On a Quasilinear Schrödinger Problem at Resonance

The function f⁢(t) enjoys the following properties:

1. f is a uniquely defined C ∞ function and invertible,

2. |f′⁢(t)|≤1 for all t∈ℝ,

3. |f⁢(t)|≤|t| for all t∈ℝ,

4. f⁢(t)/2≤t⁢f′⁢(t)≤f⁢(t) for all t≥0,

5. |f⁢(t)|≤21/4⁢|t|1/2 for all t∈ℝ,

6. |f⁢(t)⁢f′⁢(t)|≤1/2 for all t∈ℝ,

7. There exists a positive constant C such that

   | f ⁢ ( t ) | ≥ { C ⁢ | t | if ⁢ | t | ≤ 1 , C ⁢ | t | 1 / 2 if ⁢ | t | ≥ 1 .

Proof.

The proofs of (1)–(7) can be found in [10, Lemma 2.1] (see also [16, 12, 6]). ∎

If v is a critical point of Iλ, then any critical point is of class C2,α⁢(Ω). Moreover, if v∈C2,α⁢(Ω) is a critical point of Iλ , then the function u=f⁢(v) is a classical solution of (1.3).

Proof.

The proof is similar to the proof of [10, Propostion 2.6]. ∎

Setting

Λ = sup ⁡ { λ > λ 1 : (2.1) admits a nonnegative solution } ,

we have Λ>λ1.

Proof.

We shall use bifurcation theory to show that (2.1) admits positive solutions for λ>λ1 near λ1. To do this we define ℱ:C02,β⁢(Ω)×ℝ→C0,β⁢(Ω) by ℱ⁢(u,λ)=-Δ⁢u-λ⁢f⁢(u)⁢f′⁢(u). We have that ℱ⁢(0,λ)=0 for all λ. Moreover, by Lemma 2.1 (4) and since f′⁢(0)=1, we also have that ℱu⁢(0,λ1)⁢v=-Δ⁢v-λ1⁢v. Hence,

N ⁢ ( ℱ u ⁢ ( 0 , λ 1 ) ) = 〈 ϕ 1 〉 , codim ⁡ R ⁢ ( ℱ u ⁢ ( 0 , λ 1 ) ) = 1   and   ℱ λ , u ⁢ ( 0 , λ 1 ) ⁢ ϕ 1 = - ϕ 1 ∉ R ⁢ ( ℱ u ⁢ ( 0 , λ 1 ) ) .

It follows that (0,λ1) is a bifurcation point for ℱ (see [7]).

Thus, if we decompose

C 0 2 , β ⁢ ( Ω ) = 〈 ϕ 1 〉 ⊕ V , where   V = 〈 ϕ 1 〉 ⊥ ,

then by Theorem 3.1 we obtain a neighborhood U of (0,λ1) in C02,β⁢(Ω)×ℝ, and continuous functions ϕ:(-a,a)→ℝ, ψ:(-a,a)→V with ϕ⁢(0)=λ1,ψ⁢(0)=0 and

ℱ - 1 ⁢ { 0 } ∩ U = { ( α ⁢ ϕ 1 + α ⁢ ψ ⁢ ( α ) , ϕ ⁢ ( α ) ) : α ∈ ( - a , a ) } ∪ { ( 0 , λ ) : ( 0 , λ ) ∈ U } .

Set uα=α⁢ϕ1+α⁢ψ⁢(α). Note that, in particular, ψ⁢(α)→0 in C1,β⁢(Ω¯) as α→0, and that uα>0 in Ω for α sufficiently small.

Next we show that ϕ⁢(α)>λ1 for all positive and sufficiently small α.

Suppose, by contradiction, that there exists a sequence αn→0+ with ϕ⁢(αn)≤λ1. Let un be the positive solution of problem (2.1) associated with λ=ϕ⁢(αn). Then,

- ∫ Ω Δ ⁢ u n ⁢ ϕ 1 ⁢ 𝑑 x - ∫ Ω ϕ ⁢ ( α n ) ⁢ f ⁢ ( u n ) ⁢ f ′ ⁢ ( u n ) ⁢ ϕ 1 ⁢ 𝑑 x = 0 ,

equivalently, we have

λ 1 ⁢ ∫ Ω u n ⁢ ϕ 1 = ∫ Ω ϕ ⁢ ( α n ) ⁢ f ⁢ ( u n ) ⁢ f ′ ⁢ ( u n ) ⁢ ϕ 1 ⁢ 𝑑 x .

But, by (2) and (3) in Lemma 2.1, we obtain

λ 1 ⁢ ∫ Ω u n ⁢ ϕ 1 = ∫ Ω ϕ ⁢ ( α n ) ⁢ f ⁢ ( u n ) ⁢ f ′ ⁢ ( u n ) ⁢ ϕ 1 ⁢ 𝑑 x ≤ λ 1 ⁢ ∫ Ω f ⁢ ( u n ) ⁢ f ′ ⁢ ( u n ) ⁢ ϕ 1 ⁢ 𝑑 x < λ 1 ⁢ ∫ Ω u n ⁢ ϕ 1 ⁢ 𝑑 x ,

which gives a contradiction, since un>0 for all n∈ℕ. In the last inequality we used that if f′⁢(t)=1⇔t=0, then f⁢(t)⁢f′⁢(t)<t for all t>0. Therefore, ϕ⁢(α)>λ1. ∎

Let λ∈(λ1,Λ). Then, problem (2.1) admits a supersolution.

Proof.

From the definition of Λ it follows that there exists a λ0∈(λ,Λ) such that problem (2.1) admits a nonnegative solution u+. We have that u+ is a supersolution of (2.1).

Indeed, for any ϕ∈W, ϕ≥0 in Ω, by Lemma 2.1 (4), we have

The function

u - = { v in ⁢ B R ⁢ ( 0 ) , 0 in ⁢ ℝ N ∖ B R ⁢ ( 0 ) ,

is a subsolution of (2.1) and, by construction, we can see that u-<u+.

Proof.

Assume that ψ∈C0∞⁢(Ω) with ψ≥0. See Figure 1. We need to consider two cases.

Case (1): supp⁡ψ∩BR⁢(0)=∅. We have that I′⁢(u-)⁢ψ=0, so u- is solution and, therefore, a subsolution of the problem.

Case (2): supp⁡ψ∩BR⁢(0)≠∅. In this case, by Lemma 2.1 (4)–(7), we have

where C is a positive constant independent of ϵ. Notice that for ϵ sufficiently small, we can consider

∫ B R ( 0 ) ∩ { v ≤ 1 } ψ ⁢ ( x ) = ∫ B R ⁢ ( 0 ) χ { v ≤ 1 } ⁢ ψ ⁢ ( x ) ≥ ∫ B R ⁢ ( 0 ) ψ ⁢ ( x ) - ϵ α + 2 .

Therefore,

for ϵ small enough, where Co and C are positive constants independent of ϵ. ∎

Proof of Theorem 1.1.

From Lemma 2.4, for each λ∈(λ1,Λ), we have that problem (2.1) admits a supersolution u+. Moreover, by Lemma 2.5 there is a subsolution u- of problem (2.1) satisfying u-<u+ in Ω. Therefore, there exists a nonnegative solution u of problem (2.1), verifying u-≤u≤u+ in Ω, and by Proposition 2.2 we conclude that Theorem 1.1 is valid in this case.

By Lemma 2.5, there exists a subsolution u- of (2.1) . Since u-<ϕ1 in Ω, we have, also in this case, that u is a nonnegative solution of problem (2.1). ∎

Let un⇀u in W as n→∞. Then, H⁢(un)→H⁢(u) and F⁢(un)→F⁢(u) as n→∞.

Proof.

Since un⇀u in W as n→∞, by Sobolev embedding, up to a subsequence, we have un→u in Lp⁢(Ω) as n→∞ , 2≤p<2* and un⁢(x)→u⁢(x) a.e. in Ω and there exists h∈Lp such that |un|≤h a.e. in Ω. By Lemma 2.1 (1) we have f⁢(un⁢(x))→f⁢(u⁢(x)) a.e. in Ω and f′⁢(un⁢(x))→f′⁢(u⁢(x)) a.e. in Ω.

For all v∈W,

〈 H ⁢ ( u n ) - H ⁢ ( u ) , v 〉 = ∫ Ω [ f ⁢ ( u n ) ⁢ f ′ ⁢ ( u n ) - f ⁢ ( u ) ⁢ f ′ ⁢ ( u ) ] ⁢ v ⁢ 𝑑 x ,

since f⁢(un)⁢f′⁢(un)→f⁢(u)⁢f′⁢(u) a.e. in Ω and by Lemma 2.1 (6), which together with the Lebesgue dominated convergence theorem imply that

∫ Ω f ⁢ ( u n ) ⁢ f ′ ⁢ ( u n ) ⁢ v → ∫ Ω f ⁢ ( u ) ⁢ f ′ ⁢ ( u ) ⁢ v   as ⁢ n → ∞ .

We observe that

〈 F ⁢ ( u n ) - F ⁢ ( u ) , v 〉 = ∫ Ω [ g ⁢ ( x , f ⁢ ( u n ) ) ⁢ f ′ ⁢ ( u n ) - g ⁢ ( x , f ⁢ ( u ) ) ⁢ f ′ ⁢ ( u ) ] ⁢ v ⁢ 𝑑 x ,

and by (G1) we have

g ⁢ ( x , f ⁢ ( u n ) ) ⁢ f ′ ⁢ ( u n ) → g ⁢ ( x , f ⁢ ( u ) ) ⁢ f ′ ⁢ ( u )   as ⁢ n → ∞ .

By (G2) and Lemma 2.1 (5), it follows that

| g ⁢ ( x , f ⁢ ( u n ) ) ⁢ f ′ ⁢ ( u n ) | ≤ σ ⁢ ( x ) + ρ ⁢ ( x ) ⁢ | f ⁢ ( u n ) | r ≤ σ ⁢ ( x ) + C ⁢ ρ ⁢ ( x ) ⁢ | u n | r / 2 ,

and g⁢(⋅,f⁢(un))→g⁢(⋅,f⁢(u)) in L(2*)′⁢(Ω) as n→∞, because ρ∈L∞ and so σ⁢(x)+ρ⁢hr/2∈L(2*)′⁢(Ω). ∎