On the solvability of an indefinite nonlinear Kirchhoff equation via associated eigenvalue problems

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J. Differential Equations 269 (2020) 2853–2895

www.elsevier.com/locate/jde

On the solvability of an indefinite nonlinear Kirchhoff equation via associated eigenvalue problems

Han-Su Zhang

, Tiexiang Li

, Tsung-fang Wu

aSchoolofmathematics,SoutheastUniversity,Nanjing211189,PRChina

bDepartmentofAppliedMathematics,NationalUniversityofKaohsiung,Kaohsiung811,Taiwan

Received 17October2019;revised 4February2020;accepted 11February2020 Availableonline 17February2020

Abstract

Westudythenon-existence,existenceandmultiplicityofpositivesolutionstothefollowingnonlinear Kirchhoffequation:

⎧⎨

⎩

−M

R^N|∇u|²dx

u+μV (x) u=Q(x)|u|^p−2u+λf (x) uinR^N, u∈H¹

R^N

,

whereN ≥3,2< p <2^∗:= _N^2N₋₂, M (t)=at+b (a, b >0), thepotential V is anonnegativefunc- tion in R^N and the weight functionQ∈L^∞

R^N

with changes sign in := {V =0}. Wemainly prove the existence of at least two positive solutions in the cases that (i) 2< p <min

4,2^∗ and 0< λ<

1−2 [(4−p) /4]^2/p

λ₁(f);(ii) p≥4,λ≥λ₁(f)andnearλ₁(f)forμ>0 sufficiently large,whereλ₁(f)is thefirsteigenvalue of−in H₀¹()withweightfunctionf:=f|,whose correspondingpositiveprincipaleigenfunctionisdenotedby φ₁.Furthermore,wealsoinvestigatedthe non-existenceandexistenceofpositivesolutionsifa,λbelongstodifferentintervals.

©2020ElsevierInc.Allrightsreserved.

MSC:35B38;35B40;35J20;35J61

* Correspondingauthor.

E-mailaddresses:[email protected](H.-S. Zhang),[email protected](T. Li),[email protected](T.-f. Wu).

https://doi.org/10.1016/j.jde.2020.02.017

0022-0396/©2020ElsevierInc.Allrightsreserved.

Keywords:NonlinearKirchhoffequations;Neharimanifold;Eigenvalueproblem;Positivesolution;

Concentration-compactnessprinciple

1. Introduction

In this paper we are concerned the following nonlinear Kirchhoff equation:

−M

R^N|∇u|²dx

u+μV (x) u=g(x, u)inR^N, u∈H¹

R^N

, (1.1)

where N≥3, g∈R^N×R →Rbeing continuous, M(s) =as+b (a, b >0)and the parameter μ >0. We assume that the potential function V satisfies the following conditions:

(V₁) V is a nonnegative continuous function on R^N; (V₂) there exists c >0 such that the set {V < c} :=

x∈R^N :V (x) < c is nonempty and has finite Lebesgue measure;

(V₃) =int

x∈R^N :V (x)=0 is nonempty bounded domain and has a smooth boundary with =

x∈R^N :V (x)=0 .

The hypotheses (V1)−(V₃)imply that μV represents a potential well whose depth is con- trolled by μ. μV is called a steep potential well if μis sufficiently large and one expects to find solutions which localize near its bottom . This problem has found much interest after being first introduced by Bartsch and Wang [9] in the study of the existence of positive solutions for nonlinear Schrödinger equations and has been attracting much attention, see [3,7,8,33,38] and the references therein.

Kirchhoff type equations, of the form similar to Equation (1.1), originate from physics. In- deed, if we set V (x) ≡0 and replace R^Nby a bounded domain ⊂R^Nin Equation (1.1), then it becomes the following Dirichlet problem of Kirchhoff type:

− a

|∇u|²dx+b

u=g(x, u) in,

u=0 on∂, (1.2)

which is analogous to the stationary case of equations that arise in the study of string or mem- brane vibrations, namely,

u_{t t}−

⎛

⎝a

|∇u|²dx+b

⎞

⎠u=g(x, u), (1.3)

where udenotes the displacement, gis the external force and bis the initial tension while ais related to the intrinsic properties of the string (such as Young’s modulus). Equation (1.3) was first proposed by Kirchhoff [23] in 1883 to describe the transversal oscillations of a stretched string, particularly, taking into account the subsequent change in string length caused by oscillations. It is notable that Equation (1.3) is often referred to as being nonlocal because of the presence of the integral over the domain .

After the pioneering work by Pohozaev [28] and Lions [24], the qualitative analysis of non- trivial solutions for the nonlinear Kirchhoff type equations, similar to Equation (1.1), has begun to receive much attention in recent years. We refer the reader to [2,12,15,16,18–22,26,29–32,34, 37,39] and the references therein.

Let us briefly comment on some of the things that are relevant to our work. In [30], the authors introduced the steep potential well V to the Kirchhoff type equations. When the potential V satisfies the hypotheses (V1) −(V₃), the following results were obtained.

(i) N≥3: if 0 < a < a^∗and μ >0 sufficiently large, then Equation (1.1) has at least one positive solution, when g(x, u)is asymptotically linear at infinity on uand bλ⁽¹⁾₁ <1;

(ii) N=3: if 0 < a < λ⁽³⁾₁ and μ >0 sufficiently large, then Equation (1.1) has at least one positive solution, when g(x, u)is asymptotically 3-linear at infinity on u;

(iii) N=3: for any a >0 and μ >0 sufficiently large, Equation (1.1) has at least one positive solution, when g(x, u)is asymptotically 4-linear at infinity on u,

where

λ^(k)₁ =inf

⎧⎨

⎩(

|∇u|²dx)^k+12 :u∈H₀¹(),

q|u|^k+1dx=1

⎫⎬

⎭

and qis a bounded function on ¯ with q⁺≡0. After that, Xie and Ma [39] obtained the existence and concentration of positive solutions for Equation (1.1) with N=3 when potential V satisfies conditions (V1) −(V₃)and nonlinearity gsatisfies the following conditions:

(G₁) there _u exists ρ >4 such that 0 < ρG(x, u) ≤g(x, u)u for u >0, where G(x, u) =

0 g(x, s)ds;

(G2) ^G(x,u)

u³ is increasing for u >0.

In our recent papers [29,32], we concluded that when N≥3 and g(x, u)is superlinear and subcritical on u, the geometric structure of the functional Jrelated to Equation (1.1) is known to have a global minimum and a mountain pass, owing to the fourth power of the nonlocal term. By using the standard variational methods, two different positive solutions can be found, since some embedding inequalities are proved with the help of the fact of 2^∗:=_N^2N₋₂≤4.

In simple terms, when g(x, u) =Q(x)|u|^p−2uand Q ∈L^∞ R^N

is sign-changing, the cur- rent progress through the above literature is as follows:

(I ) N=3 and 4 < p <6: for any a >0 and μ >0 sufficiently large, Equation (1.1) has at least one positive solution;

(I I ) N=3 and 2 < p≤4: for a >0 small enough and μ >0 sufficiently large, Equation (1.1) has at least one positive solution;

(I I I ) N≥4 and 2 < p <2^∗: for a >0 small enough and μ >0 sufficiently large, Equation (1.1) has at least two positive solution.

Motivated by these findings, we now extend the analysis to the Kirchhoff type equation with combination of a superlinear term and a linear term, that is g(x, u) =Q(x)|u|^p−2u +λf (x)u.

Our intension here is to illustrate the difference in the solution behavior which arises from the consideration of the nonlocal and eigenvalue problem effects. The problem we consider is thus

−M

R^N|∇u|²dx

u+μV (x) u=Q(x)|u|^p−2u+λf (x) uinR^N, u∈H¹

R^N

, (Eμ,λ)

where N≥3, 2 < p <2^∗:=_N^2N₋₂, M (t)=at+b (a, b >0)and the parameters μ, λ >0. We are interested in the case the weight functions f and Qsatisfying {f >0} ∩and {Q >0} ∩ has the positive Lebesgue measures which is why we call indefinite nonlinear Kirchhoff equation in the title.

To go further, let us give some notations first. For the sake of simplicity, we always assume that b=1 in Equation (Eμ,λ). Let D^1,2

R^N

be the completing of C₀^∞ R^N

with respect to the norm u²_D1,2=

R^N|∇u|²dx. Denote by Sp, S_p()and Sthe best constants for the embeddings of H¹(R^N)in L^p(R^N), H₀¹()in L^p()and D^1,2(R^N)in L^2∗(R^N), respectively. We denote a strong convergence by “→” and a weak convergence by “”.

Now, we give the variational setting for Equation E_μ,λ

. Let

X=

⎧⎪

⎨

⎪⎩u∈H¹

R^N :

R^N

V u²dx <∞

⎫⎪

⎬

⎪⎭

be equipped with the inner product and norm u, v =

R^N

∇u∇v+V uvdx, u = u, u^1/2.

For μ >0, we also need the following inner product and norm u, vμ=

R^N

∇u∇v+μV uvdx, uμ= u, u^1/2μ .

It is clear that · ≤ ·μfor μ ≥1 and set X_μ= X,·μ

.

Note that u ∈X_μis a solution of Equation (E_μ,λ)if for any v∈X_μthere holds

M

⎛

⎜⎝

R^N

|∇u|²dx

⎞

⎟⎠

R^N

∇u∇v+μ

R^N

V (x) uv=

R^N

Q(x)|u|^p−2uv+λf (x) uv

dx.

And uis called a positive solution if uis a solution and u >0 in R^N. It is well known that Equation

E_μ,λ

is variational, and its solutions correspond to the critical point of the energy functional Jμ,λ:X_μ→R

Jμ,λ(u)=a

4u⁴_D1,2+1

2u²μ− 1 p

R^N

Q|u|^pdx−λ 2

R^N

f u²dx,

where uμ=

R^N

|∇u|²+μV u² dx1/2

is the standard norm in Xμand Xμis a subspace of H¹

R^N

(see below). Thus, if uis a critical point of Jμ,λon Xμ, then uis a solution of Equation E_μ,λ

.

Assume the following hypotheses (D):

(D₁) f ∈L^N/2 R^N

which f⁺:=max{f,0} ≡0 in ; (D₂) Q ∈L^∞

R^N

which Q⁺:=max{Q,0} ≡0 in .

Remark 1.1. Since {f >0} ∩has a positive Lebesgue measure, we can assume that λ1(f) denote the positive principal eigenvalue of the problem

−u(x)=λf(x)u(x)forx∈; u(x)=0 forx∈∂, (1.4) where f is a restriction of f on . Clearly, λ₁(f)has a corresponding positive principal eigenfunction φ1with

fφ₁²dx=1 and

|∇φ₁|²dx=λ₁(f).

We now summarize our main results as follows.

Theorem 1.1. Suppose that N=3, 4 < p <6and conditions (V1)−(V3)and (D1)−(D2)hold.

Then for each a >0and 0 < λ < λ1(f), Equation Eμ,λ

has a positive solution u⁻_μsatisfying J_μ,λ

u⁻_μ

>0for μ >0sufficiently large.

Theorem 1.2. Suppose that N=3, 4 < p <6, conditions (V1)−(V₃)and (D1)−(D₂)hold and

Qφ^p₁dx <0. Then for each a >0there exists δ0such that for every λ1(f)≤λ < λ1(f)+ δ₀, Equation

E_μ,λ

has at least two positive solutions u⁻_μ and u⁺_μ satisfying Jμ,λ

u⁺_μ

<0 <

Jμ,λ

u⁻_μ

for μ >0sufficiently large.

To consider the case N=3 and p=4, we need the following maximum problem

0:=sup

u∈X

R³Q|u|⁴dx u⁴_D1,2

>0.

Then we have the following results.

Theorem 1.3. Suppose that N=3, p=4and conditions (V1)−(V3)and (D1)−(D2)hold.

Then we have the following results.

(i) For each 0 < a < ₀ and 0 < λ < λ₁(f), Equation E_μ,λ

has a positive solution u⁻_μ satisfying Jμ,λ

u⁻_μ

>0for μ >0sufficiently large.

(ii) If 0<∞, then for each a≥0 and 0 < λ < λ1(f), Equation Eμ,λ

does not admit nontrivial solution for μ >0sufficiently large.

(iii)If 0<∞, then for each a > 0and λ ≥λ1(f), Equation Eμ,λ

has a positive solution u⁺_μ satisfying Jμ,λ

u⁺_μ

<0for μ >0sufficiently large.

(iv)If 0<∞and 0is not attained, then for a=₀and λ ≥λ₁(f), Equation E_μ,λ

has a positive solution u⁺_μ satisfying Jμ,λ

u⁺_μ

<0for μ >0sufficiently large.

Theorem 1.4. Suppose that N =3, p=4 and conditions (V1)−(V₃)and (D1)−(D₂)hold.

Then for each λ⁻₁²(f)

Qφ₁⁴dx < a < ₀there exists δ0such that for every λ1(f)≤λ <

λ₁(f)+δ₀, Equation E_μ,λ

has two positive solutions u⁻_μ and u⁺_μ satisfying Jμ,λ

u⁺_μ

<0 <

J_μ,λ u⁻_μ

for μ >0sufficiently large.

Remark 1.2. When 4 < p <6, by the hypothesis of Theorem1.2, in order to obtain the multi- plicity of the positive solution for Equation

E_μ,λ

, the weight function Qmust be sign-changing in , but when p=4, the weight function Qcan be positive in from the hypothesis of Theo- rem1.4.

To consider the case 2 < p <min{4,2^∗}, we first show that the non-existence of solutions.

Theorem 1.5. Suppose that N≥4, 2 < p <2^∗ and conditions (V1)−(V3)and (D1)−(D2) hold. Then for each 0 < λ < λ1(f)there exists

0<Aλ<1 2

(4−p) λ1(f) p (λ1(f)−λ)

(4−p)/(p−2)⎛

⎝Q_∞|{V < c}|^2∗−p2∗

S^p

⎞

⎠

2/(p−2)

such that for every a >Aλ, Equation E_μ,λ

does not admit nontrivial solution for μ >0suffi- ciently large.

To prove the existence of positive solution, we need the following conditions:

(D₃) There exist two numbers c_∗, R_∗>0 such that

|x|^p−2Q (x)≤c_∗[V (x)]^4−p for all |x|> R_∗. (D4) |{V < c}|^(6−p)/6≤ _S^pSpQ,min

p()Q_∞, where Q,min=inf_x∈Q (x) >0.

Then we have the following results.

Theorem 1.6. Suppose that N=3, 2 < p <4and conditions (V1)−(V₃)and (D1)−(D₃)hold.

Then we have the following results.

(i)There exists a0>0such that for every 0 < a < a0and 0 < λ < λ1(f), Equation E_μ,λ

has a positive solution u⁺_μ satisfying Jμ,λ

u⁺_μ

<0for μ >0sufficiently large.

(ii) For each λ ≥λ₁(f)and a >0, Equation E_μ,λ

has a positive solution u⁺_μ satisfying J_μ,λ

u⁺_μ

<0for μ >0sufficiently large.

Theorem 1.7. Suppose that N≥4, 2 < p <2^∗ and conditions (V1)−(V3)and (D1)−(D2) hold. Then we have the following results.

(i)There exists a0>0such that for every 0 < a < a0and 0 < λ < λ1(f), Equation E_μ,λ

has a positive solution u⁺_μ satisfying Jμ,λ

u⁺_μ

<0for μ >0sufficiently large.

(ii) For each a >0 and λ ≥λ1(f), Equation Eμ,λ

has a positive solution u⁺_μ satisfying J_μ,λ

u⁺_μ

<0for μ >0sufficiently large.

Fig. 1. Bifurcation diagram for Theorems1.1and1.2.

Theorem 1.8. Suppose that N ≥3, 2 < p <min{4,2^∗} and conditions (V1)−(V3) , (D1)− (D2)and (D4)hold. Then there exists a0>0 such that for every 0 < a < a0 and 0 < λ <

1−2

4−p 4

2/p

λ1(f), Equation E_μ,λ

has a positive solution u⁻_μsatisfying J_μ,λ u⁻_μ

>0 for μ >0sufficiently large.

Combining the Theorems1.6, 1.8results, we have the following multiplicity result.

Corollary 1.9. Suppose that N=3, 2 < p <4and conditions (V1)−(V₃)and (D1)−(D₄)hold.

Then there exists a0>0such that for every 0 < a < a0and 0 < λ <

1−2

4−p 4

2/p

λ₁(f), Equation

Eμ,λ

has two positive solutions u⁻_μand u⁺_μsatisfying Jμ,λ

u⁺_μ

<0 < Jμ,λ

u⁻_μ for μ >0sufficiently large.

Combining the Theorems1.7, 1.8results, we have the following multiplicity result.

Corollary 1.10. Suppose that N ≥4, 2 < p <2^∗ and conditions (V₁)−(V₃) , (D₁)−(D₂) and (D₄) hold. Then there exists a₀ >0 such that for every 0 < a < a₀ and 0 < λ <

1−2

4−p 4

2/p

λ₁(f), Equation E_μ,λ

has two positive solutions u⁻_μ and u⁺_μ satisfying Jμ,λ

u⁺_μ

<0 < Jμ,λ

u⁻_μ

for μ >0sufficiently large.

In order to make the above theoretical results more intuitive, the bifurcation diagrams of positive solutions concerning with the ranges of constant p, a, λis shown.

(I )4 < p <6.

(I I ) p=4.

(I I I )2 < p <min{4, 2^∗}.

We illustrate the finding of Theorems1.1-1.8graphically in Figs.1-3with different values of a, pand λ. These figures depict how the number of positive solutions of uchanges with the parameter λunder certain conditions. Subgraphs show the bifurcation diagram of the positive solution of uwhen ais in different ranges, respectively.

Remark 1.3. In Fig.3(a), the part marked with a question mark is not covered in this article, it is indicated by a dotted line that the exact number of positive solutions is unknown.

Fig. 2. Bifurcation diagrams for Theorems1.3and1.4, wherearel:=max{0, λ⁻²₁ (f)

Qφ⁴dx}.

Fig. 3.BifurcationdiagramsforTheorems1.7(i)and1.8(alsoforTheorems1.6and1.8)on(a)andforTheorems1.5 and1.7(ii)on(b),whereλ_rel:=

1−2_4−p 4

2/p λ₁(f).

To study the main Theorems, we shall establish their result by considering minimization on two distinct components of the Nehari manifold corresponding to Equation

E_μ,λ

. We are like- wise interested in the conditions of M and g that subsequently gives rise to the non-existence and existence of positive solutions. Our focus here, however, is on a given set of M and gso that it is possible to examine in detail the number of solutions admitted subject to the varia- tions of parameters imbedded in these functions. A similar analysis has been carried out on other elliptic equations with interesting results. Amann and Lopez-Gomez [1], Binding et al. [4,5], and Brown and Zhang [10], for example, studied the following semilinear boundary value prob- lem:

−u=λf(x) u+b(x)|u|^p−2uin,

u=0 on∂, (1.5)

where is a bounded domain with smooth boundary in R^N, λ >0 is a real parameter, 2 < p <2^∗ and f, b:→R are smooth functions which change sign in . In [4,5] by using variational methods, in Brown and Zhang [10] by using Nehari manifold and fibrering maps, and in Amann and Lopez-Gomez [1] by using global bifurcation theory. The existence and multiplicity results can be summarized as follows. It is known that

(A) there exists a positive solution to Equation (1.5) whenever 0 < λ < λ1(f);

(B) if

bφ^p₁dx <0, there exists δ0>0 such that Equation (1.5) has at least two positive solu- tions whenever λ1(f) < λ < λ1(f)+δ₀.

Results (A) and (B)can be understood in term of global bifurcation theory as the sign of

bφ^p₁dx determines the direction of bifurcation from the branch of zero solutions at the bi- furcation point at λ =λ1(f)so that bifurcation is to the left when

bφ₁^pdx >0 and to the right when

bφ₁^pdx <0; the corresponding bifurcation diagrams are shown in Fig. 1 of [10].

Furthermore, some who’s been done for this type of problem in R^N. We are only aware of the works Chabrowski and Costa [11] and Costa and Tehrani [13] which also studied the existence and multiplicity of positive solutions for Schrödinger type equations in R^N

−_pu=λf (x) u! +Q(x)" |u|^p−2uinR^N, (1.6) where λis a real parameter and p < q < Np/(N−p)and 1 < p < N. The functions f"and

"

Qdenote sign-changing potentials such that f"∈L^N/p(R^N) ∩L^∞(R^N)and Q"∈L^∞(R^N). Let λ1 "f

denote the lowest positive eigenvalue of −p and let ϕ1>0 be the associated eigen- function. When p=2, under a slightly more general assumption on the nonlinearity appearing on the right-hand side of (1.6), some results are obtained in [13] by using the Mountain-Pass Theorem and variational methods. However, in order to apply their result to Equation (1.6) they needed a “thickness” condition on the set o=

x:Q (x)" =0 . [11] by using Nehari manifold and fibrering maps which under a limits condition lim_|x|→∞Q (x)" =Q"_∞<0. Their main re- sult is almost the same as in results (A)and (B)above. However, the principal eigenvalue and eigenfunction are replaced by the problem −u(x) =λf (x)u(x)" for x∈R^N.

The approach to Equation Eμ,λ

has been inspired by the papers of [10,11] without any condition on oor lim_|x|→∞Q (x)=Q_∞<0. Moreover, since Equation

E_μ,λ

is on R^N, its variational setting is characterized by a lack of compactness. To overcome this difficulty we apply a simplified version of the steep well method of [9] and concentration compactness principle of [25]. Furthermore, the first eigenvalue of problem −u +μV (x) u =λf (x) uin R^Nis less than λ1(f), which indicates that the original method at [10,11] cannot be directly applied, thus we provide an approximation estimate of eigenvalue to prove our main results.

The plan of the paper is as follows. In Section2, we discuss the Nehari manifold and examine carefully the connection between the Nehari manifold and the fibrering maps. In Section3, we establish the non-emptiness of submanifolds and the proofs of the main theorems are given in the remaining sections. In section4, we discuss the Nehari manifold when 4 < p <6. In particular, we prove that Theorems1.1, 1.2. In Section5, we discuss the case when p=4 and prove that Theorems1.3, 1.4. In section6, we discuss the case when p <4 and prove that Theorems1.6, 1.7and 1.8.

2. Preliminaries

It follows from conditions (V1)and (V2)and similar to the argument in [30], one has

R^N

(|∇u|²+u²)dx≤

1+S⁻²|{V < c}|^N2 u²_μ

for all μ ≥μ₀:=^S_c²|{V < c}|^−N2, which implies that the imbedding Xμ→H¹(R^N)is contin- uous. Moreover, for any r∈ [2, 2^∗], there holds

R^N

|u|^rdx≤S^−r|{V < c}|^2∗−r2∗ u^rμ forμ≥μ0. (2.1)

Because the energy functional Jμ,λis not bounded below on Xμ, it is useful to consider the functional on the Nehari manifold (see [27])

Nμ,λ=#

u∈X_μ\ {0} :$

J_μ,λ (u) , u

%=0

&

. Thus, u ∈Nμ,λif and only if

au⁴_D1,2+ u²_μ=

R^N

Q|u|^pdx+λ

R^N

f u²dx.

Note that N_μ,λ contains every nonzero solution of Equation E_μ,λ

. It is useful to understand Nμ,λin terms of the stationary points of mappings of the form h_u(t ) =J_μ,λ(t u)(t >0). Such a map is known as the fibrering map. It was introduced by Drábek and Pohozaev [14], and further discussed by Brown and Zhang [10]. It is clear that, if uis a local minimizer of Jμ,λ, then huhas a local minimum at t=1. Thus, t u ∈Nμ,λif and only if h_u(t ) =0 for u ∈X\ {0}. Thus points in Nμ,λcorrespond to stationary points of the maps huand so it is natural to divide Nμ,λinto three subsets N⁺_μ,λ, N⁻_μ,λ and N⁰_μ,λcorresponding to local minima, local maxima and points of inflexion of fibrering maps. We have

h_u(t )=at³u⁴_D1,2+t

⎛

⎜⎝u²μ−λ

R^N

f u²dx

⎞

⎟⎠−t^p−1

R^N

Q|u|^pdx

and

h_u(t )=3at²u⁴_D1,2+

⎛

⎜⎝u²μ−λ

R^N

f u²dx

⎞

⎟⎠−(p−1) t^p−2

R^N

Q|u|^pdx.

Hence if we define

N⁺_μ,λ=

u∈Nμ,λ:h_u(1) >0 ; N⁰_μ,λ=

u∈Nμ,λ:h_u(1)=0 ; N⁻_μ,λ=

u∈Nμ,λ:h_u(1) <0 ,

which indicates that for u ∈Nμ,λ, we have h_u(1) =0 and u ∈N⁺_μ,λ, N⁰_μ,λ, N⁻_μ,λ if h_u(1) >

0, h_u(1) =0, h_u(1) <0, respectively. Note that for all u ∈Nμ,λ,

h_u(1)= −(p−2)

⎛

⎜⎝u²_μ−λ

R^N

f u²dx

⎞

⎟⎠−a (p−4)u⁴_D1,2

=2au⁴_D1,2−(p−2)

R^N

Q|u|^pdx

= −2

⎛

⎜⎝u²μ−λ

R^N

f u²dx

⎞

⎟⎠−(p−4)

R^N

Q|u|^pdx. (2.2)

Now, we define

⁺_μ=

⎧⎪

⎨

⎪⎩u∈X: uμ=1,u²μ−λ

R^N

f u²dx >0

⎫⎪

⎬

⎪⎭

and ⁻_μ and ⁰_μsimilarly by replacing >by <and =respectively. We also define ⁺_μ(p)=

u∈X: uμ=1, p(u) >0 and ⁻_μ(p)and ⁰_μ(p)analogously, where

_p(u)= ^RNQ|u|^pdx for 2< p <2^∗andp=4,

R^NQ|u|^pdx−au⁴_D1,2 forp=4.

Thus, if u ∈⁺_μ∩⁺_μ(p)and p≥4, hu(t ) >0 for tsmall and positive but hu(t ) → −∞as t→

∞; also hu(t )has a unique (maximum) stationary point tmax(u)and tmax(u)u ∈N⁻_μ,λ. Similarly, if u ∈⁻_μ∩⁻_μ(p)and 2 < p <2^∗, h_u(t ) <0 for t small and positive, h_u(t ) → ∞as t→ ∞ and hu(t )has a unique minimum tmin(u)so that tmin(u)u ∈N⁺_μ,λ. Finally, if u ∈⁺_μ∩⁻_μ(p), h_uis strictly increasing for all t >0. Thus, we have the following results.

Lemma 2.1. Suppose that N=3and 4 < p <6. If ⁻μ∩⁺_μ(p)= ∅and u ∈X_μ\{0}, then (i)a multiple of ulies in N⁻_μ,λif and only if _u^u

μ lies in ⁺_μ∩⁺_μ(p);

(ii)a multiple of ulies in N⁺_μ,λif and only if _u^u

μ lies in ⁻_μ∩⁻_μ(p);

(iii)when u ∈⁺_μ∩⁻_μ(p), no multiple of ulies in Nμ,λ. Lemma 2.2. Suppose that N=3and p=4. If u ∈X_μ\{0}, then (i)a multiple of ulies in N⁻_μ,λif and only if _u^u

μ lies in ⁺_μ∩⁺_μ(p);

(ii)a multiple of ulies in N⁺_μ,λif and only if _u^u

μ lies in ⁻_μ∩⁻_μ(p);

(iii)when u ∈⁺_μ∩⁻_μ(p)or ⁻μ∩⁺_μ(p), no multiple of ulies in Nμ,λ. Lemma 2.3. Suppose that N≥3and 2 < p <min{4,2^∗}. If u ∈Xμ\{0}, then (i)if _u^u

μ lies in ⁻μ∩⁺_μ(p)or ⁻_μ∩⁻_μ(p), then a multiple of ulies in N⁺_μ,λ;

(ii)when u ∈⁺_μ∩⁻_μ(p), no multiple of ulies in Nμ,λ.

The following Lemma shows that minimizers on Nμ,λare critical points for Jμ,λin Xμ. Lemma 2.4. Suppose that u0is a local minimizer for Jμ,λon Nμ,λ and that u0∈/N⁰_μ,λ. Then J_μ,λ (u0) =0in X⁻_μ¹.

Proof. The proof of Lemma2.4is essentially same as that in Brown and Zhang [10, Theorem 2.3] (or see Binding et al. [4]), so we omit it here.

Finally, we investigate the compactness condition for the functional Jμ,λ. Here we call that a C¹-functional Jμ,λ satisfies Palais-Smale condition at level β ((PS)β-condition for short) in Nμ,λ, if any sequence {u_n}⊂Nμ,λis uniformly bounded which satisfy Jμ,λ(un)=β+o (1)and J_μ,λ (u_n)=o (1)has a convergent subsequence.

Proposition 2.5. Suppose that conditions (V1) −(V₂)and (D1) −(D₂)hold. Then there exists D!₀∈Rindependent of μsuch that Jμ,λsatisfies (PS)β–condition in Nμ,λwith β <D!₀for μ >0 sufficiently large.

Proof. Let {u_n} ⊂Nμ,λ be a (PS)β–sequence for Jμ,λ with β <D!₀. Since {u_n} ⊂X_μis uni- formly bounded, i.e., there exists d0>0 such that

unμ< d0. (2.3)

Then there exist a subsequence {u_n}and u0in Xμsuch that u_n u₀weakly inX_μ;

u_n→u0strongly inL^r_loc(R^N)for 2≤r <2^∗. Then by condition (D1),

nlim→∞

R^N

f u²_ndx=

R^N

f u²₀dx. (2.4)

Now, we prove that un→u₀strongly in Xμ. Let vn=u_n−u₀. By (2.3) one has u₀μ≤lim inf

n→∞ u_nμ≤d₀, leading to

vnμ= un−u0μ≤2d0. (2.5) It follows from the condition (V1)that

R^N

v_n²dx=

{V≥c}

v²_ndx+

{V <c}

v_n²dx≤ 1

μcv_n²_μ+o (1) ,

which implies that

R^N

|v_n|^pdx≤ 1

μcv_n²_μ ^2∗−p_2∗−2

S^−2∗v_n²_D^∗1,2

^p−2

2∗−2

+o(1)

≤ 1

μc

^{2∗−p(N−2)}

4

S^{−N (p2−2)}v_n^p_μ+o(1), (2.6) where we have used the Hölder and Sobolev inequalities. Moreover, by Brezis-Lieb Lemma [6]

and condition (D2), we have

R^N

Q|v_n|^pdx=

R^N

Q|u_n|^pdx−

R^N

Q|u₀|^pdx+o(1). (2.7)

Since the sequence {u_n}is bounded in Xμ, there exists a constant A >0 such that

R^N

|∇un|²dx→Aasn→ ∞.

It indicates that for any ϕ∈C₀^∞(R^N), o(1)= $

J_μ,λ (u_n) , ϕ

%

→

R^N

∇u0∇ϕdx+

R^N

μV u0ϕdx+aA

R^N

∇u0∇ϕdx

−

R^N

f u₀ϕdx−

R^N

Q|u₀|^p−2u₀ϕdxasn→ ∞,

which shows that

u₀²_μ+aAu₀²_D1,2−

R^N

f u²₀dx−

R^N

Q|u₀|^pdx=0. (2.8)

Note that

u_n²μ+au_n⁴_D1,2−

R^N

f u²_ndx−

R^N

Q|u_n|^pdx=0. (2.9)

Then by (2.4) and (2.6)–(2.9) one has

o (1)= vn²μ+aun⁴_D1,2−aAu0²_D1,2−

R^N

Q|vn|^pdx

= v_n²_μ+au_n²_D1,2

u_n²_D1,2− u₀²_D1,2

−

R^N

Q|v_n|^pdx

= v_n²_μ+au_n²_D1,2v_n²_D1,2−

R^N

Q|v_n|^pdx. (2.10)

It follows from (2.1), (2.5), (2.6), (2.10) and condition (D2)that o (1)= v_n²_μ+au_n²_D1,2v_n²_D1,2−

R^N

Q|v_n|^pdx

≥ v_n²μ− Q_∞

⎛

⎜⎝

R^N

|v_n|^pdx

⎞

⎟⎠

p−2

p ⎛

⎜⎝

R^N

|v_n|^pdx

⎞

⎟⎠

2 p

≥

⎡

⎢⎣1− Q_∞

⎡

⎣(2d0)^p2 |{V < c}|^2∗−p2∗

S^p

⎤

⎦

p−2

p

1 μc

^{2∗−p(N−2)}

2p

S⁻

N (p−2) p

⎤

⎥⎦v_n²_μ+o (1) ,

which implies that vn→0 strongly in Xμfor μ >0 sufficiently large. Consequently, this com- pletes the proof.

3. Non-emptiness of submanifolds First, we need the following result.

Theorem 3.1. Let μn→ ∞as n → ∞and {v_n} ⊂Xwith v_nμn≤c₀for some c0>0. Then for every μ >0there exist subsequence {v_n}and v₀∈H₀¹()such that v_n v₀ in X_μ and vn→v0in L^r

R^N

for all 2 ≤r <2^∗.

Proof. Since v_nμ≤ v_nμn≤c₀ for n sufficiently large. We may assume that there exists v₀∈Xsuch that

vn v0inXμ, vn→v0a.e. inR^N, v_n→v₀inL^r_loc

R^N

for 2≤r <2^∗. By Fatou’s Lemma, we have

R^N

V v²₀dx≤lim inf

n→∞

R^N

V v_n²dx≤lim inf

n→∞

v_n²_μn μ_n =0,