Existence and Multiplicity of Nontrivial Solutions to Quasilinear Elliptic Equations

Anran Li; Chongqing Wei

doi:10.1515/ans-2015-5053

Artikel Open Access

Existence and Multiplicity of Nontrivial Solutions to Quasilinear Elliptic Equations

Anran Li und Chongqing Wei

Veröffentlicht/Copyright: 28. Juni 2016

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Abstract

In this paper, Morse theory is used to study the existence and multiplicity of nontrivial solutions for the following class of quasilinear elliptic equations:

{ - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u = g ⁢ ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ ⁡ Ω ,

where Ω⊂ℝN is a bounded open domain with smooth boundary ∂⁡Ω, N⩾3 and g is a Carathéodory function with some additional growth conditions.

Keywords: Quasilinear Elliptic Equation; Critical Group; Homological Nontrivial Critical Point,Morse Theory; Local Linking

MSC 2010: 35B38; 35D05; 35J50

1 Introduction and Main Results

Quasilinear Schrödinger equations of the following form arise in various branches of mathematical physics:

(Q) i ⁢ ∂ t ⁡ ψ = Δ ⁢ ψ + V ⁢ ( x ) ⁢ ψ - h ⁢ ( | ψ | 2 ) ⁢ ψ - Δ ⁢ [ ρ ⁢ ( | ψ | 2 ) ] ⁢ ρ ′ ⁢ ( | ψ | 2 ) ⁢ ψ ,

where ψ:ℝ×ℝN→ℂ, V=V⁢(x), x∈ℝN, is a given potential, and ρ, h are real functions. The above quasilinear equations have been derived as models of several physical phenomena corresponding to various types of ρ. The case ρ⁢(s)=s was considered by Kurihura in [16] for the superfluid film equation in plasma physics (cf. [17]). In the case where ρ⁢(s)=(1+s)1/2, equations (Q) model the self-channeling of a high-power ultra short laser in matter, see [3, 4, 7]. Equations (Q) also appears in plasma physics and fluid mechanics [2, 15, 30], in mechanics [14] and in condensed matter theory [27]. Considering the case ρ⁢(s)=s, and putting

ψ ⁢ ( t , x ) = exp ⁡ ( - i ⁢ F ⁢ t ) ⁡ u ⁢ ( x ) , F ∈ ℝ ,

we can obtain the corresponding equation

(SQ) - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u + V ⁢ ( x ) ⁢ u = h ~ ⁢ ( u ) in ⁢ ℝ N ,

where we have renamed V⁢(x)-F to be V⁢(x), and h~⁢(u)=h⁢(u2)⁢u. In the past several years a lot of attention has been paid to equations (SQ), see, for example, [8, 12, 13, 22, 23, 25, 26, 29] and the references therein. In particular, the existence of a positive ground state solution has been proved in [29] by using a constrained minimization argument, which gives a solution to (SQ) with an unknown Lagrange multiplier λ in front of the nonlinear term. In [22], by a change of variables, the quasilinear problem was transformed to a semilinear one, an Orlicz space framework was used as the working space, and the existence of positive solutions of (SQ) was proved by the Mountain Pass theorem. The same method of change of variables was also used in [8], but the usual Sobolev space H1⁢(ℝN) framework was used as the working space for the study of a different class of nonlinearities. In [26], a new perturbation approach was used to treat a class of quasilinear Schrödinger equations which include (SQ). A new existence result was given for the critical exponent case in [26].

The following quasilinear elliptic equations have been well studied by means of different techniques (see, for example, [9, 5, 10, 31] and the references therein):

(BQ) { - ∑ i , j = 1 N D j ⁢ ( a i ⁢ j ⁢ ( x , u ) ⁢ D i ⁢ u ) + 1 2 ⁢ ∑ i , j = 1 N D s ⁢ a i ⁢ j ⁢ ( x , u ) ⁢ D i ⁢ u ⁢ D j ⁢ u = g ⁢ ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ ⁡ Ω ,

where the function ai⁢j:Ω×ℝ→ℝ usually satisfies the following hypotheses:

ai⁢j=aj⁢i for all i,j,
ai⁢j⁢(x,s) is measurable about x for all s∈ℝ,
ai⁢j⁢(x,s) is of class C1 about s for all x∈Ω,
|ai⁢j⁢(x,s)|≤C, |Ds⁢ai⁢j⁢(x,s)|≤C for a.e. x∈Ω, where C is a positive constant.

Specifically, under ellipticity and semipositivity conditions, i.e., when ν>0 exists such that

∑ i , j = 1 N a i ⁢ j ⁢ ( x , s ) ⁢ ξ i ⁢ ξ j ≥ ν ⁢ ∑ i , j = 1 N ξ i 2 , ∑ i , j = 1 N s ⁢ D s ⁢ a i ⁢ j ⁢ ( x , s ) ⁢ ξ i ⁢ ξ j ≥ 0 ,

for a.e. x∈Ω, s∈ℝ and (ξ1,…,ξN)∈ℝN, Corvellec and Degiovanni [9] proved a result of Amann–Zehnder type via non-smooth Morse theory.

In this paper, we prove the existence of multiple nontrivial solutions for the quasilinear elliptic equation

(P) { - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u = g ⁢ ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ ⁡ Ω ,

where Ω⊂ℝN is a bounded open domain with smooth boundary ∂⁡Ω, N⩾3, and g:Ω×ℝ→ℝ is a Carathéodory function that satisfies the following subcritical growth condition:

|g⁢(x,t)|≤c⁢(1+|t|q-1) for some c>0 and q∈[1,22*), where 2*=2⁢N/(N-2).

Note that there is a special case, ai⁢j⁢(x,s)=(1+2⁢s2)⁢δi⁢j, which cannot be included in the above quasilinear elliptic problems (BQ). Here, we do not use the methods mentioned above to study problem (P). Instead, we transform the quasilinear elliptic problem (P) into a semilinear elliptic problems by a change of variables, which was introduced in [22, 8]. Then we can study the semilinear elliptic problem via smooth Morse theory.

Now we state the assumptions and the main results of this paper. Near the origin, we consider the following three cases:

There exist δ>0 and ν∈(0,2) such that

(1.1) g ⁢ ( x , t ) ⁢ t > 0 for a.e. ⁢ x ∈ Ω , 0 < | t | ≤ δ ,

(1.2) ν ⁢ G ⁢ ( x , t ) - g ⁢ ( x , t ) ⁢ t ≥ 0 for a.e. ⁢ x ∈ Ω , | t | ≤ δ .
There exist k∈ℕ+,δ>0 and λk<c1<c2<λk+1 such that

c 1 ⁢ | t | 2 ≤ 2 ⁢ G ⁢ ( x , t ) ≤ c 2 ⁢ | t | 2 for ⁢ t ∈ ℝ , | t | ≤ δ ,

where λk and λk+1 are the k-th and (k+1)-th eigenvalues of the Laplacian operator -Δ.
There exist δ>0 and c3∈(0,λ1) such that

2 ⁢ G ⁢ ( x , t ) ≤ c 3 ⁢ | t | 2 for a.e. ⁢ x ∈ Ω , | t | ≤ δ .

Near infinity, we consider the following two cases:

There exist M>0 and θ>4 such that

0 < θ ⁢ G ⁢ ( x , t ) ≤ g ⁢ ( x , t ) ⁢ t for a.e. ⁢ x ∈ Ω , | t | ≥ M .
We have

lim | t | → ∞ ⁡ 4 ⁢ G ⁢ ( x , t ) | t | 4 = λ 1 uniformly for a.e. ⁢ x ∈ Ω ,

(1.3) lim | t | → ∞ ⁡ ( G ⁢ ( x , t ) - 1 4 ⁢ λ 1 ⁢ | t | 4 ) = - ∞ uniformly for a.e. ⁢ x ∈ Ω .

Our main results are the following.

Theorem 1.1

If the function g satisfies (g), (g1) and (g2’), then problem (P) has at least one nontrivial weak solutions in W01,2⁢(Ω).

Theorem 1.2

If the function g satisfies (g), (g3) and (g1’), then problem (P) has at least one nontrivial weak solutions in W01,2⁢(Ω).

Theorem 1.3

If the function g satisfies (g), (g2) and (g2’), then problem (P) has at least two nontrivial weak solutions in W01,2⁢(Ω).

Remark 1.4

In [11], do Ó, Miyagaki and Moreira studied problem (P) with g⁢(x,u)=λ⁢u, and one positive solution was got for any λ∈[λ1,Λ). Then, for g⁢(x,u)=λ¯⁢u-g~⁢(x,u), where λ¯ is the “first” eigenvalue of the nonhomogeneous operator L⁢u=-Δ⁢u-Δ⁢(u2)⁢u (in fact λ¯=λ1, see Lemma 2.2), they studied problem (P) with g~⁢(x,0)≠0 and infs∈ℝ,s≠0⁡g~⁢(x,s)/s>λ¯, and one nontrivial solution was also obtained. Compared with [11], in this paper, the nonlinearity g satisfies different conditions, i.e., g⁢(x,0)=0 and g satisfies different growth conditions. Recently, in [20, 24], Liu, Wang and Sim considered bifurcation problems for more general cases of (BQ).

In this paper, as we have mentioned, we will use a change of variables and Morse theory to prove our main existence results, so in Section 2, we collect some concepts and basic results that will be used in the sequel. In Section 3 and 4, we verify the compactness condition and compute the group Cq⁢(I,0) and Cq⁢(I,∞). The proofs of the main results will be given in Section 5.

Notation

Throughout the article the letter C will denote various positive constants whose values may change from line to line but are not essential to the analysis of the problem. The norm of W01,2⁢(Ω) and Ls⁢(Ω) will be, respectively, denoted by ∥⋅∥ and ∥⋅∥Ls⁢(Ω). We denote the weak and strong convergence in X by ⇀ and →, respectively.

2 Preliminary

In this section, we will collect some concepts and basic results that will be used in the sequel.

As observed, there are some technical difficulties in applying variational methods directly to the formal functional associated to problem (P). The main difficulty is related to the fact that it is not well defined in the usual Sobolev space. To overcome this difficulty, we employ an argument developed by Liu, Wang and Wang in [22] (see also [8]). We make the change of variables v=f-1⁢(u), where f is defined by

f ′ ⁢ ( t ) = 1 1 + 2 ⁢ f 2 ⁢ ( t ) and f ⁢ ( 0 ) = 0 on ⁢ [ 0 , + ∞ ) ,

and

f ⁢ ( t ) = - f ⁢ ( - t ) on ⁢ ( - ∞ , 0 ] .

Then, we can transform the search of solutions u=u⁢(x) of problem (P) into the search of solutions v=v⁢(x) of the semilinear equation problem

(P’) { - Δ ⁢ v = 1 1 + 2 ⁢ f 2 ⁢ ( v ) ⁢ g ⁢ ( x , f ⁢ ( v ) ) , x ∈ Ω , v = 0 , x ∈ ∂ ⁡ Ω .

We can study problem (P’) in the normal Sobolev space H01⁢(Ω)=W01,2⁢(Ω) equipped with the inner product and norm

〈 u , v 〉 = ∫ Ω ∇ u ∇ v d x and ∥ u ∥ = ( ∫ Ω | ∇ u | 2 d x ) 1 / 2 ,

respectively. Under our assumptions on g, the functional I:H01⁢(Ω)→ℝ associated to (P’), given by

I ( v ) = 1 2 ∫ Ω | ∇ v | 2 d x - ∫ Ω G ( x , f ( v ) ) d x ,

is well defined and of class C1 on H01⁢(Ω). Similarly, the eigenvalue problem

(Pμ) { - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u = μ ⁢ u , x ∈ Ω , u = 0 , x ∈ ∂ ⁡ Ω

can be transformed into the problem

(Pμ’) { - Δ ⁢ v = μ ⁢ 1 1 + 2 ⁢ f 2 ⁢ ( v ) ⁢ f ⁢ ( v ) , x ∈ Ω , v = 0 , x ∈ ∂ ⁡ Ω .

Lemma 2.1

Lemma 2.1 (see [22, 8, 12, 13])

The function f enjoys the following properties:

f is a uniquely defined C ∞ function and invertible,
|f′⁢(t)|≤1 for all t∈ℝ,
|f⁢(t)|≤|t| for all t∈ℝ,
f⁢(t)/t→1 as t→0,
f⁢(t)/t→24 as t→+∞,
f⁢(t)/2≤t⁢f′⁢(t)≤f⁢(t) for all t≥0,
|f⁢(t)|≤24⁢|t| for all t∈ℝ,
the function f 2 is strictly convex,
there exists a positive constant C such that

| f ⁢ ( t ) | ≥ { C ⁢ | t | if ⁢ | t | ≤ 1 , C ⁢ | t | if ⁢ | t | ≥ 1 ,
there exist positive constants C 1 and C 2 such that

| t | ≤ C 1 ⁢ | f ⁢ ( t ) | + C 2 ⁢ | f ⁢ ( t ) | 2 for all ⁢ t ∈ ℝ ,
|f⁢(t)⁢f′⁢(t)|≤2/2 for all t∈ℝ,
for each λ>1, we have f2⁢(λ⁢t)≤λ2⁢f2⁢(t) for all t∈ℝ,
for each λ<1, we have f2⁢(λ⁢t)≥λ2⁢f2⁢(t) for all t∈ℝ.

Lemma 2.2

The “first” eigenvalue of the nonhomogeneous operator L⁢u=-Δ⁢u-Δ⁢(u2)⁢u is equal to the first eigenvalue of the Laplacian operator -Δ, that is to say

μ 1 := inf 0 ≠ v ∈ H 0 1 ⁢ ( Ω ) ⁡ ∫ Ω | ∇ v | 2 d x ∫ Ω | f ( v ) | 2 d x = inf 0 ≠ v ∈ H 0 1 ⁢ ( Ω ) ⁡ ∫ Ω | ∇ v | 2 d x ∫ Ω | v | 2 d x = λ 1 .

Proof.

On the one hand, by Lemma 2.1 (3), we can easily get that

μ 1 = inf 0 ≠ v ∈ H 0 1 ⁢ ( Ω ) ⁡ ∫ Ω | ∇ v | 2 d x ∫ Ω | f ( v ) | 2 d x ≥ inf 0 ≠ v ∈ H 0 1 ⁢ ( Ω ) ⁡ ∫ Ω | ∇ v | 2 d x ∫ Ω | v | 2 d x = λ 1 .

On the other hand, in order to prove μ1≤λ1, we need to prove that there exists a nonnegative function ϕ1∈H01⁢(Ω) which is a solution for problem (Pλ1′) (the proof of this fact is given in [11], and we will provide a sketch of the proof for the reader’s convenience).

The first eigenfunction φ1 associated to the eigenvalue λ1 of the Laplacian operator is a supersolution of (Pλ1′). In fact,

∫ Ω ∇ ⁡ φ 1 ⁢ ∇ ⁡ ϕ ⁢ d ⁢ x - λ 1 ⁢ ∫ Ω f ⁢ ( φ 1 ) ⁢ f ′ ⁢ ( φ 1 ) ⁢ ϕ ⁢ 𝑑 x = λ 1 ⁢ ∫ Ω φ 1 ⁢ ϕ ⁢ 𝑑 x - λ 1 ⁢ ∫ Ω f ⁢ ( φ 1 ) ⁢ f ′ ⁢ ( φ 1 ) ⁢ ϕ ⁢ 𝑑 x

= λ 1 ⁢ ∫ Ω ( 1 - f ⁢ ( φ 1 ) φ 1 ⁢ f ′ ⁢ ( φ 1 ) ) ⁢ φ 1 ⁢ ϕ ⁢ 𝑑 x

≥ 0 ,

where we have used the fact that f⁢(φ1)⁢f′⁢(φ1)≤φ1 and φ1>0, ϕ≥0 in Ω.

We will prove that there exists a nonnegative function which is a subsolution of (Pλ1′). Since φ1>0 in Ω, there exists R>0 and c1>0 such that φ1>c1 in B2⁢R(0)⊂⊂Ω. Define v⁢(x)=ϵα⁢u⁢(ϵ⁢x), ϵ>0, α>2, where u is a solution of the problem

{ - Δ ⁢ u = 1 in ⁢ B ϵ ⁢ R ⁢ ( 0 ) ⊂ Ω ⊂ ℝ N , u = 0 on ⁢ ∂ ⁡ B ϵ ⁢ R ⁢ ( 0 ) .

Thus, v satisfies

{ - Δ ⁢ v = ϵ α + 2 in ⁢ B R ⁢ ( 0 ) ⊂ Ω ⊂ ℝ N , v = 0 on ⁢ ∂ ⁡ B R ⁢ ( 0 ) ,

and by the maximum principle we have v>0. The function

w = { v in ⁢ B R ⁢ ( 0 ) ⊂ Ω , 0 in ⁢ Ω ∖ B R ⁢ ( 0 ) ,

is a subsolution for (Pλ1′) and, by construction, we can see that w<φ1. In fact, assuming that ψ∈C0∞⁢(Ω) with ψ≥0, we need to consider two cases: If supp⁡ψ∩BR⁢(0)=∅, then we have

∫ B R ⁢ ( 0 ) ∇ ⁡ v ⁢ ∇ ⁡ ψ ⁢ d ⁢ x - λ 1 ⁢ ∫ B R ⁢ ( 0 ) f ⁢ ( v ) ⁢ f ′ ⁢ ( v ) ⁢ ψ ⁢ 𝑑 x = 0 .

If supp⁡ψ∩BR⁢(0)≠∅, then we have

∫ B R ⁢ ( 0 ) ∇ ⁡ v ⁢ ∇ ⁡ ψ ⁢ d ⁢ x - λ 1 ⁢ ∫ B R ⁢ ( 0 ) f ⁢ ( v ) ⁢ f ′ ⁢ ( v ) ⁢ ψ ⁢ 𝑑 x = ϵ α + 2 ⁢ ∫ B R ⁢ ( 0 ) ψ ⁢ 𝑑 x - λ 1 ⁢ ∫ B R ⁢ ( 0 ) f ⁢ ( v ) ⁢ f ′ ⁢ ( v ) ⁢ ψ ⁢ 𝑑 x

= ϵ α + 2 ⁢ ∫ B R ⁢ ( 0 ) ψ ⁢ 𝑑 x - λ 1 ⁢ ∫ B R ( 0 ) ∩ { v > 1 } f ⁢ ( v ) ⁢ f ′ ⁢ ( v ) ⁢ ψ ⁢ 𝑑 x - λ 1 ⁢ ∫ B R ( 0 ) ∩ { v ≤ 1 } f ⁢ ( v ) ⁢ f ′ ⁢ ( v ) ⁢ ψ ⁢ 𝑑 x

≤ ϵ α + 2 ⁢ ∫ B R ⁢ ( 0 ) ψ ⁢ 𝑑 x - λ 1 ⁢ C 2 ⁢ ( ∫ B R ( 0 ) ∩ { v > 1 } ψ ⁢ 𝑑 x + ∫ B R ( 0 ) ∩ { v ≤ 1 } v ⁢ ψ ⁢ 𝑑 x )

(2.1) ≤ ϵ α + 2 ⁢ ∫ B R ⁢ ( 0 ) ψ ⁢ ( x ) ⁢ 𝑑 x - λ 1 ⁢ C 2 ⁢ ∫ B R ( 0 ) ∩ { v ≤ 1 } v ⁢ ( x ) ⁢ ψ ⁢ ( x ) ⁢ 𝑑 x ,

where C is a positive constant independent of ϵ.

Notice that for ϵ>0 small enough, we can consider

(2.2) ∫ B R ( 0 ) ∩ { v ≤ 1 } ψ ⁢ ( x ) ⁢ 𝑑 x = ∫ B R ⁢ ( 0 ) χ { v ≤ 1 } ⁢ ψ ⁢ ( x ) ⁢ 𝑑 x = ∫ B R ⁢ ( 0 ) ψ ⁢ ( x ) ⁢ 𝑑 x .

In fact, we can extend u by setting it zero outside the ball Bϵ⁢R⁢(0), which is C1,γ⁢(B2⁢R⁢(0)), γ∈(0,1). By the definition of v, for R fixed, we can choose ϵ, sufficiently small, such that

| v ( x ) | = ϵ α | u ( ϵ x ) | ≤ ϵ α max B 2 ⁢ R | u ( y ) | = C R ϵ α ≤ 1 .

Therefore, (2.1) and (2.2) imply that

∫ B R ⁢ ( 0 ) ∇ ⁡ v ⁢ ∇ ⁡ ψ ⁢ d ⁢ x - λ 1 ⁢ ∫ B ϵ ⁢ R ⁢ ( 0 ) f ⁢ ( v ) ⁢ f ′ ⁢ ( v ) ⁢ ψ ⁢ 𝑑 x ≤ ϵ α + 2 ⁢ ∫ B R ⁢ ( 0 ) ψ ⁢ ( x ) ⁢ 𝑑 x - λ 1 ⁢ C 2 ⁢ ∫ B R ( 0 ) ∩ { v ≤ 1 } v ⁢ ( x ) ⁢ ψ ⁢ ( x ) ⁢ 𝑑 x

≤ ϵ α + 2 ⁢ ∫ B R ⁢ ( 0 ) ψ ⁢ ( x ) ⁢ 𝑑 x - λ 1 ⁢ C ′ 2 ⁢ ϵ α ⁢ ∫ B R ⁢ ( 0 ) ψ ⁢ ( x ) ⁢ 𝑑 x

< 0 .

Then, we can get

μ 1 ≤ ∫ Ω | ∇ ϕ 1 | 2 d x ∫ Ω | f ( ϕ 1 ) | 2 d x = λ 1 ⁢ ∫ Ω f ⁢ ( ϕ 1 ) ⁢ f ′ ⁢ ( ϕ 1 ) ⁢ ϕ 1 ⁢ 𝑑 x ∫ Ω | f ( ϕ 1 ) | 2 d x ≤ λ 1 ⁢ ∫ Ω | f ( ϕ 1 ) | 2 d x ∫ Ω | f ( ϕ 1 ) | 2 d x = λ 1 .

Thus, μ1=λ1, i.e., the first eigenvalue of (Pμ′) is also the first eigenvalue of the Laplacian operator. ∎

In order to use the Morse theory to solve problem (P), we will list some concepts and results on Morse theory.

Let X be a real Banach space, I∈C1⁢(X,ℝ) and 𝒦={u∈X:I′⁢(u)=0}. Let also u∈𝒦 be an isolated critical point of I with I⁢(u)=c∈ℝ, and U be a neighborhood of u containing the unique critical point u. The group

C q ⁢ ( I , u ) := H q ⁢ ( I c ∩ U , I c ∩ U ∖ { u } ) , q ∈ ℤ ,

is called the q-th critical group of I at u, where Ic={u∈X:I⁢(u)≤c} and Hq⁢(⋅,⋅) denotes the q-th singular relative homology group with integer coefficients. For a<inf⁡I⁢(𝒦), the group

C q ⁢ ( I , ∞ ) := H q ⁢ ( X , I a ) , q ∈ ℤ ,

is called the critical group of I at infinity (cf. [6, 28, 1]).

In applications, we require the functional I to satisfy the following compactness condition.

Definition 2.3

The functional I satisfies the Palais–Smale condition at the level c∈ℝ ((PS)c in short) if any sequence {un}⊂X satisfying I⁢(un)→c, I′⁢(un)→0 in X* (the dual space of X) as n→∞, has a convergent subsequence. The functional I satisfies the (PS) condition if it satisfies the (PS)c condition for all c∈ℝ.

Definition 2.4

Definition 2.4 ([18])

Assume that I∈C1⁢(X,ℝ) has a critical point u=0 with I⁢(0)=0. We say that I has a local linking at 0 with respect to a direct sum decomposition X=X-⊕X+, κ=dim⁡X-<∞, if there exists small r>0 such that

I ⁢ ( u ) > 0 for ⁢ u ∈ X + ⁢ with ⁢ 0 < ∥ u ∥ ≤ r , I ⁢ ( u ) ≤ 0 for ⁢ u ∈ X - ⁢ with ⁢ ∥ u ∥ ≤ r .

Lemma 2.5

Lemma 2.5 ([19, 21])

Assume that I∈C1⁢(X,R) has a critical point u=0 with I⁢(0)=0. If I has a local linking at 0 with respect to a direct sum decomposition X=X-⊕X+, κ=dim⁡X-<∞, then Cκ⁢(I,0)≇0; that is, 0 is an homological nontrivial critical point of I.

Lemma 2.6

Lemma 2.6 ([21])

Let X be a real Banach space and let I∈C1⁢(X,R) satisfy the (PS) condition and be bounded from below. If I has a critical point that is homological nontrivial and is not the minimizer of I, then I has at least three critical points.

We refer the reader to [6, 28, 1] and the references therein for more information about Morse theory.

3 Critical Groups at Zero

In this section, we compute the critical groups of I at zero.

First of all, we verify the compactness condition for the functional I.

Lemma 3.1

If g satisfies (g) and (g1), then we have

(3.1) C q ⁢ ( I , 0 ) ≅ 0 for all ⁢ q ∈ ℤ .

Proof.

By definition, we write

C q ⁢ ( I , 0 ) := H q ⁢ ( B ρ ⁢ ( 0 ) ∩ I 0 , B ρ ⁢ ( 0 ) ∩ I 0 ∖ { 0 } ) , where B ρ ⁢ ( 0 ) = { u ∈ W 0 1 , 2 ⁢ ( Ω ) : ∥ u ∥ ≤ ρ }

and ρ>0 is to be chosen later. We will prove (3.1) by constructing a deformation mapping for the topological pairs (Bρ⁢(0),Bρ⁢(0)∖{0}) and (I0∩Bρ⁢(0),I0∩Bρ⁢(0)∖{0}). For this purpose, we need to analyze the local properties of I near zero.

A direct calculation, using (1.1) and (1.2), shows that there exists a constant C>0 such that

(3.2) G ⁢ ( x , t ) ≥ C ⁢ | t | ν for ⁢ a . e . x ∈ Ω , | t | ≤ δ .

From (g) and (3.2), it follows that

(3.3) G ⁢ ( x , t ) ≥ C ⁢ | t | ν - C ⁢ | t | q for a.e. ⁢ x ∈ Ω , t ∈ ℝ

for some q∈(4,22*) and C>0. Hence, for v∈W01,2⁢(Ω) and 0<s<1, by (3), (7) and (13) of Lemma 2.1, we have

I ⁢ ( s ⁢ v ) = s 2 2 ⁢ ∫ Ω | ∇ ⁡ v | 2 ⁢ 𝑑 x - ∫ Ω G ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ 𝑑 x

≤ s 2 2 ⁢ ∥ v ∥ 2 - ∫ Ω [ C ⁢ | f ⁢ ( s ⁢ v ) | ν - C ⁢ | f ⁢ ( s ⁢ v ) | q ] ⁢ 𝑑 x

≤ s 2 2 ⁢ ∥ v ∥ 2 - C ⁢ s ν ⁢ ∥ f ⁢ ( v ) ∥ L ν ⁢ ( Ω ) ν + C ⁢ s q / 2 ⁢ ∥ v ∥ L q / 2 ⁢ ( Ω ) q / 2 .

Since ν<2<q/2, for given v∈W01,2⁢(Ω) with v≠0, there exists s0=s0⁢(v)>0 such that

(3.4) I ⁢ ( s ⁢ v ) < 0 , s ∈ ( 0 , s 0 ) .

Let v∈W01,2⁢(Ω) be such that I⁢(v)=0. Then, by (g), (g1) and Lemma 2.1 (6), we have

d d ⁢ s ⁢ I ⁢ ( s ⁢ v ) | s = 1 = 〈 I ′ ⁢ ( s ⁢ v ) , v 〉 | s = 1

= ∫ Ω | ∇ v | 2 d x - ∫ Ω g ( x , f ( v ) ) f ′ ( v ) v d x

= ( 1 - ν 2 ) ⁢ ∥ v ∥ 2 + ∫ Ω [ ν ⁢ G ⁢ ( x , f ⁢ ( v ) ) - g ⁢ ( x , f ⁢ ( v ) ) ⁢ f ′ ⁢ ( v ) ⁢ v ] ⁢ 𝑑 x

≥ ( 1 - ν 2 ) ⁢ ∥ v ∥ 2 + ∫ { | f | < δ } [ ν ⁢ G ⁢ ( x , f ⁢ ( v ) ) - g ⁢ ( x , f ⁢ ( v ) ) ⁢ f ⁢ ( v ) ] ⁢ 𝑑 x - ∫ { | f | ≥ δ } | ν ⁢ G ⁢ ( x , f ⁢ ( v ) ) - g ⁢ ( x , f ⁢ ( v ) ) ⁢ f ′ ⁢ ( v ) ⁢ v | ⁢ 𝑑 x

≥ ( 1 - ν 2 ) ∥ v ∥ 2 - C ∫ Ω | f ( v ) | q d x (for some q ∈ ( 4 , 22 * ) )

≥ ( 1 - ν 2 ) ∥ v ∥ 2 - C ∫ Ω | v | q / 2 d x (by Lemma 2.1 (7))

≥ ( 1 - ν 2 ) ⁢ ∥ v ∥ 2 - C ⁢ ∥ v ∥ q / 2 .

Then there exists ρ>0 such that

(3.5) d d ⁢ s ⁢ I ⁢ ( s ⁢ v ) | s = 1 > 0 for ⁢ v ∈ W 0 1 , 2 ⁢ ( Ω ) ⁢ with ⁢ I ⁢ ( v ) = 0 ⁢ and ⁢ 0 < ∥ v ∥ ≤ ρ .

Now we fix ρ>0. Then from (3.5) it follows that

(3.6) I ⁢ ( s ⁢ v ) ⁢ < 0 for ⁢ s ∈ ( 0 , 1 ) , u ∈ W 0 1 , 2 ⁢ ( Ω ) ⁢ with ⁢ I ⁢ ( v ) ⁢ < 0 ⁢ and ∥ ⁢ v ∥ ≤ ρ .

In fact, if ∥v∥≤ρ and I⁢(v)<0, then there exists τ∈(0,1) such that I⁢(s⁢v)<0 for all s∈(1-τ,1), by the continuity of I. Suppose that there exists some s0∈(0,1-τ] such that I⁢(s0⁢v)=0 and I⁢(s⁢v)<0 for s0<s<1. Denote v0=s0⁢v. Then by (3.5) we have dd⁢s⁢I⁢(s⁢v0)|s=1>0, but I⁢(s0⁢v)-I⁢(s⁢v)>0 implies that

d d ⁢ s ⁢ I ⁢ ( s ⁢ v 0 ) | s = 1 = s 0 ⁢ d d ⁢ s ⁢ I ⁢ ( s ⁢ v ) | s = s 0 ≤ 0 .

This contradiction shows that (3.6) holds.

Now we define a mapping T:Bρ⁢(0)→[0,1] as follows:

T ⁢ ( v ) = { 1 for ⁢ v ∈ B ρ ⁢ ( 0 ) ⁢ with ⁢ I ⁢ ( v ) ≤ 0 , s for ⁢ v ∈ B ρ ⁢ ( 0 ) ⁢ with ⁢ I ⁢ ( v ) > 0 , I ⁢ ( s ⁢ v ) = 0 , s < 1 .

By (3.4), (3.5) and (3.6), the mapping T is well defined and if I⁢(v)>0, then there exists a unique T⁢(v)∈(0,1) such that

(3.7) I ⁢ ( T ⁢ ( v ) ⁢ v ) = 0 , I ⁢ ( s ⁢ v ) < 0 for ⁢ s ∈ ( 0 , T ⁢ ( v ) ) , I ⁢ ( s ⁢ v ) > 0 for ⁢ s ∈ ( T ⁢ ( v ) , 1 ) .

From (3.5), (3.7) and the implicit function theorem, it follows that the mapping T is continuous in v. Define a mapping η:[0,1]×Bρ⁢(0)→Bρ⁢(0) by

η ⁢ ( s , v ) = ( 1 - s ) ⁢ v + s ⁢ T ⁢ ( v ) ⁢ v , s ∈ [ 0 , 1 ] , v ∈ B ρ ⁢ ( 0 ) .

It is easy to see that η is a continuous deformation from (Bρ⁢(0),Bρ⁢(0)∖{0}) to (I0∩Bρ⁢(0),I0∩(Bρ⁢(0)∖{0})). By the homotopy invariance of the homology group, we have

C q ⁢ ( I , 0 ) := H q ⁢ ( I 0 ∩ B ρ ⁢ ( 0 ) , I 0 ∩ B ρ ⁢ ( 0 ) ∖ { 0 } ) ≅ H q ⁢ ( B ρ ⁢ ( 0 ) , B ρ ⁢ ( 0 ) ∖ { 0 } ) ≅ 0 , q ∈ ℤ ,

since Bρ⁢(0)∖{0} is contractible. ∎

Lemma 3.2

If g satisfies (g2), then I has the local linking at the origin with respect to W01,2⁢(Ω)=V⊕W, where V=⊕j=1kKer⁡(-Δ-λj) and Cκ⁢(I,0)≠0, κ=dim⁡V.

Proof.

First, I has a local linking at the origin.

Take v∈V. It is easily seen that ∥v∥≤ρ⇒|v⁢(x)|≤δ, x∈Ω for ρ>0 small. So from (g2) it follows that for ∥v∥≤ρ,

I ( v ) = 1 2 ∫ Ω | ∇ v | 2 d x - ∫ Ω G ( x , f ( v ) ) d x

≤ λ k 2 ∫ Ω | v | 2 d x - ∫ Ω G ( x , f ( v ) ) d x

≤ λ k 2 ∫ Ω | v | 2 d x - c 1 2 ∫ Ω | f ( v ) | 2 d x

Since

lim t → 0 ⁡ f ⁢ ( t ) ⁢ t t 2 = 1 > λ k c 1 ,

we have

f ⁢ ( t ) ⁢ t ≥ λ k c 1 ⁢ t 2 for ⁢ | t | ⁢ small enough.

Thus, from (3.3) it follows that

I ( v ) ≤ λ k 2 ∫ Ω | v | 2 d x - c 1 2 λ k c 1 ∫ Ω | v | 2 d x ≤ 0 for ∥ v ∥ ≤ ρ .

Taking w∈W, we have

∫ Ω | ∇ w | 2 d x ≥ λ k + 1 ∫ Ω | w | 2 d x ,

and then

I ( w ) = 1 2 ∫ Ω | ∇ w | 2 d x - ∫ Ω G ( x , f ( w ) ) d x

= 1 2 ⁢ ∫ Ω ( | ∇ ⁡ w | 2 - c 2 ⁢ | f ⁢ ( w ) | 2 ) ⁢ 𝑑 x - ∫ Ω ( G ⁢ ( x , f ⁢ ( w ) ) - c 2 2 ⁢ | f ⁢ ( w ) | 2 ) ⁢ 𝑑 x

≥ 1 2 ∫ Ω ( | ∇ w | 2 - c 2 | w | 2 ) d x - C ∫ Ω | w | s d x (for some s ∈ ( 2 , 2 * ) )

≥ 1 2 ⁢ ( 1 - c 2 λ k + 1 ) ⁢ ∥ w ∥ 2 - C ⁢ ∥ w ∥ s .

So, when w∈W and 0<∥w∥≤ρ for ρ>0 small, it follows that I⁢(w)>0.

Finally, by Lemma 2.5, we can get that Cκ⁢(I,0)≠0, κ=dim⁡V. ∎

Lemma 3.3

If g satisfies (g3), then Cq⁢(I,0)≅δq,0⁢Z for all q∈Z.

Proof.

For any v∈W01,2⁢(Ω), it follows from (g3) that

I ( v ) = 1 2 ∫ Ω | ∇ v | 2 d x - ∫ Ω G ( x , f ( v ) ) d x

= 1 2 ⁢ ∥ v ∥ 2 - ∫ { | f ( v ) | < δ } G ⁢ ( x , f ⁢ ( v ) ) ⁢ 𝑑 x - ∫ { | f ( v ) | ≥ δ } G ⁢ ( x , f ⁢ ( v ) ) ⁢ 𝑑 x

≥ 1 2 ∥ v ∥ 2 - c 3 2 ∫ Ω | v | 2 d x - C ∫ Ω | v | s d x (for some s ∈ ( 2 , 2 * ) )

≥ 1 2 ⁢ ( 1 - c 3 λ 1 ) ⁢ ∥ v ∥ 2 - C ⁢ ∥ v ∥ s .

Hence, v=0 is a local minimizer of I and so Cq⁢(I,0)≅δq,0⁢ℤ for all q∈ℤ. ∎

4 Compactness and Critical Groups at Infinity

In this section, we verify the compactness condition and compute Cq⁢(I,∞).

Since the function g satisfies the subcritical growth condition (g), a standard argument shows the following lemma.

Lemma 4.1

Let g satisfy (g). Then any bounded sequence {vn}⊂W01,2⁢(Ω) such that I′⁢(vn)→0 in (W01,2⁢(Ω))* as n→∞ has a convergent subsequence.

Proof.

Let {vn} be a bounded sequence satisfying I′⁢(vn)→0⁢in⁢(W01,2⁢(Ω))* as n→∞. Up to a subsequence, we may assume that there is some v0∈W01,2⁢(Ω) such that

v n ⇀ v 0 in ⁢ W 0 1 , 2 ⁢ ( Ω ) ⁢ as ⁢ n → ∞ , v n → v 0 in ⁢ L p ⁢ ( Ω ) , p ∈ ( 1 , 2 * ) ⁢ , as ⁢ n → ∞ .

Since I′⁢(vn)→0 in (W01,2⁢(Ω))*, by Lemma 2.1, we have

I ′ ⁢ ( v n ) ⁢ ( v n - v ) → 0 , I ′ ⁢ ( v 0 ) ⁢ ( v n - v 0 ) → 0 as ⁢ n → ∞

and

| ∫ Ω ( f ( v n ) f ′ ( v n ) - f ( v 0 ) f ′ ( v 0 ) ) ( v n - v 0 ) d x | ≤ ∫ Ω | ( f ( v n ) f ′ ( v n ) - f ( v 0 ) f ′ ( v 0 ) ) ( v n - v 0 ) | d x

≤ ∫ Ω 2 ⁢ | v n - v 0 | ⁢ 𝑑 x (by Lemma 2.1 (11))

≤ C ( ∫ Ω | v n - v 0 | 2 d x ) 1 / 2 → 0 as n → ∞ .

Therefore,

∥ v n - v 0 ∥ 2 = I ′ ⁢ ( v n ) ⁢ ( v n - v ) - I ′ ⁢ ( v 0 ) ⁢ ( v n - v 0 ) + ∫ Ω ( f ⁢ ( v n ) ⁢ f ′ ⁢ ( v n ) - f ⁢ ( v 0 ) ⁢ f ′ ⁢ ( v 0 ) ) ⁢ ( v n - v 0 ) ⁢ 𝑑 x → 0 as ⁢ n → ∞ . ∎

Thus, in order to verify the compactness of the functional I, we just need to prove that all the (PS) sequences for I are bounded.

Lemma 4.2

If g satisfies (g) and (g1’) , then all the (PS) sequences for I are bounded.
If g satisfies (g) and (g2’) , then the functional I is coercive.

Proof.

(i) Let {vn}⊂W01,2⁢(Ω) such that

I ⁢ ( v n ) → c , I ′ ⁢ ( v n ) → 0 in ⁢ ( W 0 1 , 2 ⁢ ( Ω ) ) * ⁢ as ⁢ n → ∞ .

By (g), (g1’) and Lemma 2.1 (6), we have

θ 2 ⁢ c + o ⁢ ( 1 ) + o ⁢ ( ∥ v n ∥ ) = θ 2 ⁢ I ⁢ ( v n ) - 〈 I ′ ⁢ ( v n ) , v n 〉

≥ θ 4 ⁢ ∥ v n ∥ 2 - θ 2 ⁢ ∫ Ω G ⁢ ( x , f ⁢ ( v n ) ) ⁢ 𝑑 x - ( ∥ v n ∥ 2 - ∫ Ω g ⁢ ( x , f ⁢ ( v n ) ) ⁢ f ′ ⁢ ( v n ) ⁢ v n ⁢ 𝑑 x )

≥ ( θ 4 - 1 ) ⁢ ∥ v n ∥ 2 + ∫ Ω ( g ⁢ ( x , f ⁢ ( v n ) ) ⁢ f ′ ⁢ ( v n ) ⁢ v n - θ 2 ⁢ G ⁢ ( x , f ⁢ ( v n ) ) ) ⁢ 𝑑 x

= ( θ 4 - 1 ) ⁢ ∥ v n ∥ 2 + ∫ { | f ( v n ) | > M } ( g ⁢ ( x , f ⁢ ( v n ) ) ⁢ f ′ ⁢ ( v n ) ⁢ v n - θ 2 ⁢ G ⁢ ( x , f ⁢ ( v n ) ) ) ⁢ 𝑑 x

+ ∫ { | f ( v n ) | ≤ M } ( g ⁢ ( x , f ⁢ ( v n ) ) ⁢ f ′ ⁢ ( v n ) ⁢ v n - θ 2 ⁢ G ⁢ ( x , f ⁢ ( v n ) ) ) ⁢ 𝑑 x

≥ ( θ 4 - 1 ) ⁢ ∥ v n ∥ 2 + ∫ { | f ( v n ) | > M } ( 1 2 ⁢ g ⁢ ( x , f ⁢ ( v n ) ) ⁢ f ⁢ ( v n ) - θ 2 ⁢ G ⁢ ( x , f ⁢ ( v n ) ) ) ⁢ 𝑑 x - C

≥ ( θ 4 - 1 ) ⁢ ∥ v n ∥ 2 - C .

Then {vn} is bounded in W01,2⁢(Ω). Furthermore, the (PS) condition follows from Lemma 4.1.

(ii) We will show that under conditions (g) and (g2’), I is coercive on W01,2⁢(Ω), i.e., I⁢(v)→+∞ as ∥v∥→∞. Hence, the (PS) sequence of I must be bounded. Denote

H ⁢ ( x , t ) = G ⁢ ( x , t ) - 1 4 ⁢ λ 1 ⁢ | t | p .

Then (1.3) implies

(4.1) lim | t | → ∞ ⁡ H ⁢ ( x , t ) = - ∞ uniformly for a.e. ⁢ x ∈ Ω .

Rewrite I as

I ( v ) = 1 2 ∫ Ω | ∇ v | 2 d x - λ 1 4 ∫ Ω | f ( v ) | 4 d x - ∫ Ω H ( x , f ( v ) ) d x .

Assume that I is not coercive on W01,2⁢(Ω). Then there exists a sequence {vn}∈W01,2⁢(Ω) such that

(4.2) ∥ v n ∥ → ∞ as ⁢ n → ∞ , but I ⁢ ( v n ) ≤ C ^ for some ⁢ C ^ ∈ ℝ .

Denote

w n = v n ∥ v n ∥ , n ∈ ℕ .

Then ∥wn∥=1. We may assume that there exists w0∈W01,2⁢(Ω) such that as n→∞,

(4.3) { w n ⇀ w 0 in ⁢ W 0 1 , 2 ⁢ ( Ω ) , w n → w 0 in ⁢ L 2 ⁢ ( Ω ) , w n ⁢ ( x ) → w 0 ⁢ ( x ) for a.e. ⁢ x ∈ Ω .

Now, using (4.1) and (4.2), we deduce

C ^ ∥ v n ∥ 2 ≥ I ⁢ ( v n ) ∥ v n ∥ 2 ≥ 1 2 - 1 4 ⁢ λ 1 ⁢ ∫ Ω | f ⁢ ( v n ) | 4 ∥ v n ∥ 2 ⁢ 𝑑 x - ∫ Ω H ⁢ ( x , f ⁢ ( v n ) ) ∥ v n ∥ 2 ⁢ 𝑑 x

(4.4) ≥ 1 2 - 1 2 λ 1 ∫ Ω | w n | 2 d x - C ∥ v n ∥ 2

for some C>0. From (4.3) and (4.4), it follows that

(4.5) lim sup n → ∞ ∫ Ω | ∇ w n | 2 d x ≤ λ 1 ∫ Ω | w 0 | 2 d x .

On the other hand, using the Poincaré inequality and the weak lower semicontinuity of the norm, we have that

(4.6) λ 1 ∫ Ω | w 0 | 2 d x ≤ ∫ Ω | ∇ w 0 | 2 d x ≤ lim inf n → ∞ ∫ Ω | ∇ w n | 2 d x .

From (4.5), (4.6) and the uniform convexity of W01,2⁢(Ω), we have

w n → w 0 as ⁢ n → ∞ ⁢ in ⁢ W 0 1 , 2 ⁢ ( Ω )

and

∫ Ω | ∇ w 0 | 2 d x = λ 1 ∫ Ω | w 0 | p d x .

Hence, ∥w0∥=1, and so w0=±φ1. Take w0=φ1. Then vn⁢(x)→+∞ a.e. in Ω. The Fatou lemma and (4.1) imply that

C ^ ≥ I ⁢ ( v n ) ≥ - ∫ Ω H ⁢ ( x , f ⁢ ( v n ) ) ⁢ 𝑑 x → + ∞ as n → ∞ .

This contradiction shows that I is coercive on W01,2⁢(Ω). Then all the (PS) sequences for I are bounded. Furthermore, the (PS) condition follows from Lemma 4.1. ∎

Finally, we will compute the critical group Cq⁢(I,∞) under conditions (g1’) and (g2’).

Lemma 4.3

If g satisfies (g) and (g1’), then Cq⁢(I,∞)≅0 for all q∈ℤ.
If g satisfies (g) and (g2’), then Cq⁢(I,∞)≅δq,0⁢ℤ for all q∈ℤ.

Proof.

(i) Let S∞ be the unit sphere in W01,2⁢(Ω). By (g1’) and Lemma 2.1 (9), for any v∈S∞, we have that

I ( s v ) = 1 2 ∫ Ω | ∇ ( s v ) | 2 d x - ∫ Ω G ( x , f ( s v ) ) d x

≤ 1 2 s 2 ∥ v ∥ 2 - C ∫ Ω | f ( s v ) | θ d x + C

≤ 1 2 s 2 ∥ v ∥ 2 - C ∫ { | s v | > 1 } | s v | θ / 2 d x - C ∫ { | s v | ≤ 1 } | s v | θ d x + C

≤ 1 2 s 2 ∥ v ∥ 2 - C ∫ Ω | s v | θ / 2 d x + C

≤ 1 2 ⁢ s 2 ⁢ ∥ v ∥ 2 - C ⁢ s θ / 2 ⁢ ∥ v ∥ θ / 2 + C → - ∞ as ⁢ s → ∞ .

Now using (g1’) again, we get that there exists a constant A>0 such that for any a>A, if I⁢(s⁢v)≤-a for some s>0 and v∈S∞, then dd⁢s⁢I⁢(s⁢v)<0. In fact,

d d ⁢ s ⁢ I ⁢ ( s ⁢ v ) = 〈 I ′ ⁢ ( s ⁢ v ) , v 〉

= ∫ Ω ∇ ⁡ ( s ⁢ v ) ⁢ ∇ ⁡ v ⁢ d ⁢ x - ∫ Ω g ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ f ′ ⁢ ( s ⁢ v ) ⁢ v ⁢ 𝑑 x

= 1 s ⁢ ( ∥ s ⁢ v ∥ 2 - ∫ Ω g ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ f ′ ⁢ ( s ⁢ v ) ⁢ s ⁢ v ⁢ 𝑑 x )

= 1 s ⁢ ( ∥ s ⁢ v ∥ 2 - ∫ Ω g ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ f ′ ⁢ ( s ⁢ v ) ⁢ s ⁢ v ⁢ 𝑑 x - 2 ⁢ ∫ Ω G ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ 𝑑 x + 2 ⁢ ∫ Ω G ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ 𝑑 x )

= 1 s ⁢ [ ( ∥ s ⁢ v ∥ 2 - 2 ⁢ ∫ Ω G ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ 𝑑 x ) + ( 2 ⁢ ∫ Ω G ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ 𝑑 x - ∫ Ω g ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ f ′ ⁢ ( s ⁢ v ) ⁢ s ⁢ v ⁢ 𝑑 x ) ]

≤ 1 s [ ( ∥ s v ∥ 2 - 2 ∫ Ω G ( x , f ( s v ) ) d x )

+ ( 2 ∫ { | f ( s v ) | ≥ M } G ( x , f ( s v ) ) d x - 1 2 ∫ { | f ( s v ) | ≥ M } g ( x , f ( s v ) ) f ( s v ) d x + C ( M , q , Ω ) ) ]

≤ 1 s ⁢ [ ( ∥ s ⁢ v ∥ 2 - 2 ⁢ ∫ Ω G ⁢ ( x , f ⁢ ( s ⁢ v ) ) ⁢ 𝑑 x ) + C ⁢ ( M , q , Ω ) ]

= 1 s ⁢ [ 2 ⁢ I ⁢ ( s ⁢ v ) + C ⁢ ( M , q , Ω ) ] .

Therefore, for any fixed a>A:=C⁢(M,q,Ω)/2, we get that

I ⁢ ( s ⁢ v ) ≤ - a ⟹ d d ⁢ s ⁢ I ⁢ ( s ⁢ v ) < 0 .

Hence, for any a>A, there exists a unique Y:Y⁢(v)>0 such that

(4.7) I ⁢ ( Y ⁢ ( v ) ⁢ v ) = - a for ⁢ v ∈ S ∞ .

By (4.7) and the implicit function theorem, Y is a continuous function from S∞ to ℝ. Now define

h ⁢ ( v ) = { 1 if ⁢ I ⁢ ( v ) < - a , 1 ∥ v ∥ ⁢ Y ⁢ ( v ∥ v ∥ ) if ⁢ I ⁢ ( v ) > - a , v ≠ 0 .

Then h∈C⁢(W01,2⁢(Ω),ℝ). Define the map

ω ⁢ ( s , v ) = ( 1 - s ) ⁢ v + s ⁢ h ⁢ ( v ) ⁢ v .

Clearly, ω is continuous, and for all v∈W01,2⁢(Ω)∖{0} with I⁢(v)>-a, by (4.7), I⁢(ω⁢(1,v))=I⁢(h⁢(v)⁢v)=-a. Therefore, ω⁢(1,v)∈I-a for v∈W01,2⁢(Ω)∖{0} and ω⁢(s,v)=v for s∈[0,1], v∈I-a. So I-a is a strong deformation retract of W01,2⁢(Ω)∖{0}. Hence,

C q ⁢ ( I , ∞ ) = H q ⁢ ( W 0 1 , 2 ⁢ ( Ω ) , I - a ) ≅ H q ⁢ ( W 0 1 , 2 ⁢ ( Ω ) , W 0 1 , 2 ⁢ ( Ω ) ∖ { 0 } ) ≅ H q ⁢ ( B ∞ , S ∞ ) ≅ 0 , q ∈ ℤ .

(ii) Since, by Lemma 4.2 (ii), we know that I is coercive and continuous, I is bounded from below. Thus, Cq⁢(I,∞)≅δq,0⁢ℤ for all q∈ℤ. ∎

5 Proofs of the Main Results

Proof of Theorem 1.1.

By Lemma 4.3 (ii), we have that Cq⁢(I,∞)≅δq,0⁢ℤ, q∈ℤ, so I has a critical point v* with Cq⁢(I,v*)≅δq,0⁢ℤ, q∈ℤ. In fact, v* is a global minimizer of I. By Lemma 3.1, we have Cq⁢(I,0)≅0, q∈ℤ. Hence, v*≠0 and problem (P’) has at least one nontrivial solution v*∈W01,2⁢(Ω). Thus, problem (P) also has at least one nontrivial solution in W01,2⁢(Ω). ∎

Proof of Theorem 1.2.

From Lemma 4.3 (i) and Lemma 3.3, we have Cq⁢(I,∞)≅0, q∈ℤ and Cq⁢(I,0)≅δq,0⁢ℤ, q∈ℤ, and hence Cq⁢(I,∞)≢Cq⁢(I,0). Thus, Morse theory implies that problem (P’) has at least one nontrivial solution. Therefore, problem (P) also has at least one nontrivial solution. ∎

Proof of Theorem 1.3.

Lemma 4.1 and Lemma 4.2 (ii) imply that I satisfies the (PS) condition, is bounded from below and has a global minimizer. Since Ck⁢(I,0)≇0, 0 is not the minimizer of I and is homological nontrivial. It follows from Lemma 2.6 that I has at least two nontrivial critical points. Thus, problem (P) also has at least two nontrivial solutions in W01,2⁢(Ω). ∎

Communicated by Zhi-Qiang Wang

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11526126

Award Identifier / Grant number: 11571209

Funding statement: This work is supported by the Natural Science Foundation of China (grant nos. 11526126 and 11571209).

Acknowledgements

The authors are grateful to the anonymous referees for their helpful comments and suggestions.

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Received: 2015-06-22

Accepted: 2016-05-25

Published Online: 2016-06-28

Published in Print: 2016-11-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

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Quasilinear Elliptic Equation; Critical Group; Homological Nontrivial Critical Point,Morse Theory; Local Linking

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