Multiple solutions to multi-critical Schrödinger equations

Theorem 1.1

Suppose ( V 1 ) and ( V 2 ) hold, then for every 0 < μ < μ 1 , there exists λ ( μ ) > 0 such that problem (1.1) has at least one ground state positive solution for each λ ⩾ λ ( μ ) .

Theorem 1.2

Suppose (V1) and (V2) hold. Let { u n } be a sequence of solutions of (1.1) such that 0 < μ < μ 1 , λ n → ∞ , and I λ n , μ ( u n ) → c < m ( R N ) as n → ∞ . Then, { u n } concentrates at a solution u of (1.14), that is, u n → u in E as n → ∞ .

Theorem 1.3

Suppose (V1) and (V2) hold and N ⩾ 4 . Then there exists 0 < μ ∗ < μ 1 , for each 0 < μ ⩽ μ ∗ , there is Λ ( μ ) > 0 such that problem (1.1) has at least cat Ω ( Ω ) positive solutions whenever λ ⩾ Λ ( μ ) .

Lemma 2.1

Let λ n ⩾ 1 and u n ∈ E be such that  ‖ u n ‖ λ n 2 ≤ K for λ n → ∞ . Then there is a u ∈ H 0 1 ( Ω ) such that, up to a subsequence, u n ⇀ u weakly in E and u n → u in L 2 ( R N ) as n → ∞ . Moreover, if u = 0 , we have u n → 0 in L p ( R N ) for 2 ≤ p < 2 N N − 2 as n → ∞ .

Proof

Since for λ n ⩾ 1 , ‖ u n ‖ 2 ⩽ ‖ u n ‖ λ n 2 ≤ K , we may assume u n ⇀ u weakly in E and u n → u in L loc 2 ( R N ) . Set C m ≔ x : V ( x ) ⩾ 1 m for m ∈ N . Then

and for all m ∈ N ,

Hence, u n → 0 in L 2 ( C m ) as n → ∞ . By a diagonal process, we may assume that u n → 0 in L 2 ( R N \ Ω ¯ ) as n → ∞ .

In order to show that u n → u in L 2 ( R N ) , we set for R > 0 that

and

where M 0 is given in ( V 2 ) . Since A ( R ) ⊂ C m if m large, we have

By the Hölder inequality, for 1 < p < N N − 2 , 1 p + 1 p ′ = 1 there holds

where o R ( 1 ) → 0 as R → ∞ . Then, for ε > 0 , there exists N > 0 such that for n > N ,

It yields that u n → u in L 2 ( R N ) .

Now, we turn to the last assertion. If u n ⇀ 0 , for 2 < p < 2 N N − 2 and any subset Ω ˜ ⊂ R N , the interpolation inequality yields

for n large, where θ = N ( p − 2 ) 2 p . Choosing Ω ˜ to be A ( R ) and B ( R ) , respectively, we obtain u n → 0 in L p ( R N ) as n → ∞ .□

Lemma 2.2

For every 0 < μ < μ 1 , there exists λ ( μ ) > 0 such that

if λ ⩾ λ ( μ ) . Moreover, for all u ∈ E and λ ⩾ λ ( μ ) there exists a constant C > 0 such that

where C ≤ ( μ + μ 1 ) / ( 2 + μ + μ 1 ) .

Lemma 3.1

If μ ∈ ( 0 , μ 1 ) and λ ⩾ λ ( μ ) , then every ( PS ) c sequence { u n } ⊂ E for I λ , μ is bounded in E .

Proof

Since 2 < p < 2 min ∗ , by Lemma 2.2 we have

This implies that { u n } is bounded in E .□

Lemma 3.2

For all μ ∈ ( 0 , μ 1 ) and λ ⩾ λ ( μ ) , the functional I λ , μ satisfies that

1. I λ , μ ( 0 ) = 0 and there exist ρ , δ > 0 , such that inf ‖ u ‖ λ = ρ I λ , μ ( u ) ≥ δ > 0 ;

2. There exists e ∈ E such that ‖ e ‖ λ > ρ and I λ , μ ( e ) ≤ 0 .

Proof

By the Hardy-Littlewood-Sobolev inequality, we have

Then, ( i ) follows by choosing ‖ u ‖ λ = ρ sufficiently small. ( i i ) is valid since for any fixed positive u ∈ E \ { 0 } , I λ , μ ( t u ) → − ∞ as t → + ∞ .□

Lemma 3.3

There exist τ = τ ( μ ) > 0 and a constant σ ( μ ) > 0 such that the mountain pass level c λ , μ of I λ , μ satisfies that

for all λ > 0 .

Proof

It is proved in [17] that c μ , Ω < m ( R N ) and that there exists a critical point u ∈ H 0 1 ( Ω ) of I μ , Ω such that I μ , Ω ( u ) = c μ , Ω . Extend the function u to R N such that u = 0 outside Ω , then u ∈ N λ , μ , which implies

So there exists τ = τ ( μ ) > 0 such that

for all λ > 0 .

Next, for each u ∈ N λ , μ ,

where C depending only on μ , yields that there exists σ > 0 independent of λ such that ‖ u ‖ λ ⩾ σ , and then

Hence,

the conclusion follows.□

Proposition 3.4

There exist μ ∈ ( 0 , μ 1 ) and λ ( μ ) > 0 such that I λ , μ satisfies the ( PS ) c condition for

whenever λ ⩾ λ ( μ ) .

Proof

It is known from Lemma 3.1 that the ( PS ) c sequence { u n } of I λ , μ is bounded in E . We may assume that

as n → ∞ . By the Hardy-Littlewood-Sobolev inequality and Hölder inequality,

for i = 1 , 2 , … , k . Thus, for every φ ∈ H 1 ( R N ) ,

that is, u 0 is a weak solution of problem (1.1).

Let w n = u n − u 0 . The Brézis-Lieb-type lemma in [11]