Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

Guofeng Che; Tsung-fang Wu

doi:10.1515/anona-2021-0213

Open Access Published by De Gruyter November 11, 2021

Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

Guofeng Che and Tsung-fang Wu

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0213

Abstract

We study the following Kirchhoff type equation:

−a+b∫RN|∇u|2dxΔu+u=k(x)|u|p−2u+m(x)|u|q−2u in RN,

where N=3, a,b>0 , 1<q<2<p<min{4,2∗} , 2≤=2N/(N − 2), k ∈ C (ℝ^N) is bounded and m ∈ L^p/(p−q)(ℝ^N). By imposing some suitable conditions on functions k(x) and m(x), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H1(RN)↪Lr(RN)(2≤r<2∗) ; then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.

Keywords: Multiple positive solutions; Kirchhoff type equation; Indefinite nonlinearities

MSC 2010: 35J20; 35B09

1 Introduction

We investigate the existence of multiple positive solutions to the Kirchhoff type equation with indefinite nonlinearities:

(p) −a+b∫RN|∇u|2dxΔu+u=k(x)|u|p−2u+m(x)|u|q−2u in RN,

where N ≥ 3, a,b>0 , 1<q<2<p<min{4,2∗} , 2≤=2N/(N − 2), and functions k(x) and m(x) satisfy the following conditions.

(H₁) k ∈ C (ℝ^N) is a bounded function in ℝ^N;

(H₂) k is sign–changing in ℝ^N and Ω1={x∈RN:k(x)>0} is a bounded domain;

(H₃) m∈Lq∗(RN) and m⁺=max{m(x), 0} ≡ ≠ 0, where q∗=pp−q .

Problem (P) is a variant type of the Kirchhoff problem as follows:

(1.1) −a+b∫Ω|∇u|2dxΔu=f(x,u), in Ω.u=0, on ∂Ω,

which is related to the stationary analogue of the equation:

(1.2) utt−a+b∫Ω|∇u|2dxΔu=f(x,u), in Ω,u=0, on ∂Ω,

where Ω is a bounded domain in ℝ^N. Such problems are nonlocal owing to the appearance of ∫Ω|∇u|2dxΔu , which makes Eq. (1.1) and Eq. (1.2) are no longer pointwise identities. In [19], Kirchhoff firstly introduced Eq. (1.2) as an extension of the classical D'Alembert’s wave equation for free vibrations of elastic strings. The changes in length of the string produced by transverse vibrations are considered by the Kirchhoff’s model. Moreover, such nonlocal problem are also appeared in other fields, for instance, biological systems. For more details about the backgrounds of problems (1.1) and (1.2), one can be referred to [2,4,12,20].

In [22], Lions firstly proposed an abstract framework of Eq. (1.1), from then on, lots of researchers began to study Eq. (1.1) in general dimension, see [11,16,25,26,28,39] and the references therein. More precisely, Zhang and Perera [39] obtained the existence of a positive solution, a negative solution and a sign–changing solutions for Eq. (1.1) with N ≥ 1 by using invariant sets of descent flow. Pei and Ma [26] got three positive solutions for Eq. (1.1) with N=1, 2, 3 via the minimax method and the Morse theory. By using the theory developed in [27], Ricceri [28] obtained three positive solutions for Eq. (1.1) with N ≥ 4.

Recently, many papers [6, 8,29,30,38,32,15,31] study the Kirchhoff equation on the whole space:

(1.3) −a+b∫RN|∇u|2dxΔu+V(x)u=f(x,u), in RN,u∈H1(RN), in RN.

When the potential function V(x) satisfies the following assumptions:

(V₁) V ∈ C (ℝ^N) with V(x) ≥ 0 in ℝ^N and there exists c0>0 such that the set {V<c0}:={x∈RN| V(x)<c0} has finite positive Lebesgue measure, where ∣·∣ is the Lebesgue measure;

(V₂) Ω=int{x∈RN ∣ V(x)=0} is nonempty and has smooth boundary with Ω‾=V−1(0) . Sun and Wu [29] obtained the existence, nonexistence and concentration of nontrivial solutions for Eq. (1.4) with N=3 and V being replaced by λV, λ>0 . Later, when N ≥ 4, V is replaced by λV, λ>0 , and f satisfies the superlinear condition, Sun et al. [30] proved that Eq. (1.4) possessed two positive solutions. When N=3, V satisfies (V₁)−(V₂) and f satisfies the classical Ambrosetti–Rabinowitz type condition, Xie and Ma [38] proved that Eq. (1.4) has at least a positive solution. Moreover, the concentration behavior is also studied by the authors. In [32], Sun and Zhang obtained the uniqueness of the positive ground state solution for Eq. (1.4) with N=3, f(x,u)=d|u|p−1u, 3<p<5 , and V(x) ≡ c, where c and d are positive constants. Besides, by using the uniqueness result and the concentration–compactness lemma [23], the authors also got the existence and concentration theorems for the following Kirchhoff problem:

(1.4) −ε2a+εb∫R3|∇u|2dxΔu+V(x)u=K(x)|u|p−1u, in R3.

In [15], by using the Schwartz symmetric arrangement, Guo proved that Eq. (1.4) possesses a positive ground state solution when V is continuous and f does not satisfies the classical Ambrosetti–Rabinowitz type condition.

For the elliptic problems with concave–convex nonlinearities, there are also several results. For example, one can be referred to [1,37,35,24,36] and the references therein. Indeed, in [1], Ambrosetti et al. firstly introduced the elliptic problem involving the concave–convex nonlinearities:

(1.5) −Δu=λuq−1+up−1inΩ,u>0 inΩ,u∈H01(Ω),

where 1<q<2<p≤2∗ (2≤=∞ if N=1, 2; 2≤=2N/(N − 2) if N ≥ 3), λ>0 and Ω⊂ℝ^N(N ≥ 1) is bounded. With the aid of variational methods, the authors established the existence, nonexistence and multiplicity of solutions for Eq. (1.6). Later, Wu [37] obtained the existence of multiple positive solutions for the following problem:

(1.6) −Δu+u=f(x)uq−1+g(x)up−1inRN,u>0 inRN,u∈H1(RN),

where 1<q<2<p≤2∗ (2≤=∞ if N=1, 2; 2≤=2N/(N − 2) if N ≥ 3), f is sign–changing and g is positive in ℝ^N.

To the best of our knowledge, for the Kirchhoff problem with concave–convex terms, there are few results [5, 8,9,10, 21]. In [9], Chen et al. considered a class of Kirchhoff problem on the bounded domain Ω⊂ℝ^N(N ≥ 1):

(1.7) −a∫Ω|∇u|2dx+bΔu=λf(x)|u|q−2u+g(x)|u|p−2uinΩ,u=0 on∂Ω,

where 1<q<2<p≤2∗ (2≤=∞ if N=1, 2; 2≤=2N/(N − 2) if N ≥ 3). By giving different scopes on a and λ, the authors proved that Eq. (1.8) possesses multiple positive solutions. By using the Nehari manifold technique, Liao et al. [21] obtained two nonnegative solutions for Eq. (1.8). When f(x)=g(x) ≡ 1 in Eq. (1.8), by constraining the energy functional of the problem on a subset of the nodal Nehari set, Chen and Ou [10] proved that there exists a constant λ∗>0 such that for any λ<λ∗ , Eq. (1.8) has a nodal solution u_λ with positive energy. In [8], we obtained three positive solutions for the Kirchhoff type problem with steep potential well.

Motivated by the above facts, more precisely by [9], it is quite natural for us to ask that whether Eq.(P) can have multiple positive solutions when 1<q<2<p≤min{4,2∗} and the domain is unbounded? As far as we know, such problem has never been discussed in the available literature. In our paper, we will give a definite answer to the above question. Under some suitable conditions on functions k and m, we will obtain three positive solutions for Eq.(P). It is worthy pointing out, in [8], we considered Eq.(P) with steep potential well and positive nonlinearities, which we overcome the compactness of the Sobolev embedding by giving some inequalities. While, in the present paper, since the potential function is a constant and the nonlinearities are sigh–changing, the standard method of recovering the compactness does not work, motivated by [17], a novel method will be used to get the Palais–Smale(PS for short) condition.

Before introducing our main conclusions, we first recall a well known result (c.f. [34]). Suppose that wΩ1 is the positive ground state solution to the nonlinear elliptic equation:

(EΩ₁) −Δu+u=k(x)|u|p−2uu∈H01Ω1 in Ω1

where Ω₁ is defined by the condition (H₂).

Obviously,

(1.8) ∥wΩ1∥H12:=∫Ω1|∇wΩ1|2+wΩ12dx=∫Ω1k(x)wΩ1pdx,

and

infu∈MΩ1JΩ1(u)=JΩ1(wΩ1)=p−22p∫Ω1k(x)wΩ1pdx,

where JΩ1 is the energy functional of problem (EΩ1) , defined by

(1.9) JΩ1(u)=12∫Ω1|∇u|2+u2dx−1p∫Ω1k(x)|u|pdx,

and the corresponding Nehari manifold is given by:

MΩ1={u∈H01(Ω1)∖{0} | 〈JΩ1′(u),u〉=0}.

In order to simplify the calculation, in the present paper, we hypothesize that a=1 in problem (P). Denote kmax:=supx∈Ω1‾k(x) and A(p)=24−p1/(p−2) . Let S_r, Sr,Ω1 and S be the best Sobolev constants for the embedding of H¹(ℝ^N) in L^r(ℝ^N), H01(Ω1) in L^r(Ω₁) and D^1,2(ℝ^N) in L2∗(RN) with 2 ≤ r ≤ 2≤, respectively. For any 2 ≤ r ≤ +∞, we shall also denote by ∣·∣_r the L^r–norm. If we take a subsequence of a sequence {u_n}, we may denote it again by {u_n},

We now summarize our main results below.

Theorem 1.1

Assume that functions k and m satisfy hypotheses (H₁)−(H₃). Then there exists a constant Π1>0 such that for each 0<b+|m|q∗<Π1 , Eq.(P) admits two positive solutions ub1 , ub∗ , satisfying

∥ub1∥H1<(2−q)Spp(p−q)kmax1/(p−2)<∥ub∗∥H1<A(p)∥wΩ1∥H1,

and

Ib(ub1)<0<Ib(ub∗),

where I_b is the corresponding energy functional of Eq.(P) defined in Section 2.

Theorem 1.2

Assume that functions k and m satisfy hypotheses (H₁)−(H₃). In addition, we assume that (H₄) The weight function m changes sign in ℝ^N and Ω2={x∈RN:m(x)>0} is a bounded domain. Then there exists a constant Π2>0 such that for each 0<b+|m|q∗<Π2 , Eq.(P) admits three positive solutions ub1 , ub2 and ub∗ satisfying

∥ub1∥H1<(2−q)Spp(p−q)kmax1/(p−2)<∥ub∗∥H1<∥ub2∥H1,

and

maxIb(ub1),Ib(ub2)<0<Ib(ub∗).

Remark 1.1

Theorem 1.2 seems to be the first result about the Kirchhoff type equation with constant potential and indefinite nonlinearities, which has three positive solutions in the whole space ℝ^N, N ≥ 3. We also remark that, in the previous papers, to obtain three positive solutions for the Kirchhoff problem, the domain is usually required to be bounded. For the unbounded domain ℝ^N, N ≥ 3, as far as we are concerned, there are few results except [8], which we obtained three positive solutions for the Kirchhoff type equation with steep potential well.

In order to obtain our main results, we will make use of variational methods. Since we consider Eq.(P) in the whole space ℝ^N, the embedding from H¹(ℝ^N) into Ls(RN)(2<s<2∗) is not compact. Moreover, the appearance of the nonlocal term ∫RN|∇u|2dxΔu makes it hard to prove that the (PS) condition holds. Precisely, for any (PS) sequence {u_n}, if u_n⇀ u in H¹(ℝ^N), we do not know whether there holds

∫RN|∇un|2dx∫RN∇un∇vdx→∫RN|∇u|2dx∫RN∇u∇vdx, for~all v∈H1(RN).

Besides, the weight function k is sign–changing, which also makes it much more complicated to recover the compactness. Note that in order to overcome the lack of compactness of the Sobolev embedding, there are some existed strategies, such as constructing a convergent Cerami sequence [18]. While in this paper, inspired by [17, 20], we will make use of a novel method to verify that the (PS) condition holds. It is worthy pointing out, due to 2<p<min{4,2∗} , the energy functional of Eq.(P) is usually not coercive and bounded below, while, in this paper, under the hypotheses (H₁)–(H₄), we can prove that I_b is coercive and bounded below in ℝ^N, thus, two positive solutions are obtained. To get the third positive solution for Eq.(P), motivated by [31], the filtration of the Nehari manifold will be utilized:

M b ( c ) = { u ∈ M b : I b ( u ) < c } for some c > 0 depending on p , q , k , m ,

where Mb is the corresponding Nehari manifold that can be divided into Mb(1)(c) and Mb(2)(c) , where

M b ( 1 ) ( c ) = { u ∈ M b ( c ) : ∥ u ∥ H 1 < c 1 } and M b ( 2 ) ( c ) = { u ∈ M b ( c ) : ∥ u ∥ H 1 > c 2 } ,

for some ci>0, i=1,2 . By a simple computation, we know that I_b is bounded from below on Mb(1)(c) under the assumptions (H₁)–(H₃), and then the third positive solution for Eq.(P) is obtained.

The remainder of this paper is organized as follows. In Section 2, we recall some preliminaries. In Section 3, we first propose some novel techniques to recover the compactness of the Sobolev embedding, then obtain two positive solutions with negative energy for Eq.(P). After introducing the filtration of Nehari manifold in Section 4, we obtain the third positive solution with positive energy for Eq.(P) and prove our results in Section 5.

2 Preliminaries

For any u ∈ H¹(ℝ^N), let

(2.1) Ib(u)=12∥u∥H12+b4∫RN|∇u|2dx2−1p∫RNk(x)|u|pdx−1q∫RNm(x)|u|qdx.

Then I_b is a well defined C¹ functional with the following derivative:

(2.2) 〈Ib′(u),v〉=〈u,v〉H1+b∫RN|∇u|2dx∫RN∇u∇vdx−∫RNk(x)|u|p−2uvdx−∫RNm(x)|u|q−2uvdx.

Define the Nehari manifold

Mb:=u∈H1(RN)∖{0}:〈Ib′(u),u〉=0.

Thus u∈Mb if and only if

∥u∥H12+b∫RN|∇u|2dx2=∫RNk(x)|u|pdx+∫RNm(x)|u|qdx.

Note that Mb is closely related to the behavior of the fibering map [13,3]: hb,u:t→Ib(tu),t>0 . From (2.1), it is easy to know that

hb,u(t)=Ib(tu)=t22∥u∥H12+bt44∫RN|∇u|2dx2−tpp∫RNk(x)|u|pdx−tqq∫RNm(x)|u|qdx.

Thus

hb,u′(t)=t∥u∥H12+bt3∫RN|∇u|2dx2−tp−1∫RNk(x)|u|pdx−tq−1∫RNm(x)|u|qdx,

and

hb,u′′(t)=∥u∥H12+3bt2∫RN|∇u|2dx2−(p−1)tp−2∫RNk(x)|u|pdx−(q−1)tq−2∫RNm(x)|u|qdx.

A straightforward calculation shows that

thb,u′(t)=∥tu∥H12+b∫RN|∇tu|2dx2−∫RNk(x)|tu|pdx−∫RNm(x)|tu|qdx.

Obviously, tu∈Mb holds if and only if h′_{b, u}(t)=0. Hence, points in Mb correspond to the stationary points of the maps h_{b, u}, then it is natural to divide Mb into three parts corresponding to the local minima, local maxima and points of inflection. Following [33], we define

Mb+:=u∈Mb:hb,u′′(1)>0,

Mb0:=u∈Mb:hb,u′′(1)=0,

and

Mb−:=u∈Mb:hb,u′′(1)<0.

Similar to the arguments of Brown and Zhang [3, Theorem 2.3], we may get the following conclusion.

Lemma 2.1

Suppose that u₀ is a local minimizer for I_b on Mb and u0∉Mb0 . Then Ib′(u0)=0 .

For each u∈Mb , there holds

(2.3) h b , u ″ ( 1 ) = − 2 ∥ u ∥ H 1 2 + 4 − p ∫ R N k ( x ) | u | p d x + 4 − q ∫ R N m ( x ) | u | q d x

(2.4) = 2 − q ∥ u ∥ H 1 2 + 4 − q b ∫ R N | ∇ u | 2 d x 2 − p − q ∫ R N k ( x ) | u | p d x = − p − 2 ∥ u ∥ H 1 2 + ( 4 − p ) b ∫ R N | ∇ u | 2 d x 2 + p − q ∫ R N m ( x ) | u | q d x .

For each u∈Mb− , in view of (2.4), there holds

(2.5) 2−q∥u∥H12<2−q∥u∥H12+4−qb∫RN|∇u|2dx2<p−qkmaxSp−p∥u∥H1p,

since 1<q<2<p<min{4,2∗} . From (2.5), for every u∈Mb− , we derive

(2.6) ∥u∥H1>ρ0:=2−qSppp−qkmax1/p−2.

From (2.3) and (2.6), for any u∈Mb− , there holds

Ib(u)>p−24p∥u∥H12−4−qp−q4pq∫RNm(x)|u|qdx≥p−24p2−qSppp−qkmax2−q/p−2−4−qp−q|m|q∗4pqSpq∥u∥H1q>0,

provided that

(2.7) |m|q∗<Γ∗:=qp−24−qSp2p−q(p−q)/(p−2)2−qkmax(2−q)/(p−2).

Therefore, we may have the following result.

Lemma 2.2

Assume that hypotheses (H₁)−(H₃) hold. Then for every 0<|m|q∗<Γ∗ , I_b is coercive and bounded from below on Mb− , where Γ_≤ is defined by (2.7). Moreover, for every u∈Mb− , there holds Ib(u)>C0 for some C0:=C0(p,q,k,m)>0 .

3 Existence of two negative–energy positive solutions

Lemma 3.1

Under the conditions (H₁)−(H₃), there exists a constant Γ0>0 such that for every 0<|m|q∗≤Γ0 , we obtain

inf{Ib(u) | u∈H1(RN) with ∥u∥H1=ρ0}≥0.

Proof. For any u ∈ H¹(ℝ^N) with ∥u∥H1=ρ0 , in view of (2.1) and the Hölder inequality, there holds

(3.1) Ib(u)≥12∥u∥H12−kmaxpSpp∥u∥H1p−|m|q∗qSpq∥u∥H1q=ρ0q12ρ02−q−kmaxpSppρ0p−q−|m|q∗qSpq.

Define a function l:[0,+∞)→R as follows:

l(t)=12t2−q−kmaxpSpptp−q.

A straightforward calculations implies that

maxt≥0l(t)=l(t0)=12p−2p−qp(2−q)Spp2(p−q)kmax(2−q)/(p−2),

where

t0=p(2−q)Spp2(p−q)kmax1/(p−2)>0.

Note that

ρ0=(2−q)Spp(p−q)kmax1/(p−2)<t0,

and

l(ρ0)=(p−2)(p+2−q)2p(p−q)(2−q)Spp(p−q)kmax(2−q)/(p−2)>0.

Thus, for every u ∈ H¹(ℝ^N) satisfying ∥u∥H1=ρ0 , there hold

Ib(u)≥ρ0ql(ρ0)−|m|q∗qSpq≥0,

and

(3.2) |m|q∗≤Γ0:=q(p−2)(p+2−q)Spq2p(p−q)(2−q)Spp(p−q)kmax(2−q)/(p−2).

Consequently, the proof is complete. □

Lemma 3.2

Assume that hypotheses (H₁)−(H₄) hold. Then I_b is bounded from below and coercive on H¹(ℝ^N). Furthermore, for every b>0 , there exists Rb>ρ0 such that inf{I_b(u) ∣ u ∈ H¹(ℝ^N) with ∥u∥H1≥Rb}≥0 .

Proof. Denote |u|r,Ωir=∫Ωi|u|rdx for i=1, 2 and 1 ≤ r ≤ +∞, where Ω₁ and Ω₂ are given by conditions (H₂) and (H₄). From the Sobolev and Hölder inequalities, for any 2 ≤ r ≤ 2≤ and i=1, 2, we obtain

(3.3) ∫Ωi|u|rdx≤|u|2∗,Ωir|Ωi|2∗−r/2∗≤S−r|Ωi|2∗−r/2∗∥u∥D1,2r.

In view of (3.3) and the Hölder inequality again, we derive

(3.4) Ib(u)=b4∥u∥D1,24+12∥u∥H12−1p∫RNk(x)|u|pdx−1q∫RNm(x)|u|qdx≥b4∥u∥D1,24+12∥u∥H12−1p∫Ω1k(x)|u|pdx−1q∫Ω2m(x)|u|qdx≥b4∥u∥D1,24+12∥u∥H12−kmaxp∫Ω1|u|pdx−|m|q∗q∫Ω2|u|pdxq/p≥b4∥u∥D1,24+12∥u∥H12−kmax|Ω1|2∗−p/2∗pSp∥u∥D1,2p−|m|q∗|Ω2|q2∗−p/p2∗qSq∥u∥D1,2q.

Thus, for every b>0 , there exists Rˆb>0 such that

Ib(u)≥12∥u∥H12 for every ∥u∥D1,2≥Rˆb.

Let

(3.5) Rb=Rˆb2+4kmax|Ω1|2∗−p/2∗pSpRˆbp+|m|q∗|Ω2|q2∗−p/p2∗qSqRˆbq.

Next, we prove that

(3.6) Ib(u)≥14∥u∥H12 for every ∥u∥H1≥Rb.

For any u ∈ H¹(ℝ^N) satisfying ∥u∥H1≥Rb . If ∥u∥D1,2≥Rˆb , (3.6) holds. If ∥u∥D1,2<Rˆb , then it suffices to prove that Ib(u)≥14∥u∥H12 when

(3.7) ∫ R N u 2 d x ≥ 4 k max | Ω 1 | 2 ∗ − p / 2 ∗ p S p R ^ b p + | m | q ∗ | Ω 2 | q 2 ∗ − p / p 2 ∗ q S q R ^ b q .

Note that, by (3.3), for any r ∈ [2, 2≤] and i=1, 2, there holds

(3.8) ∫Ωi|u|rdx≤S−r|Ωi|2∗−r/2∗∥u∥D1,2r≤S−r|Ωi|2∗−r/2∗Rˆbr.

Then from (3.7) and (3.8), we obtain

(3.9) Ib(u)≥b4∥u∥D1,24+12∥u∥D1,22+12∫RNu2dx−1p∫Ω1k(x)|u|pdx−1q∫Ω2m(x)|u|qdx≥14∥u∥H12+14∫RNu2dx−kmaxp∫Ω1|u|pdx−|m|q∗q∫Ω2|u|pdxq/p≥14∥u∥H12+14∫RNu2dx−kmax|Ω1|2∗−p/2∗pSpRˆbp−|m|q∗|Ω2|q2∗−p/p2∗qSqRˆbq≥14∥u∥H12.

From (3.9), it is obvious that I_b is bounded from below and coercive on H¹(ℝ^N) and inf{I_b(u) ∣ u ∈ H¹(ℝ^N) with ∥u∥H1≥Rb}≥0 . The proof is complete. □

Lemma 3.3

Under the conditions (H₁)−(H₃), if {u_n} is a bounded (PS) sequence, then {u_n} has a convergent sequence.

Proof. Let {u_n} be a (PS) sequence satisfying ∥un∥H1≤M , where M is a positive constant. Then there exists u ∈ H¹(ℝ^N) such that

(3.10) un⇀u in H1RN

(3.11) un→u in LlocrRN for 2≤r<2∗,

(3.12) un→u a.e. in RN.

Denote ω_n=k(x)∣u_n∣^p−2u_n, and ω=k(x)∣u∣^p−2u. Then ω_n → ω a.e. in ℝ^N. Moreover, since {u_n}_n∈ℕ is bounded in L^p(ℝ^N) for 2<p<min{4,2∗} and k(x) is bounded in ℝ^N, thus {ω_n}_n∈ℕ is bounded in L^p/(p−1)(ℝ^N) and ω_n⇀ω in L^p/(p−1)(ℝ^N) for 2<p<min{4,2∗} . Then for any v ∈ H¹(ℝ^N), it follows from (H₃) and (3.10) that

(3.13) ∫RNk(x)|un|p−2unvdx→∫RNk(x)|u|p−2uvdx,

and

(3.14) ∫RN∇un∇v+unvdx→∫RN∇u∇v+uvdx,

as n → ∞. Similarly, we obtain

(3.15) ∫RNm(x)|un|q−2unvdx→∫RNm(x)|u|q−2uvdx as n→∞.

Note that {u_n} is bounded in H¹(ℝ^N), thus we obtain

∫RN|∇un|2dx→A as n→∞,

for some A>0 . Define

(3.16) Jb(u)=12∥u∥H12+bA2∫RN|∇u|2dx−1p∫RNk(x)|u|pdx−1q∫RNm(x)|u|qdx.

Then it follows from (3.15)–(3.16) that

(3.17) o(1)=〈Ib′(un),v〉→1+bA∫RN∇u∇vdx+∫RNuvdx−∫RNk(x)|u|p−2uvdx−∫RNm(x)|u|q−2uvdx=〈Jb′(un),v〉=0 for~any~v∈H1(RN).

Note that from (2.1) and (3.16), we can also easily obtain

(3.18) 〈Ib′(un),un〉+o(1)=〈Jb′(un),un〉=o(1).

Denote v_n=u_n−u, then by (3.10) v_n⇀ 0 in H¹(ℝ^N), it follows from Brézis–Lieb Lemma that

(3.19) ∥un∥H12=∥u∥H12+∥vn∥H12+o(1),

(3.20) ∫RNk(x)|un|pdx=∫RNk(x)|u|pdx+∫RNk(x)|vn|pdx+o(1),

and

(3.21) ∫RNm(x)|un|qdx=∫RNm(x)|u|qdx+∫RNm(x)|vn|qdx+o(1).

In view of (3.19)–(3.21), there holds

〈Jb′(un),un〉=〈Jb′(u),u〉+∥vn∥H12+bA∫RN|∇vn|2dx−∫RNk(x)|vn|pdx−∫RNm(x)|vn|qdx+o(1),

which, together with (3.17)–(3.18), implies that

(3.22) un−uH12+bA∫RN∇un−u2 dx−∫RNk(x)un−up dx−∫RNm(x)un−uq dx→0 as n→∞.

From 2<p<min{4,2∗} , (3.11) and (H₂), we have

(3.23) k(x)≤0for anyx∈RN∖Ω1and∫Ω1k(x)|un−u|pdx→0asn→∞,

On the other hand, since m∈Lq∗(RN) , thus for every ε>0 , there exists R∗>0 such that

(3.24) ∫|x|>R∗|m|q∗dx<εq∗.

Then from (3.24), we obtain

(3.25) ∫|x|>R∗m(x)|un−u|qdx≤∫|x|>R∗|m|q∗dx1q∗∫|x|>R∗|un−u|pdxqp≤Cε,

since u_n and u are bounded in L^p(ℝ^N) for 2≤p<min{4,2∗} .

Note that from (3.11), we may also have

(3.26) ∫|x|≤R∗m(x)|un−u|qdx≤∫|x|≤R∗|m|q∗dx1q∗∫|x|≤R∗|un−u|pdxqp→0

as n → ∞. In view of (3.25) and (3.26), there holds

(3.27) ∫RNm(x)|un−u|qdx=∫|x|≤R∗m(x)|un−u|qdx+∫|x|>R∗m(x)|un−u|qdx→0 as n→∞.

It is now deduced from (3.21)–(3.23) and (3.27) that

(3.28) 0≤lim supn→∞∥un−u∥H12≤lim supn→∞∥un−u∥H12+bA∫RN|∇un−u|2dx≤lim infn→∞∫RNk(x)|un−u|pdx+∫RNm(x)|un−u|qdx≤limn→∞∫Ω1k(x)|un−u|pdx+∫RNm(x)|un−u|qdx=0.

Thus, u_n → u in H¹(ℝ^N) by (3.028). The proof is complete. □

Theorem 3.4

Assume that hypotheses (H₁)−(H₃) hold. Then the functional I_b has a local minimizer ub1∈H1(RN) for each 0<|m|q∗≤Γ0 , where Γ₀ is given by (3.2). Moreover,

(i) ub1 is a positive solution of Eq.(P);

(ii) Ib(ub1)<0 and ∥ub1∥H1<ρ0 .

Proof. In view of the hypothesis (H₃), there exists ω ∈ H¹(ℝ^N)\{0} satisfying

∫RNm(x)|ω|qdx>0.

When t>0 is sufficiently small, there holds

(3.29) Ib(tω)=t22∥ω∥H12+bt44∫RN|∇ω|2dx2−tpp∫RNk(x)|ω|pdx−tqq∫RNm(x)|ω|qdx<0.

Thus, from (3.29) and Lemma 3.1, we obtain

(3.30) θm:=inf{Ib(u) | u∈B‾ρ0(0)}<0.

From the Ekeland variational principle [14], there exists a sequence {un}⊂B‾ρ0(0) satisfying

Ibun=θm+o(1) and Ib′un=o(1).

It is easy to know that I_b satisfies the (PS)θm –condition in B‾ρ0(0) by Lemma 3.3. As a result, there exists ub1∈B‾ρ0(0) satisfying un→ub1 as n → +∞. This implies that ub1 is a local minimizer on Bρ0(0) that satisfies Ib(ub1)=θm<0 and ∥ub1∥H1<ρ0 by Lemma 3.1. Since Ib(ub(1))=Ib(|ub1|)=θm , we may hypothesize that ub1 is a positive solution of Eq.(P). The proof is complete. □

Theorem 3.5

Assume that hypotheses (H₁)−(H₄) hold. Then there exists a constant b0>0 such that for any 0<b<b0 , Eq.(P) admits a positive solution ub2 satisfying Ib(ub2)<0 and ρ0<∥ub2∥H1<Rb .

Proof. Since k ∈ C (ℝ^N) and kmax:=supx∈Ω1‾k(x) , thus there exists a domain Ω₀⊂Ω₁ such that for every x ∈ Ω₀, there holds k(x)≥kmax2 . Let I₀(u)≔ I_b(u) with b=0. Then for every ξ∈H01(Ω1) , we know that limt→+∞I0(tξ)tp≤−kmax2p∫Ω0|ξ|pdx<0 , which shows that I₀(tξ) → −∞ as t → +∞. Hence, there exists ζ ∈ H¹(ℝ^N) with ∥ζ∥H1>ρ0 such that I0(ζ)<0 . Since I_b(ζ) → I₀(ζ) as b → 0⁺, then there exists b0>0 such that for every 0<b<b0 , there holds Ib(ζ)<0 . It is easy to know that ∥ζ∥H1<Rb by Lemma 3.2. The rest proof is analogous to that of Theorem 3.4, we omit it here. □

4 The filtration of Nehari manifold

Set

B(p,q,k,m)=p(4−q)2q/(p−2)|m|q∗p−2qSp,Ω1q∫Ω1k(x)wΩ1pdx(2−q)/2.

For any u∈Mb with

(4.1) ∥u∥H1>ρ0=2−qSppp−qkmax1/p−2,

and

Ib(u)<p−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx,

the Sobolev and Hölder inequalities imply that

p−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx>Ib(u)=12∥u∥H12+b4∫RN|∇u|2dx2−1p∫RNk(x)|u|pdx−1q∫RNm(x)|u|qdx≥p−22p∥u∥H12−4−pb4p∥u∥H14−p−qpqSpq|m|q∗∥u∥H1q>p−22p∥u∥H12−4−pb4p+p−qpqSpqρ0q−4|m|q∗∥u∥H14,

since ∥u∥H1q<ρ0q−4∥u∥H14 . This implies that for

0<(4−p)4pb+p−qpqSpqρ0q−4|m|q∗<Γ1,

there are two constants C₁, C₂ satisfying 0<C1<A(p)∥wΩ1∥H1<C2 such that

∥u∥H1<C1or∥u∥H1>C2,

where

(4.2) Γ1:=minp−28pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx,p−24pA2(p)∫Ω1k(x)wΩ1pdx.

Thus, it holds

(4.3) Mbp−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx=u∈Mb|∥u∥H1>ρ0andIb(u)<p−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx=Mb(1)∪Mb(2),

where

Mb(1):=u∈Mbp−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx | ∥u∥H1<C1,

and

Mb(2):=u∈Mbp−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx | ∥u∥H1>C2.

Moreover, we get

(4.4) ∥ u ∥ H 1 < C 1 < A ( p ) w Ω 1 H 1 for all u ∈ M b ( 1 ) ,

and

∥ u ∥ H 1 > C 2 > A ( p ) w Ω 1 H 1 for all u ∈ M b ( 2 ) .

It follows from (2.3),(4.1) and (4.4) that

(4.5) hb,u′′(1)≤−(p−2)∥u∥H12+(4−p)b∥u∥H14+p−qSpq|m|q∗∥u∥H1q<−(p−2)∥u∥H12+(4−p)b+p−qSpqρ0q−4|m|q∗∥u∥H14<0,

provided that

(4−p)b4+p−qqSpqρ0q−4|m|q∗<p−24A2(p)∥wΩ1∥H12.

Hence, we get the following result.

Lemma 4.1

Under the conditions (H₁)−(H₃), there is a constant Π0>0 such that for every 0<b+|m|q∗<Π0 , Mb(1)⊂Mb− is a C¹ sub–manifold. Furthermore, the local minimizer of the functional I_b in Mb(1) is a critical point of I_b in H¹(ℝ^N).

Lemma 4.2

Under the conditions (H₁)−(H₃), there exists a constant Π₁ ∈ (0, Π₀) such that for any 0<b+|m|q∗<Π1 , there is a constant tb− satisfying tb−<tˉb such that tb−wΩ1(x)∈Mb(1) , where

tˉb=24−p1/(p−2)if∫Ω1m(x)wΩ1qdx≥0,24−p1+∫Ωm(x)wΩ1qdx∫Ω1k(x)wΩ1pdx1/(p−2)if∫Ω1m(x)wΩ1qdx<0.

Proof. Let

f(t)=t−2∥wΩ1∥H12−tp−4∫Ω1k(x)wΩ1pdx−tq−4∫Ω1m(x)wΩ1qdxfort>0.

Then

hb,wΩ1′(t)=t∥wΩ1∥H12+bt3∫Ω1|∇wΩ1|2dx2−tp−1∫Ω1k(x)wΩ1pdx−tq−1∫Ω1m(x)wΩ1qdx=t3f(t)+b∫Ω1|∇wΩ1|2dx2.

It is obvious that twΩ1∈Mb if and only if hb,wΩ1′(t)=0 , i.e., f(t)+b∫Ω1|∇wΩ1|2dx2=0 . In view of (1.8), we obtain

f(t)=t−2∫Ω1|∇wΩ1|2+wΩ12dx−tp−4∫Ω1k(x)wΩ1pdx−tq−4∫Ω1m(x)wΩ1qdx=tq−4t2−q−tp−q−Ck,m∫Ω1k(x)wΩ1pdxfort>0,

where

Ck,m:=∫Ω1m(x)wΩ1qdx∫Ω1k(x)wΩ1pdx≤|m|q∗Sp,Ω1q∫Ω1k(x)wΩ1pdx(2−q)/2.

Define f₁(t)=t^2−q−t^p−q−C_{k, m}, then f(t)>0 if and only if f1(t)>0 , and f′₁(t)=0 exactly at

t1=2−qp−q1/(p−2),

and

f(t1)=t1q−4t12−q−t1p−q−Ck,m∫Ω1k(x)wΩ1pdx=t1q−4p−2p−q2−qp−q(2−q)/(p−2)−Ck,m∫Ω1k(x)wΩ1pdx>0,

provided that

|m|q∗<Γ2:=(p−2)Sp,Ω1qp−q2−qp−q(2−q)/(p−2)∫Ω1k(x)wΩ1pdx(2−q)/2.

Next, we consider the problem in the following two cases.

Case (I):∫Ω1m(x)wΩ1qdx≥0 . Let

f˜(t)=t−2∥wΩ1∥H12−tp−4∫Ω1k(x)wΩ1pdx, t>0.

Obviously, f(t)≤f˜(t) for t > 0. In view of (1.8), there holds

f˜(t)=t−2∫Ω1|∇wΩ1|2+wΩ12dx−tp−4∫Ω1k(x)wΩ1pdx=t−2−tp−4∫Ω1k(x)wΩ1pdx.

Evidently,

f ~ 1 = 0 , lim t → 0 + f ~ ( t ) = ∞ and lim t → ∞ f ~ ( t ) = 0.

Moreover

inft>0f˜(t)=f˜(t˜0)=−p−24−p4−p22/(p−2)∫Ω1k(x)wΩ1pdx<0,

where

t˜0:=24−p1/(p−2)>1.

Observe that

p−24−p4−p22/(p−2)∫Ω1k(x)wΩ1pdx∫Ω1|∇wΩ1|2dx2≥p−24−p4−p22/(p−2)∫Ω1k(x)wΩ1pdx∥wΩ1∥H14=p−24−p4−p22/(p−2)1∫Ω1k(x)wΩ1pdx.

Thus, for each

0<b<b∗(1):=p−24−p4−p22/(p−2)1∫Ω1k(x)wΩ1pdx,

we obtain

inft>0f˜(t)=−p−24−p4−p22/(p−2)∫Ω1k(x)wΩ1pdx<−b∫Ω1|∇wΩ1|2dx2.

This shows that there exist two constants tb+ and tb− that satisfy

tb−<24−p1/(p−2)<tb+

such that

f(tb±)+b∫Ω1|∇wΩ1|2dx2=0, f′(tb−)<0 and f′(tb+)>0.

That is, tb±wΩ1∈Mb . Besides, it is evident that

hb,tb−wΩ1′′(1)=(tb−)5f′(tb−)<0,

and

hb,tb+wΩ1′′(1)=(tb+)5f′(tb+)>0.

These imply that tb±wΩ1∈Mb± .

Case (II):∫Ω1m(x)wΩ1qdx<0 . Note that

f(t)=t−2∫Ω1|∇wΩ1|2+wΩ12dx−tp−4∫Ω1k(x)wΩ1pdx+tq−4∫Ω1m(x)wΩ1qdx=t−2−tp−4∫Ω1k(x)wΩ1pdx+tq−4∫Ω1m(x)wΩ1qdx.

Define

fˆ(t)=t−2∫Ω1k(x)wΩ1pdx+∫Ω1m(x)wΩ1qdx−tp−4∫Ω1k(x)wΩ1pdx.

It is obvious that

(4.6) f(t)<fˆ(t) for t>1,

and fˆ(1)=f(1)=|∫Ω1m(x)wΩ1qdx|>0 and limt→∞fˆ(t)=0 . Furthermore, we obtain

(4.7) inf t > 0 f ^ ( t ) = f ^ ( t ^ 0 ) = − p − 2 ( 4 − p ) t ^ 0 2 ∫ Ω 1 k ( x ) w Ω 1 p d x + ∫ Ω 1 m ( x ) w Ω 1 q d x < 0 ,

where

(4.8) tˆ0:=24−p1+∫Ω1m(x)wΩ1qdx∫Ω1k(x)wΩ1pdx1/(p−2)>1.

Observe that

p − 2 ( 4 − p ) t ^ 0 2 ∫ Ω 1 k ( x ) w Ω 1 p d x + ∫ Ω 1 m ( x ) w Ω 1 q d x 1 ∫ Ω 1 | ∇ w Ω 1 | 2 d x 2 ≥ p − 2 ( 4 − p ) t ^ 0 2 ∫ Ω 1 k ( x ) w Ω 1 p d x + ∫ Ω 1 m ( x ) w Ω 1 q d x 1 ∥ w Ω 1 ∥ H 1 4 = p − 2 ( 4 − p ) t ^ 0 2 ∫ Ω 1 k ( x ) w Ω 1 p d x + ∫ Ω 1 m ( x ) w Ω 1 q d x 1 ∫ Ω 1 k ( x ) w Ω 1 p d x 2 = p − 2 4 − p 4 − p 2 2 / ( p − 2 ) 1 + ∫ Ω 1 m ( x ) w Ω 1 q d x ∫ Ω 1 k ( x ) w Ω 1 p d x ( p − 4 ) / ( p − 2 ) 1 ∫ Ω 1 k ( x ) w Ω 1 p d x := b ∗ ( 2 ) .

It is obvious that b∗(2)<b∗(1) since 2<p<min{4,2∗} . Thus, for each 0<b<b∗(2) , it follows from (4.7) that

(4.9) fˆ(tˆ0)=−p−2(4−p)tˆ02∫Ω1k(x)wΩ1pdx+∫Ω1m(x)wΩ1qdx<−b∫Ω1|∇wΩ1|2dx2.

From (4.6), (4.8) and (4.9), we derive

(4.10) f(tˆ0)<fˆ(tˆ0)<−b∫Ω1|∇wΩ1|2dx2.

In view of (4.6) and (4.10), there are two constants tb+ and tb− that satisfy

(4.11) 1<tb−<tˆ0<tb+

such that

f(tb±)+b∫Ω1|∇wΩ1|2dx2=0, f′(tb−)<0 and f(tb+)>0.

That is, tb±wΩ1∈Mb . A straightforward calculation shows that

hb,tb−wΩ1′′(1)=(tb−)5f′(tb−)<0,

and

hb,tb+wΩ1′′(1)=(tb+)5f′(tb+)>0.

These imply that tb±wΩ1∈Mb± .

Next we prove that tb−wΩ1∈Mb(1) .

If ∫Ω1m(x)wΩ1qdx<0 , it follows from (4.11) that

(4.12) I b ( t b − w Ω 1 ) = 1 4 ( t b − ) 2 1 − 4 − p p ( t b − ) p − 2 ∫ Ω 1 k ( x ) w Ω 1 p d x + 4 − q 4 q ( t b − ) q ∫ Ω 1 m ( x ) w Ω 1 q d x ≤ 1 4 ( t b − ) 2 1 − 4 − p p ( t b − ) p − 2 ∫ Ω 1 k ( x ) w Ω 1 p d x

+4−q4q24−p1+∫Ω1m(x)wΩ1qdx∫Ω1k(x)wΩ1pdxq/(p−2)∫Ω1m(x)wΩ1qdx :=I1+I2,

where

I1=14(tb−)21−4−pp(tb−)p−2∫Ω1k(x)wΩ1pdx,

and

I2=4−q4q24−p1+∫Ω1m(x)wΩ1qdx∫Ω1k(x)wΩ1pdxq/(p−2)∫Ω1m(x)wΩ1qdx.

Define a function by

g(t)=t241−4−pptp−2, for 2<p<min{4,2∗},

where

1 < t < t ^ 0 := 2 4 − p 1 + ∫ Ω 1 m ( x ) w Ω 1 q d x ∫ Ω 1 k ( x ) w Ω 1 p d x 1 / ( p − 2 ) .

By a direct calculation, we obtain

max0<t≤tˆ0g(t)=gt∗=p−24p24−p2/(p−2),

where

(4.13) t∗=24−p1/(p−2)∈(1,tˆ0).

Hence,

max0<t≤tˆ0g(t)∫Ω1k(x)wΩ1pdx≤(p−2)A2(p)2p∥wΩ1∥H12,

that is,

(4.14) I1≤(p−2)A2(p)2p∥wΩ1∥H12.

On the other hand, by 0<b+|m|q∗<Π1 and (4.11), it holds

(4.15) I 2 = 4 − q 4 q 2 4 − p 1 + ∫ Ω 1 m ( x ) w Ω 1 q d x ∫ Ω 1 k ( x ) w Ω 1 p d x q / ( p − 2 ) ∫ Ω 1 m ( x ) w Ω 1 q d x ≤ 4 − q 4 q 2 4 − p 1 + | m | q ∗ | w Ω 1 | p q ∫ Ω 1 k ( x ) w Ω 1 p d x q / ( p − 2 ) | m | q ∗ | w Ω 1 | p q ≤ 4 − q 2 q 2 4 − p q / ( p − 2 ) | m | q ∗ S p , Ω 1 − q ∫ Ω 1 k ( x ) w Ω 1 p d x q / 2 if q ≤ p − 2 4 − q 4 q 4 4 − p q / ( p − 2 ) | m | q ∗ S p , Ω 1 − q ∫ Ω 1 k ( x ) w Ω 1 p d x q / 2 if q > p − 2

≤4−q2q44−pq/(p−2)|m|q∗Sp,Ω1−q∫Ω1k(x)wΩ1pdxq/2≤p−22pA2(p)B(p,q,k,m)∫Ω1k(x)wΩ1pdx.

It follows from (4.12), (4.14) and (4.15) that tb−wΩ1∈Mb1 .

Moreover, if ∫Ω1m(x)wΩ1qdx≥0 , then by (H₂), we have

Ib(tb−wΩ1)=14(tb−)2∥wΩ1∥H12−4−p4p(tb−)p∫Ω1k(x)wΩ1pdx−4−q4q(tb−)q∫Ω1m(x)wΩ1qdx≤14(tb−)2∥wΩ1∥H12−4−p4p(tb−)p∫Ω1k(x)wΩ1pdx=14(tb−)21−4−pp(tb−)p−2∫Ω1k(x)wΩ1pdx.

Let

η(t)=t241−4−pptp−2, 2<p<min{4,2∗},

where

1<t≤t˜0:=24−p1/(p−2).

A simple calculation shows that

max0<t≤t˜0η(t)=ηt˜0=p−24p24−p2/(p−2).

Then,

max0<t≤t˜0η(t)∫Ω1k(x)wΩ1pdx<p−22pA2(p)∫Ω1k(x)wΩ1pdx,

that is,

Ib(tb−wΩ1)<p−22pA2(p)∫Ω1k(x)wΩ1pdx≤p−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx.

The proof is complete. □

5 Proofs of main results

Lemma 5.1

Under the conditions (H₁)−(H₃), there exist a constant 𝜎 > 0 and a differentiable function t*:B (0, 𝜎)⊂ H¹(ℝ^N) → ℝ⁺ such that for any u∈Mb(1) , there holds

t∗(0)=1 and t∗(v)(u−v)∈Mb(1)

for all v ∈ B (0, σ), and

〈(t∗)′(0),φ〉=ψ(u,φ)Γ(u),

where

(5.1) ψ(u,φ)=2∫RN(∇u∇φ+uφ)dx+4b∫RN|∇u|2dx∫RN∇u∇φdx−p∫RNk(x)|u|p−2uφdx−q∫RNm(x)|u|q−2uφdx,

and

(5.2) Γ(u)=∥u∥H12−(p−1)∫RNk(x)|u|pdx−(q−1)∫RNm(x)|u|qdx

for all φ ∈ H¹(ℝ^N).

Proof. The proof of Lemma 5.1 is analogous to [7, Lemma 3.1] and [24, Lemma 3.1], we omit it here. □

Define

α b − = inf u ∈ M b ( 1 ) I b ( u ) .

In view of Lemma 2.2 and Mb(1)⊂Mb− , we derive

(5.3) 0<C0≤αb−<p−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx.

Proposition 5.2

Assume that hypotheses (H₁)−(H₃) hold. Then there exists a sequence {un}⊂Mb(1) such that

(5.4) Ib(un)=αb−+o(1) and I b′(un)=o(1).

Proof. The proof of Proposition 5.2 is similar to [7, Proposition 3.2] and [24, Proposition 2], we omit it here. □

By similar arguments to those of Lemma 3.3, we may get the following conclusion.

Proposition 5.3

Assume that hypotheses (H₁) −(H₃) hold. Then for any 0<d<p−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx , I_b satisfies the (PS)_d–condition in Mb(1) .

Theorem 5.4

Assume that hypotheses (H₁)−(H₃) hold. Then for any 0<b+|m|q∗<Π1 , where Π1>0 is given by Lemma 4.2, Eq.(P) admits a positive solution ub∗∈Mb(1) satisfying Ib(ub∗)>0 and

(2−q)Spp(p−q)kmax1/(p−2)<ub∗H1<A(p)wΩ1H1.

Proof. By Proposition 5.2, there is a sequence {un}⊂Mb(1) satisfying

Ib(un)=αb−+o(1) and ~Ib′(un)=o(1).

Clearly, 0<αb−<p−22pA2(p)(1+B(p,q,k,m))∫Ω1k(x)wΩ1pdx , and for small b, I_b satisfies the (PS)αb− –condition in Mb(1) by Proposition 5.3. Thus there exists ub∗∈H1(RN)∖{0} such that un→ub∗ in H¹(ℝ^N) . Observe that {un}⊂Mb(1) and

∥un∥H1<C1<A(p)wΩ1H1 for all n=1,2,...,

then from Fatou’s lemma, we have

(5.5) ∥ u b ∗ ∥ H 1 ≤ lim inf n → ∞ ∥ u n ∥ H 1 ≤ C 1 < A ( p ) w Ω 1 H 1 .

(4.5) and (5.5) imply that

hb,ub∗′′(1)<0,

provided that

4−p4b+p−qqSpqρ04−q|m|q∗<Γ1,

where Γ₁ is defined by (4.2). Thus, ub∗∈Mb− . On the other hand, observe that

αb−=Ib(ub∗)≤Ib(tb−wΩ1)≤p−22pA2(p)1+B(p,q,k,m)wΩ1H12,

then ub∗∈Mb(1) . Furthermore, Ib(ub∗)=Ib(|ub∗|)=αb− . Thus, from Lemma 2.1, we may assume that ub∗ is a positive solution of Eq.(P). The proof is complete. □

Proof of Theorem 1.1: From Theorems 3.4 and 5.4, there exists a constant Π1>0 such that for any 0<b+|m|q∗<Π1 , Eq.(P) admits two positive solutions ub1 and ub∗ that satisfy

∥ub1∥H1<(2−q)Spp(p−q)kmax1/(p−2)<∥ub∗∥H1<A(p)wΩ1H1,

and

Ib(ub1)<0<Ib(ub∗).

The proof is complete.

Proof of Theorem 1.2: From Theorems 3.4, 3.5 and 5.4, there exists a constant Π2=min{b0,Π1}>0 such that for any 0<b+|m|q∗<Π2 , Eq.(P) admits three positive solutions ub1 , ub2 and ub∗ that satisfy

∥ub1∥H1<ρ0:=(2−q)Spp(p−q)kmax1/(p−2)<min∥ub∗∥H1,∥ub2∥H1,

and

maxIb(ub1),Ib(ub2)<0<Ib(ub∗).

Then by Lemma 2.2 and (4.3), ub∗∈Mb(2) . Hence,

∥ub1∥H1<ρ0<∥ub∗∥H1<∥ub2∥H1.

The proof is complete.

Acknowledgments

G. Che was supported by the National Natural Science Foundation of China (Grant No. 12001114) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110275). T. F. Wu was supported by the Ministry of Science and Technology, Taiwan (110-2115-M-390-006-MY2).

Conflict of interest

Conflict of interest statement: Authors state no conflict of interest.

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Received: 2021-07-30

Accepted: 2021-10-10

Published Online: 2021-11-11

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

Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

Abstract

1 Introduction

Theorem 1.1

Theorem 1.2

Remark 1.1

2 Preliminaries

Lemma 2.1

Lemma 2.2

3 Existence of two negative–energy positive solutions

Lemma 3.1

Lemma 3.2

Lemma 3.3

Theorem 3.4

Theorem 3.5

4 The filtration of Nehari manifold

Lemma 4.1

Lemma 4.2

5 Proofs of main results

Lemma 5.1

Proposition 5.2

Proposition 5.3

Theorem 5.4

Acknowledgments

References

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