

It follows by the maximum principle that v≥0 on U0, a contradiction.

On the other hand,

{ - Δ Φ ⁢ ( v - c k ) = λ ⁢ f k ⁢ ( x , v ) ≤ 0 in ⁢ U c k , v - c k = 0 on ⁢ ∂ ⁡ U c k ,

and, again by the maximum principle, we have a contradiction. (We observe that the inequalities above are meant in the distributional sense.) We conclude that 0≤v≤ck and, by using the definition of fk and Lemma 3.1, we obtain that v∈C1,α⁢(Ω¯) is a positive solution of (3.3) with ∥v∥∞<ck.

The converse is immediate. ∎

The energy functional Ik⁢(λ,⋅) is coercive and weakly sequentially lower semicontinuous.

Proof.

Using the Hölder and Poincaré inequalities, we have

I k ⁢ ( λ , u ) ≥ ∫ Ω Φ ⁢ ( | ∇ ⁡ u | ) ⁢ 𝑑 x - λ ⁢ C ⁢ ∫ Ω | u | ⁢ 𝑑 x ≥ ∫ Ω Φ ⁢ ( | ∇ ⁡ u | ) ⁢ 𝑑 x - C ⁢ ∥ u ∥ Φ ≥ ∫ Ω Φ ⁢ ( | ∇ ⁡ u | ) ⁢ 𝑑 x - C ⁢ ∥ ∇ ⁡ u ∥ Φ .

We recall that (see [9, Lemma 3.14])

∫ Ω Φ ⁢ ( | ∇ ⁡ u | ) ⁢ 𝑑 x ∥ ∇ ⁡ u ∥ Φ → ∞   as ⁢ ∥ ∇ ⁡ u ∥ Φ → ∞ .

Using this fact and the last inequality, we have Ik⁢(λ,u)→∞ as ∥u∥1,Φ→∞.

Now, assume that un⇀u in W01,Φ⁢(Ω). Using compact embeddings and the Lebesgue theorem, we have

∫ Ω F k ⁢ ( x , u n ) ⁢ 𝑑 x → ∫ Ω F k ⁢ ( x , u ) ⁢ 𝑑 x .

In addition, using the fact that the norm function is weakly sequentially lower semicontinuous, we have

By Lemma 3.3, for each λ>0, there is uλ,k∈W01,Φ⁢(Ω) such that

I k ⁢ ( λ , u λ , k ) = min ⁡ { I ⁢ ( u ) | u ∈ W 0 1 , Φ ⁢ ( Ω ) }

and, so, Qk⁢(λ) is not empty.

The operator -ΔΦ:W01,Φ⁢(Ω)→W-1,Φ~⁢(Ω) is of type S+.

Proof.

Assume that un⇀u in W01,Φ⁢(Ω) and lim sup⁡〈-ΔΦ⁢un,un-u〉≤0. Since -ΔΦ is monotone, we have

From (3.4), from the fact that -ΔΦ is strictly monotone, and from a well-known result by Dal Maso and Murat [6] it follows that

Applying Young’s inequality, namely,

and the Δ2-condition, we have