Existence and multiplicity of positive solutions for the following semilinear elliptic equation: βˆ’Ξ”π‘’+𝑒=π‘Ž(π‘₯)|𝑒|π‘βˆ’2𝑒+πœ†π‘(π‘₯)|𝑒|π‘žβˆ’2𝑒 in ℝ𝑁, π‘’βˆˆπ»1(ℝ𝑁), are established, where πœ†>0,1<π‘ž<2<𝑝<2βˆ—(2βˆ—=2𝑁/(π‘βˆ’2) if 𝑁β‰₯3,2βˆ—=∞ if 𝑁=1,2), π‘Ž, 𝑏 satisfy suitable conditions, and 𝑏 maybe changes sign in ℝ𝑁. The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.

1. Introduction

In this paper, we deal with the multiplicity of positive solutions for the following semilinear elliptic equation:βˆ’Ξ”π‘’+𝑒=π‘Ž(π‘₯)π‘’π‘βˆ’1+πœ†π‘(π‘₯)π‘’π‘žβˆ’1inℝ𝑁,𝑒>0inℝ𝑁,π‘’βˆˆπ»1ℝ𝑁,(πΈπ‘Ž,πœ†π‘) where πœ†>0, 1<π‘ž<2<𝑝<2βˆ— (2βˆ—=2𝑁/(π‘βˆ’2) if 𝑁β‰₯3, 2βˆ—=∞ if 𝑁=1,2) and π‘Ž,𝑏 are measurable functions and satisfy the following conditions: (π‘Ž1)0<π‘ŽβˆˆπΏβˆž(ℝ𝑁), where lim|π‘₯|β†’βˆžπ‘Ž(π‘₯)=1, and there exist 𝐢0>0 and 𝛿0>0 such that π‘Ž(π‘₯)β‰₯1βˆ’πΆ0π‘’βˆ’π›Ώ0|π‘₯|βˆ€π‘₯βˆˆβ„π‘.(1.1)(𝑏1)π‘βˆˆπΏπ‘žβˆ—(ℝ𝑁)(π‘žβˆ—=𝑝/(π‘βˆ’π‘ž)), 𝑏+=max{𝑏,0}β‰’0, π‘βˆ’=max{βˆ’π‘,0} is bounded and π‘βˆ’ has a compact support 𝐾 in ℝ𝑁. (𝑏2)There exist 𝐢1>0, 0<𝛿1<min{𝛿0,π‘ž} and 𝑅0>0 such that 𝑏+(π‘₯)βˆ’π‘(π‘₯)β‰₯𝐢1π‘’βˆ’π›Ώ1|π‘₯|βˆ€|π‘₯|β‰₯𝑅0.(1.2)

Semilinear elliptic equations with concave-convex nonlinearities in bounded domains are widely studied. For example, Ambrosetti et al. [1] considered the following equation:βˆ’Ξ”π‘’=π‘’π‘βˆ’1+πœ†π‘’π‘žβˆ’1inΞ©,𝑒>0inΞ©,𝑒=0onπœ•Ξ©,(πΈπœ†) where πœ†>0, 1<π‘ž<2<𝑝<2βˆ—. They proved that there exists πœ†0>0 such that (πΈπœ†) admits at least two positive solutions for all πœ†βˆˆ(0,πœ†0), has one positive solution for πœ†=πœ†0 and no positive solution for πœ†>πœ†0. Actually, Adimurthi et al. [2], Damascelli et al. [3], Korman [4], Ouyang and Shi [5], and Tang [6] proved that there exists πœ†0>0 such that (πΈπœ†) in the unit ball 𝐡𝑁(0;1) has exactly two positive solutions for πœ†βˆˆ(0,πœ†0), has exactly one positive solution for πœ†=πœ†0 and no positive solution exists for πœ†>πœ†0. For more general results of (πΈπœ†) (involving sign-changing weights) in bounded domains; see, the work of Ambrosetti et al. in [7], of Garcia Azorero et al. in [8], of Brown and Wu in [9], of Brown and Zhang in [10], of Cao and Zhong in [11], of de Figueiredo et al. in [12], and their references.

However, little has been done for this type of problem in ℝ𝑁. We are only aware of the works [13–17] which studied the existence of solutions for some related concave-convex elliptic problems (not involving sign-changing weights). Furthermore, we do not know of any results for concave-convex elliptic problems involving sign-changing weight functions except [18, 19]. Wu in [18] have studied the multiplicity of positive solutions for the following equation involving sign-changing weights:βˆ’Ξ”π‘’+𝑒=π‘“πœ†(π‘₯)π‘’π‘žβˆ’1+π‘”πœ‡(π‘₯)π‘’π‘βˆ’1inℝ𝑁,𝑒>0inℝ𝑁,π‘’βˆˆπ»1ℝ𝑁,(πΈπ‘“πœ†,π‘”πœ‡) where 1<π‘ž<2<𝑝<2βˆ— the parameters πœ†,πœ‡β‰₯0. He also assumed that π‘“πœ†(π‘₯)=πœ†π‘“+(π‘₯)+π‘“βˆ’(π‘₯) is sign chaning and π‘”πœ‡(π‘₯)=π‘Ž(π‘₯)+πœ‡π‘(π‘₯), where π‘Ž and 𝑏 satisfy suitable conditions and proved that (πΈπ‘“πœ†,π‘”πœ‡) has at least four positive solutions.

In a recent work [19], Hsu and Lin have studied (πΈπ‘Ž,πœ†π‘) in ℝ𝑁 with a sign-changing weight function. They proved there exists πœ†0>0 such that (πΈπ‘Ž,πœ†π‘) has at least two positive solutions for all πœ†βˆˆ(0,πœ†0) provided that π‘Ž, 𝑏 satisfy suitable conditions and 𝑏 maybe changes sign in ℝ𝑁.

Continuing our previous work [19], we consider (πΈπ‘Ž,πœ†π‘) in ℝ𝑁 involving a sign-changing weight function with suitable assumptions which are different from the assumptions in [19].

In order to describe our main result, we need to defineΞ›0=ξ‚΅2βˆ’π‘ž(π‘βˆ’π‘ž)β€–π‘Žβ€–πΏβˆžξ‚Ά(2βˆ’π‘ž)/(π‘βˆ’2)ξ‚΅π‘βˆ’2‖‖𝑏(π‘βˆ’π‘ž)+β€–β€–πΏπ‘žβˆ—ξ‚Άπ‘†π‘π‘(2βˆ’π‘ž)/2(π‘βˆ’2)+π‘ž/2>0,(1.3) where β€–π‘Žβ€–πΏβˆž=supπ‘₯βˆˆβ„π‘π‘Ž(π‘₯), ‖𝑏+β€–πΏπ‘žβˆ—βˆ«=(ℝ𝑁|𝑏+(π‘₯)|π‘žβˆ—π‘‘π‘₯)1/π‘žβˆ— and 𝑆𝑝 is the best Sobolev constant for the imbedding of 𝐻1(ℝ𝑁) into 𝐿𝑝(ℝ𝑁).

Theorem 1.1. Assume that (π‘Ž1), (𝑏1)-(𝑏2) hold. If πœ†βˆˆ(0,(π‘ž/2)Ξ›0), (πΈπ‘Ž,πœ†π‘) admits at least two positive solutions in 𝐻1(ℝ𝑁).

This paper is organized as follows. In Section 2, we give some notations and preliminary results. In Section 3, we establish the existence of a local minimum. In Section 4, we prove the existence of a second solution of (πΈπ‘Ž,πœ†π‘).

At the end of this section, we explain some notations employed. In the following discussions, we will consider 𝐻=𝐻1(ℝ𝑁) with the norm βˆ«β€–π‘’β€–=(ℝ𝑁(|βˆ‡π‘’|2+𝑒2)𝑑π‘₯)1/2. We denote by 𝑆𝑝 the best constant which is given by𝑆𝑝=infπ‘’βˆˆπ»β§΅{0}‖𝑒‖2ξ€·βˆ«β„π‘|𝑒|𝑝𝑑π‘₯2/𝑝.(1.4) The dual space of 𝐻 will be denoted by π»βˆ—. βŸ¨β‹…,β‹…βŸ© denote the dual pair between π»βˆ— and 𝐻. We denote the norm in 𝐿𝑠(ℝ𝑁) by ‖⋅‖𝐿𝑠 for 1β‰€π‘ β‰€βˆž. 𝐡𝑁(π‘₯;π‘Ÿ) is a ball in ℝ𝑁 centered at π‘₯ with radius π‘Ÿ. π‘œπ‘›(1) denotes π‘œπ‘›(1)β†’0 as π‘›β†’βˆž. 𝐢, 𝐢𝑖 will denote various positive constants, the exact values of which are not important.

2. Preliminary Results

Associated with (1.3), the energy functional π½πœ†βˆΆπ»β†’β„π‘ defined byπ½πœ†(1𝑒)=2‖𝑒‖2βˆ’1π‘ξ€œβ„π‘π‘Ž(π‘₯)|𝑒|π‘πœ†π‘‘π‘₯βˆ’π‘žξ€œβ„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯,(2.1) for all π‘’βˆˆπ» is considered. It is well-known that π½πœ†βˆˆπΆ1(𝐻,ℝ) and the solutions of (πΈπ‘Ž,πœ†π‘) are the critical points of π½πœ†.

Since π½πœ† is not bounded from below on 𝐻, we will work on the Nehari manifold. For πœ†>0 we defineπ’©πœ†=ξ€½ξ«π½π‘’βˆˆπ»β§΅{0}βˆΆξ…žπœ†ξ¬ξ€Ύ.(𝑒),𝑒=0(2.2) Note that π’©πœ† contains all nonzero solutions of (πΈπ‘Ž,πœ†π‘) and π‘’βˆˆπ’©πœ† if and only ifξ«π½ξ…žπœ†(𝑒),𝑒=‖𝑒‖2βˆ’ξ€œβ„π‘π‘Ž(π‘₯)|𝑒|π‘ξ€œπ‘‘π‘₯βˆ’πœ†β„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯=0.(2.3)

Lemma 2.1. π½πœ† is coercive and bounded from below on π’©πœ†.

Proof. If π‘’βˆˆπ’©πœ†, then by (𝑏1), (2.3), and the HΓΆlder and Sobolev inequalities, one has π½πœ†(𝑒)=π‘βˆ’22𝑝‖𝑒‖2ξ‚΅βˆ’πœ†π‘βˆ’π‘žξ‚Άξ€œπ‘π‘žβ„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯(2.4)β‰₯π‘βˆ’22𝑝‖𝑒‖2ξ‚΅βˆ’πœ†π‘βˆ’π‘žξ‚Άπ‘†π‘π‘žπ‘βˆ’π‘ž/2‖‖𝑏+β€–β€–πΏπ‘žβˆ—β€–π‘’β€–π‘ž.(2.5) Since π‘ž<2<𝑝, it follows that π½πœ† is coercive and bounded from below on π’©πœ†.

The Nehari manifold is closely linked to the behavior of the function of the form πœ‘π‘’βˆΆπ‘‘β†’π½πœ†(𝑑𝑒) for 𝑑>0. Such maps are known as fibering maps and were introduced by DrΓ‘bek and Pohozaev in [20] and are also discussed by Brown and Zhang in [10]. If π‘’βˆˆπ», we have πœ‘π‘’π‘‘(𝑑)=22‖𝑒‖2βˆ’π‘‘π‘π‘ξ€œβ„π‘π‘Ž(π‘₯)|𝑒|𝑝𝑑𝑑π‘₯βˆ’π‘žπ‘žπœ†ξ€œβ„π‘π‘(π‘₯)|𝑒|π‘žπœ‘π‘‘π‘₯,ξ…žπ‘’(𝑑)=𝑑‖𝑒‖2βˆ’π‘‘π‘βˆ’1ξ€œβ„π‘π‘Ž(π‘₯)|𝑒|𝑝𝑑π‘₯βˆ’π‘‘π‘žβˆ’1πœ†ξ€œβ„π‘π‘(π‘₯)|𝑒|π‘žπœ‘π‘‘π‘₯,π‘’ξ…žξ…ž(𝑑)=‖𝑒‖2βˆ’(π‘βˆ’1)π‘‘π‘βˆ’2ξ€œβ„π‘π‘Ž(π‘₯)|𝑒|𝑝𝑑π‘₯βˆ’(π‘žβˆ’1)π‘‘π‘žβˆ’2πœ†ξ€œβ„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯.(2.6) It is easy to see that π‘‘πœ‘ξ…žπ‘’(𝑑)=‖𝑑𝑒‖2βˆ’ξ€œβ„π‘π‘Ž(π‘₯)|𝑑𝑒|π‘ξ€œπ‘‘π‘₯βˆ’πœ†β„π‘π‘(π‘₯)|𝑑𝑒|π‘žπ‘‘π‘₯,(2.7) and so, for π‘’βˆˆπ»β§΅{0} and 𝑑>0, πœ‘ξ…žπ‘’(𝑑)=0 if and only if π‘‘π‘’βˆˆπ’©πœ† that is, the critical points of πœ‘π‘’ correspond to the points on the Nehari manifold. In particular, πœ‘ξ…žπ‘’(1)=0 if and only if π‘’βˆˆπ’©πœ†. Thus, it is natural to split π’©πœ† into three parts corresponding to local minima, local maxima, and points of inflection. Accordingly, we define 𝒩+πœ†=ξ€½π‘’βˆˆπ’©πœ†βˆΆπœ‘π‘’ξ…žξ…žξ€Ύ,𝒩(1)>00πœ†=ξ€½π‘’βˆˆπ’©πœ†βˆΆπœ‘π‘’ξ…žξ…žξ€Ύ,𝒩(1)=0βˆ’πœ†=ξ€½π‘’βˆˆπ’©πœ†βˆΆπœ‘π‘’ξ…žξ…žξ€Ύ,(1)<0(2.8) and note that if π‘’βˆˆπ’©πœ†, that is, πœ‘ξ…žπ‘’(1)=0, thenπœ‘π‘’ξ…žξ…ž(1)=(2βˆ’π‘ž)‖𝑒‖2ξ€œβˆ’(π‘βˆ’π‘ž)β„π‘π‘Ž(π‘₯)|𝑒|𝑝𝑑π‘₯,(2.9)=(2βˆ’π‘)‖𝑒‖2ξ€œβˆ’(π‘žβˆ’π‘)πœ†β„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯.(2.10)

We now derive some basic properties of 𝒩+πœ†, 𝒩0πœ†, and π’©βˆ’πœ†.

Lemma 2.2. Suppose that 𝑒0 is a local minimizer for π½πœ† on π’©πœ† and 𝑒0βˆ‰π’©0πœ†, then π½ξ…žπœ†(𝑒0)=0 in π»βˆ—.

Proof. See the work of Brown and Zhang in [10, Theorem 2.3].

Lemma 2.3. If πœ†βˆˆ(0,Ξ›0), then 𝒩0πœ†=βˆ….

Proof. We argue by contradiction. Suppose that there exists πœ†βˆˆ(0,Ξ›0) such that 𝒩0πœ†β‰ βˆ…. Then for π‘’βˆˆπ’©0πœ† by (2.9) and the Sobolev inequality, we have 2βˆ’π‘žπ‘βˆ’π‘žβ€–π‘’β€–2=ξ€œβ„π‘π‘Ž(π‘₯)|𝑒|𝑝𝑑π‘₯β‰€β€–π‘Žβ€–πΏβˆžπ‘†π‘βˆ’π‘/2‖𝑒‖𝑝,(2.11) and so ‖𝑒‖β‰₯2βˆ’π‘ž(π‘βˆ’π‘ž)β€–π‘Žβ€–πΏβˆžξ‚Ά1/(π‘βˆ’2)𝑆𝑝𝑝/2(π‘βˆ’2).(2.12) Similarly, using (2.10), HΓΆlder and Sobolev inequalities, we have ‖𝑒‖2=πœ†π‘βˆ’π‘žξ€œπ‘βˆ’2ℝ𝑁𝑏(π‘₯)|𝑒|π‘žπ‘‘π‘₯β‰€πœ†π‘βˆ’π‘žβ€–β€–π‘π‘βˆ’2+β€–β€–πΏπ‘žβˆ—π‘†π‘βˆ’π‘ž/2β€–π‘’β€–π‘ž(2.13) which implies β€–ξ‚΅πœ†π‘’β€–β‰€π‘βˆ’π‘žβ€–β€–π‘π‘βˆ’2+β€–β€–πΏπ‘žβˆ—ξ‚Ά1/(2βˆ’π‘ž)π‘†π‘βˆ’π‘ž/2(2βˆ’π‘ž).(2.14) Hence, we must have ξ‚΅πœ†β‰₯2βˆ’π‘ž(π‘βˆ’π‘ž)β€–π‘Žβ€–πΏβˆžξ‚Ά(2βˆ’π‘ž)/(π‘βˆ’2)ξ‚΅π‘βˆ’2‖‖𝑏(π‘βˆ’π‘ž)+β€–β€–πΏπ‘žβˆ—ξ‚Άπ‘†π‘π‘(2βˆ’π‘ž)/2(π‘βˆ’2)+π‘ž/2=Ξ›0(2.15) which is a contradiction.

In order to get a better understanding of the Nehari manifold and fibering maps, we consider the function πœ“π‘’βˆΆβ„+→ℝ defined by πœ“π‘’(𝑑)=𝑑2βˆ’π‘žβ€–π‘’β€–2βˆ’π‘‘π‘βˆ’π‘žξ€œβ„π‘π‘Ž(π‘₯)|𝑒|𝑝𝑑π‘₯for𝑑>0.(2.16) Clearly, π‘‘π‘’βˆˆπ’©πœ† if and only if πœ“π‘’βˆ«(𝑑)=πœ†β„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯. Moreover,πœ“ξ…žπ‘’(𝑑)=(2βˆ’π‘ž)𝑑1βˆ’π‘žβ€–π‘’β€–2βˆ’(π‘βˆ’π‘ž)π‘‘π‘βˆ’π‘žβˆ’1ξ€œβ„π‘π‘Ž(π‘₯)|𝑒|𝑝𝑑π‘₯for𝑑>0,(2.17) and so it is easy to see that if π‘‘π‘’βˆˆπ’©πœ†, then π‘‘π‘žβˆ’1πœ“ξ…žπ‘’(𝑑)=πœ‘π‘’ξ…žξ…ž(𝑑). Hence, π‘‘π‘’βˆˆπ’©+πœ† (or π‘‘π‘’βˆˆπ’©βˆ’πœ†) if and only if πœ“ξ…žπ‘’(𝑑)>0 (or πœ“ξ…žπ‘’(𝑑)<0).

Let π‘’βˆˆπ»β§΅{0}. Then, by (2.17), πœ“π‘’ has a unique critical point at 𝑑=𝑑max(𝑒), where𝑑max((𝑒)=2βˆ’π‘ž)‖𝑒‖2∫(π‘βˆ’π‘ž)β„π‘π‘Ž(π‘₯)|𝑒|𝑝ξƒͺ𝑑π‘₯1/(π‘βˆ’2)>0,(2.18) and clearly πœ“π‘’ is strictly increasing on (0,𝑑max(𝑒)) and strictly decreasing on (𝑑max(𝑒),∞) with limπ‘‘β†’βˆžπœ“π‘’(𝑑)=βˆ’βˆž. Moreover, if πœ†βˆˆ(0,Ξ›0), then πœ“π‘’ξ€·π‘‘maxξ€Έ=(𝑒)2βˆ’π‘žξ‚Άπ‘βˆ’π‘ž(2βˆ’π‘ž)/(π‘βˆ’2)βˆ’ξ‚΅2βˆ’π‘žξ‚Άπ‘βˆ’π‘ž(π‘βˆ’π‘ž)/(π‘βˆ’2)‖𝑒‖2(π‘βˆ’π‘ž)/(π‘βˆ’2)ξ€·βˆ«β„π‘π‘Ž(π‘₯)|𝑒|𝑝𝑑π‘₯(2βˆ’π‘ž)/(π‘βˆ’2)=β€–π‘’β€–π‘žξ‚΅π‘βˆ’2π‘βˆ’π‘žξ‚Άξ‚΅2βˆ’π‘žξ‚Άπ‘βˆ’π‘ž2βˆ’π‘ž/π‘βˆ’2ξƒ©β€–π‘’β€–π‘βˆ«β„π‘π‘Ž(π‘₯)|𝑒|𝑝ξƒͺ𝑑π‘₯(2βˆ’π‘ž)/(π‘βˆ’2)β‰₯β€–π‘’β€–π‘žξ‚΅π‘βˆ’2π‘βˆ’π‘žξ‚Άξ‚΅2βˆ’π‘žξ‚Άπ‘βˆ’π‘ž(2βˆ’π‘ž)/(π‘βˆ’2)𝑆𝑝𝑝(2βˆ’π‘ž)/2(π‘βˆ’2)‖‖𝑏>πœ†+β€–β€–πΏπ‘žβˆ—π‘†π‘βˆ’π‘ž/2β€–π‘’β€–π‘žξ€œβ‰₯πœ†β„π‘π‘+(π‘₯)|𝑒|π‘žξ€œπ‘‘π‘₯β‰₯πœ†β„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯.(2.19)

Therefore, we have the following lemma.

Lemma 2.4. Let πœ†βˆˆ(0,Ξ›0) and π‘’βˆˆπ»β§΅{0}. (i)If πœ†βˆ«β„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯≀0, then there exists a unique π‘‘βˆ’=π‘‘βˆ’(𝑒)>𝑑max(𝑒) such that π‘‘βˆ’π‘’βˆˆπ’©βˆ’πœ†, πœ‘π‘’ is inceasing on (0,π‘‘βˆ’) and decreasing on (π‘‘βˆ’,∞). Moreover, π½πœ†(π‘‘βˆ’π‘’)=sup𝑑β‰₯0π½πœ†(𝑑𝑒).(2.20)(ii)If πœ†βˆ«β„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯>0, then there exist unique 0<𝑑+=𝑑+(𝑒)<𝑑max(𝑒)<π‘‘βˆ’=π‘‘βˆ’(𝑒) such that 𝑑+π‘’βˆˆπ’©+πœ†, π‘‘βˆ’π‘’βˆˆπ’©βˆ’πœ†, πœ‘π‘’ is decreasing on (0,𝑑+), inceasing on (𝑑+,π‘‘βˆ’) and decreasing on (π‘‘βˆ’,∞)π½πœ†ξ€·π‘‘+𝑒=inf0≀𝑑≀𝑑max(𝑒)π½πœ†(𝑑𝑒),π½πœ†(π‘‘βˆ’π‘’)=sup𝑑β‰₯𝑑+π½πœ†(𝑑𝑒).(2.21)(iii)π’©βˆ’πœ†={π‘’βˆˆπ»β§΅{0}βˆΆπ‘‘βˆ’(𝑒)=(1/‖𝑒‖)π‘‘βˆ’(𝑒/‖𝑒‖)=1}.(iv)There exists a continuous bijection between π‘ˆ={π‘’βˆˆπ»β§΅{0}βˆΆβ€–π‘’β€–=1} and π’©βˆ’πœ†. In particular, π‘‘βˆ’ is a continuous function for π‘’βˆˆπ»β§΅{0}.

Proof. See the work of Hsu and Lin in [19, Lemma 2.5].

We remark that it follows Lemma 2.4, π’©πœ†=𝒩+πœ†βˆͺπ’©βˆ’πœ† for all πœ†βˆˆ(0,Ξ›0). Furthermore, by Lemma 2.4 it follows that 𝒩+πœ† and π’©βˆ’πœ† are non-empty and by Lemma 2.1 we may defineπ›Όπœ†=infπ‘’βˆˆπ’©πœ†π½πœ†(𝑒),𝛼+πœ†=infπ‘’βˆˆπ’©+πœ†π½πœ†(𝑒),π›Όβˆ’πœ†=infπ‘’βˆˆπ’©βˆ’πœ†π½πœ†(𝑒).(2.22)

Theorem 2.5. (i) If πœ†βˆˆ(0,Ξ›0), then we have π›Όπœ†β‰€π›Ό+πœ†<0.
(ii) If πœ†βˆˆ(0,(π‘ž/2)Ξ›0), then π›Όβˆ’πœ†>𝑑0 for some 𝑑0>0.
In particular, for each πœ†βˆˆ(0,(π‘ž/2)Ξ›0), we have 𝛼+πœ†=π›Όπœ†<0<π›Όβˆ’πœ†.

Proof. See the work of Hsu and Lin in [19, Theorem 3.1].

Remark 2.6. (i) If πœ†βˆˆ(0,Ξ›0), then by (2.9), HΓΆlder and Sobolev inequalities, for each π‘’βˆˆπ’©+πœ† we have ‖𝑒‖2<π‘βˆ’π‘žπœ†ξ€œπ‘βˆ’2ℝ𝑁𝑏(π‘₯)|𝑒|π‘žβ‰€π‘‘π‘₯π‘βˆ’π‘žπ‘βˆ’2πœ†β€–π‘β€–πΏπ‘žβˆ—π‘†π‘βˆ’π‘ž/2β€–π‘’β€–π‘žβ‰€π‘βˆ’π‘žΞ›π‘βˆ’20β€–π‘β€–πΏπ‘žβˆ—π‘†π‘βˆ’π‘ž/2β€–π‘’β€–π‘ž,(2.23) and so β€–ξ‚΅π‘’β€–β‰€π‘βˆ’π‘žΞ›π‘βˆ’20β€–π‘β€–πΏπ‘žβˆ—π‘†π‘βˆ’π‘ž/2ξ‚Ά1/(2βˆ’π‘ž)βˆ€π‘’βˆˆπ’©+πœ†.(2.24)
 (ii) If πœ†βˆˆ(0,(π‘ž/2)Ξ›0), then by Lemma 2.4(i), (ii) and Theorem 2.5(ii), for each π‘’βˆˆπ’©βˆ’πœ† we haveπ½πœ†(𝑒)=sup𝑑β‰₯0π½πœ†(𝑑𝑒)β‰₯π›Όβˆ’πœ†>0.(2.25)

3. Existence of a Positive Solution

First, we define the Palais-Smale (simply by (PS)) sequences, (PS)-values, and (PS)-conditions in 𝐻 for π½πœ† as follows.

Definition 3.1. (i) For π‘βˆˆβ„, a sequence {𝑒𝑛} is a (PS)𝑐-sequence in 𝐻 for π½πœ† if π½πœ†(𝑒𝑛)=𝑐+π‘œπ‘›(1) and π½ξ…žπœ†(𝑒𝑛)=π‘œπ‘›(1) strongly in π»βˆ— as π‘›β†’βˆž.
(ii) π‘βˆˆβ„ is a (PS)-value in 𝐻 for π½πœ† if there exists a (PS)𝑐-sequence in 𝐻 for π½πœ†.
(iii) π½πœ† satisfies the (PS)𝑐-condition in 𝐻 if any (PS)𝑐-sequence {𝑒𝑛} in 𝐻 for π½πœ† contains a convergent subsequence.

Now we will ensure that there are (PS)𝛼+πœ†-sequence and (PS)π›Όβˆ’πœ†-sequencein on π’©πœ† and π’©βˆ’πœ†, respectively, for the functional π½πœ†.

Proposition 3.2. If πœ†βˆˆ(0,(π‘ž/2)Ξ›0), then (i)there exists a (PS)π›Όπœ†-sequence {𝑒𝑛}βŠ‚π’©πœ† in 𝐻 for π½πœ†.(ii)there exists a (PS)π›Όβˆ’πœ†-sequence {𝑒𝑛}βŠ‚π’©βˆ’πœ† in 𝐻 for π½πœ†.

Proof. See Wu [21, Proposition 9].

Now, we establish the existence of a local minimum for π½πœ† on 𝒩+πœ†.

Theorem 3.3. Assume (π‘Ž1) and (𝑏1) hold. If πœ†βˆˆ(0,(π‘ž/2)Ξ›0), then there exists π‘’πœ†βˆˆπ’©+πœ† such that (i)π½πœ†(π‘’πœ†)=π›Όπœ†=𝛼+πœ†<0, (ii)π‘’πœ† is a positive solution of (πΈπ‘Ž,πœ†π‘), (iii)β€–π‘’πœ†β€–β†’0 as πœ†β†’0+.

Proof. From Proposition 3.2(i) it follows that there exists {𝑒𝑛}βŠ‚π’©πœ† satisfying π½πœ†ξ€·π‘’π‘›ξ€Έ=π›Όπœ†+π‘œπ‘›(1)=𝛼+πœ†+π‘œπ‘›(1),π½ξ…žπœ†ξ€·π‘’π‘›ξ€Έ=π‘œπ‘›(1)inπ»βˆ—.(3.1) By Lemma 2.1 we infer that {𝑒𝑛} is bounded on 𝐻. Passing to a subsequence (Still denoted by {𝑒𝑛}), there exists π‘’πœ†βˆˆπ» such that as π‘›β†’βˆžπ‘’π‘›β‡€π‘’πœ†weaklyin𝑒𝐻,π‘›βŸΆπ‘’πœ†almosteverywhereinℝ𝑁,π‘’π‘›βŸΆπ‘’πœ†stronglyin𝐿𝑠locξ€·β„π‘ξ€Έβˆ€1≀𝑠<2βˆ—.(3.2) By (𝑏1), Egorov theorem and HΓΆlder inequality, we have πœ†ξ€œβ„π‘||𝑒𝑏(π‘₯)𝑛||π‘žξ€œπ‘‘π‘₯=πœ†β„π‘||𝑒𝑏(π‘₯)πœ†||π‘žπ‘‘π‘₯+π‘œπ‘›(1)asπ‘›βŸΆβˆž.(3.3) By (3.1) and (3.2), it is easy to see that π‘’πœ† is a solution of (πΈπ‘Ž,πœ†π‘). From π‘’π‘›βˆˆπ’©πœ† and (2.4), we deduce that πœ†ξ€œβ„π‘||𝑒𝑏(π‘₯)𝑛||π‘žπ‘‘π‘₯=π‘ž(π‘βˆ’2)‖‖𝑒2(π‘βˆ’π‘ž)𝑛‖‖2βˆ’π‘π‘žπ½π‘βˆ’π‘žπœ†ξ€·π‘’π‘›ξ€Έ.(3.4) Let π‘›β†’βˆž in (3.4). By (3.1), (3.3) and π›Όπœ†<0, we get πœ†ξ€œβ„π‘||𝑒𝑏(π‘₯)πœ†||π‘žπ‘‘π‘₯β‰₯βˆ’π‘π‘žπ›Όπ‘βˆ’π‘žπœ†>0.(3.5) Thus, π‘’πœ†βˆˆπ’©πœ† is a nonzero solution of (πΈπ‘Ž,πœ†π‘).
Next, we prove that π‘’π‘›β†’π‘’πœ† strongly in 𝐻 and π½πœ†(π‘’πœ†)=π›Όπœ†. From the fact 𝑒𝑛,π‘’πœ†βˆˆπ’©πœ† and applying Fatou's lemma, we getπ›Όπœ†β‰€π½πœ†ξ€·π‘’πœ†ξ€Έ=π‘βˆ’2‖‖𝑒2π‘πœ†β€–β€–2βˆ’π‘βˆ’π‘žπœ†ξ€œπ‘π‘žβ„π‘||𝑒𝑏(π‘₯)πœ†||π‘žπ‘‘π‘₯≀liminfπ‘›β†’βˆžξ‚΅π‘βˆ’2‖‖𝑒2𝑝𝑛‖‖2βˆ’π‘βˆ’π‘žπœ†ξ€œπ‘π‘žβ„π‘||𝑒𝑏(π‘₯)𝑛||π‘žξ‚Άπ‘‘π‘₯≀liminfπ‘›β†’βˆžπ½πœ†ξ€·π‘’π‘›ξ€Έ=π›Όπœ†.(3.6) This implies that π½πœ†(π‘’πœ†)=π›Όπœ† and limπ‘›β†’βˆžβ€–π‘’π‘›β€–2=β€–π‘’πœ†β€–2. Standard argument shows that π‘’π‘›β†’π‘’πœ† strongly in 𝐻. By Theorem 2.5, for all πœ†βˆˆ(0,(π‘ž/2)Ξ›0) we have that π‘’πœ†βˆˆπ’©πœ† and π½πœ†(π‘’πœ†)=𝛼+πœ†<π›Όβˆ’πœ† which implies π‘’πœ†βˆˆπ’©+πœ†. Since π½πœ†(π‘’πœ†)=π½πœ†(|π‘’πœ†|) and |π‘’πœ†|βˆˆπ’©+πœ†, by Lemma 2.2 we may assume that π‘’πœ† is a nonzero nonnegative solution of (πΈπ‘Ž,πœ†π‘). By Harnack inequality [22] we deduce that π‘’πœ†>0 in ℝ𝑁. Finally, by (2.10), HΓΆlder and Sobolev inequlities, β€–β€–π‘’πœ†β€–β€–2βˆ’π‘ž<πœ†π‘βˆ’π‘žβ€–β€–π‘π‘βˆ’2+β€–β€–πΏπ‘žβˆ—π‘†π‘βˆ’π‘ž/2,(3.7) and thus we conclude the proof.

4. Second Positive Solution

In this section, we will establish the existence of the second positive solution of (πΈπ‘Ž,πœ†π‘) by proving that π½πœ† satisfies the (PS)π›Όβˆ’πœ†-condition.

Lemma 4.1. Assume that (π‘Ž1) and (𝑏1) hold. If {𝑒𝑛}βŠ‚π» is a (PS)𝑐-sequence for π½πœ†, then {𝑒𝑛} is bounded in 𝐻.

Proof. See the work of Hsu and Lin in [19, Lemma 4.1].

Let us introduce the problem at infinity associated with (πΈπ‘Ž,πœ†π‘):βˆ’Ξ”π‘’+𝑒=π‘’π‘βˆ’1inℝ𝑁,π‘’βˆˆπ»,𝑒>0inℝ𝑁.(𝐸∞) We state some known results for problem (𝐸∞). First of all, we recall that by Lions [23] has studied the following minimization problem closely related to problem (𝐸∞):π‘†βˆžξ€½π½=inf∞(𝑒)βˆΆπ‘’βˆˆπ»,𝑒≒0,(𝐽∞)ξ…žξ€Ύ(𝑒)=0>0,(4.1) where 𝐽∞(𝑒)=(1/2)‖𝑒‖2βˆ«βˆ’(1/𝑝)ℝ𝑁|𝑒|𝑝𝑑π‘₯. Note that a minimum exists and is attained by a ground state 𝑀0>0 in ℝ𝑁 such that π‘†βˆž=π½βˆžξ€·π‘€0ξ€Έ=sup𝑑β‰₯0π½βˆžξ€·π‘‘π‘€0ξ€Έ=ξ‚΅12βˆ’1𝑝𝑆𝑝𝑝/(π‘βˆ’2),(4.2) where 𝑆𝑝=infπ‘’βˆˆπ»β§΅{0}‖𝑒‖2∫/(ℝ𝑁|𝑒|𝑝𝑑π‘₯)2/𝑝. Gidas et al. [24] showed that for every πœ€>0, there exist positive constants πΆπœ€, 𝐢2 such that for all π‘₯βˆˆβ„π‘,πΆπœ€exp(βˆ’(1+πœ€)|π‘₯|)≀𝑀0(π‘₯)≀𝐢2exp(βˆ’|π‘₯|).(4.3) We define𝑀𝑛(π‘₯)=𝑀0(π‘₯βˆ’π‘›π‘’),where𝑒=(0,0,…,0,1)isaunitvectorinℝ𝑁.(4.4) Clearly, 𝑀𝑛(π‘₯)∈𝐻.

Lemma 4.2. Let Ξ© be a domain in ℝ𝑁. If π‘“βˆΆΞ©β†’β„ satisfies ξ€œΞ©||𝑓(π‘₯)π‘’πœŽ|π‘₯|||𝑑π‘₯<∞forsome𝜎>0,(4.5) then ξ‚΅ξ€œΞ©π‘“(π‘₯)π‘’βˆ’πœŽ|π‘₯βˆ’Μƒπ‘₯|𝑒𝑑π‘₯𝜎|Μƒπ‘₯|=ξ€œΞ©π‘“(π‘₯)π‘’πœŽβŸ¨π‘₯,Μƒπ‘₯⟩/|Μƒπ‘₯|𝑑π‘₯+π‘œ(1)as||||Μƒπ‘₯⟢∞.(4.6)

Proof. We know 𝜎|Μƒπ‘₯|β‰€πœŽ|π‘₯|+𝜎|π‘₯βˆ’Μƒπ‘₯|. Then, |||𝑓(π‘₯)π‘’βˆ’πœŽ|π‘₯βˆ’Μƒπ‘₯|π‘’πœŽ|Μƒπ‘₯||||≀||𝑓(π‘₯)π‘’πœŽ|π‘₯|||.(4.7) Since βˆ’πœŽ|π‘₯βˆ’Μƒπ‘₯|+𝜎|Μƒπ‘₯|=𝜎⟨π‘₯,Μƒπ‘₯⟩/|Μƒπ‘₯|+π‘œ(1) as |Μƒπ‘₯|β†’βˆž, then the lemma follows from the Lebesgue dominated convergence theorem.

Lemma 4.3. Under the assumptions (π‘Ž1), (𝑏1)-(𝑏2) and πœ†βˆˆ(0,Ξ›0). Then there exists a number 𝑛0βˆˆβ„• such that for 𝑛β‰₯𝑛0sup𝑑β‰₯0π½πœ†ξ€·π‘‘π‘€π‘›ξ€Έ<π‘†βˆž.(4.8) In particular, π›Όβˆ’πœ†<π‘†βˆž for all πœ†βˆˆ(0,Ξ›0).

Proof. (i) First, since ‖𝑀𝑛‖=‖𝑀0β€– for all π‘›βˆˆβ„• and π½πœ† is continuous in 𝐻 and π½πœ†(0)=0, we infer that there exists 𝑑1>0 such that π½πœ†ξ€·π‘‘π‘€π‘›ξ€Έ<π‘†βˆžξ€Ίβˆ€π‘›βˆˆβ„•,π‘‘βˆˆ0,𝑑1ξ€».(4.9)
 (ii) Since lim|π‘₯|β†’βˆžπ‘Ž(π‘₯)=1, there exists 𝑛1βˆˆβ„• such that if 𝑛β‰₯𝑛1, we get π‘Ž(π‘₯)β‰₯1/2 for π‘₯βˆˆπ΅π‘(𝑛𝑒;1). Then, for 𝑛β‰₯𝑛1π½πœ†ξ€·π‘‘π‘€π‘›ξ€Έ=𝑑22‖‖𝑀𝑛‖‖2βˆ’π‘‘π‘π‘ξ€œβ„π‘||π‘€π‘Ž(π‘₯)𝑛||𝑝𝑑𝑑π‘₯βˆ’π‘žπ‘žξ€œβ„π‘||π‘€πœ†π‘(π‘₯)𝑛||π‘žβ‰€π‘‘π‘‘π‘₯22‖‖𝑀0β€–β€–2βˆ’π‘‘π‘π‘ξ€œπ΅π‘(0;1)||π‘€π‘Ž(π‘₯+𝑛𝑒)0||𝑝𝑑𝑑π‘₯+π‘žπ‘žπœ†β€–π‘βˆ’β€–πΏβˆžξ€œβ„π‘||𝑀𝑛||π‘žβ‰€π‘‘π‘‘π‘₯22‖‖𝑀0β€–β€–2βˆ’π‘‘π‘ξ€œ2𝑝𝐡𝑁(0;1)||𝑀0||𝑝𝑑𝑑π‘₯+π‘žπ‘žπœ†β€–π‘βˆ’β€–πΏβˆžξ€œβ„π‘||𝑀0||π‘žπ‘‘π‘₯βŸΆβˆ’βˆžasπ‘‘βŸΆβˆž.(4.10) Thus, there exists 𝑑2>0 such that for any 𝑑>𝑑2 and 𝑛>𝑛1 we get π½πœ†ξ€·π‘‘π‘€π‘›ξ€Έ<0.(4.11)
 (iii) By (i) and (ii), we need to show that there exists 𝑛0 such that for 𝑛β‰₯𝑛0sup𝑑1≀𝑑≀𝑑2π½πœ†ξ€·π‘‘π‘€π‘›ξ€Έ<π‘†βˆž.(4.12) We know that sup𝑑β‰₯0𝐽∞(𝑑𝑀0)=π‘†βˆž. Then, 𝑑1≀𝑑≀𝑑2, we have π½πœ†ξ€·π‘‘π‘€π‘›ξ€Έ=12‖‖𝑑𝑀𝑛‖‖2βˆ’1π‘ξ€œβ„π‘ξ€·π‘Ž(π‘₯)𝑑𝑀𝑛𝑝1𝑑π‘₯βˆ’π‘žξ€œβ„π‘ξ€·πœ†π‘(π‘₯)π‘‘π‘€π‘›ξ€Έπ‘žβ‰€π‘‘π‘‘π‘₯22‖‖𝑀0β€–β€–2βˆ’π‘‘π‘π‘ξ€œβ„π‘π‘€π‘0𝑑𝑑π‘₯+π‘π‘ξ€œβ„π‘(1βˆ’π‘Ž(π‘₯))𝑀𝑝𝑛𝑑𝑑π‘₯βˆ’π‘žπ‘žξ€œβ„π‘πœ†π‘(π‘₯)π‘€π‘žπ‘›π‘‘π‘₯β‰€π‘†βˆž+𝑑𝑝2π‘ξ€œβ„π‘(1βˆ’π‘Ž)+(π‘₯)𝑀𝑝𝑛𝑑𝑑π‘₯βˆ’π‘ž1π‘žξ€œβ„π‘πœ†π‘+(π‘₯)π‘€π‘žπ‘›π‘‘π‘‘π‘₯+π‘ž2π‘žξ€œβ„π‘πœ†π‘βˆ’(π‘₯)π‘€π‘žπ‘›π‘‘π‘₯.(4.13) Suppose π‘Ž satisfies (π‘Ž1), we get (1βˆ’π‘Ž)+(π‘₯)≀𝐢0π‘’βˆ’π›Ώ0|π‘₯| for all π‘₯βˆˆβ„π‘ and some positive constant 𝛿0. By (4.3) and Lemma 4.3, there exists 𝑛2>𝑛1 such that for any 𝑛β‰₯𝑛2ξ€œβ„π‘(1βˆ’π‘Ž)+(π‘₯)𝑀𝑝𝑛𝑑π‘₯≀𝐢3π‘’βˆ’min{𝛿0,𝑝}𝑛.(4.14) By (𝑏1) and (4.3), we get ξ€œβ„π‘πœ†π‘βˆ’(π‘₯)π‘€π‘žπ‘›π‘‘π‘₯β‰€πœ†β€–π‘βˆ’β€–πΏβˆžπΆ2ξ€œπΎπ‘’βˆ’π‘ž|π‘₯βˆ’π‘›π‘’|𝑑π‘₯β‰€πœ†πΆ3π‘’βˆ’π‘žπ‘›.(4.15) By (𝑏2), (4.3) and Lemma 4.3, we have ξ€œβ„π‘πœ†π‘+(π‘₯)π‘€π‘žπ‘›π‘‘π‘₯β‰₯πœ†πΆ1πΆπœ€ξ€œ|π‘₯|β‰₯𝑅0π‘’βˆ’π›Ώ1|π‘₯|π‘’βˆ’π‘ž(1+πœ€)|π‘₯βˆ’π‘›π‘’|𝑑π‘₯β‰₯πœ†πΆπœ€π‘’βˆ’π›Ώ1𝑛.(4.16) Since 0<𝛿1<min{𝛿0,π‘ž}≀min{𝛿0,𝑝} and πœ†βˆˆ(0,Ξ›0) and using (4.13)–(4.16), we have there exists 𝑛0>𝑛2 such that for all 𝑛β‰₯𝑛0, then sup𝑑1≀𝑑≀𝑑2π½πœ†ξ€·π‘‘π‘€π‘›ξ€Έ<π‘†βˆžξ€œ,πœ†β„π‘||𝑀𝑏(π‘₯)𝑛||π‘žπ‘‘π‘₯>0.(4.17) This implies that if πœ†βˆˆ(0,Ξ›0), then for all 𝑛β‰₯𝑛0 we get sup𝑑β‰₯0π½πœ†ξ€·π‘‘π‘€π‘›ξ€Έ<π‘†βˆž.(4.18) From π‘Ž(π‘₯)>0 for all π‘₯βˆˆβ„π‘ and (4.17), we have ξ€œβ„π‘||π‘€π‘Ž(π‘₯)𝑛0||π‘ξ€œπ‘‘π‘₯>0,ℝ𝑁||𝑀𝑏(π‘₯)𝑛0||π‘žπ‘‘π‘₯>0.(4.19) Combining this with Lemma 2.4(ii), from the definition of π›Όβˆ’πœ† and sup𝑑β‰₯0π½πœ†(𝑑𝑀𝑛0)<π‘†βˆž, for all πœ†βˆˆ(0,Ξ›0), we obtain that there exists 𝑑0>0 such that 𝑑0𝑀𝑛0βˆˆπ’©βˆ’πœ† and π›Όβˆ’πœ†β‰€π½πœ†ξ€·π‘‘0𝑀𝑛0≀sup𝑑β‰₯0π½πœ†ξ€·π‘‘π‘€π‘›0ξ€Έ<π‘†βˆž.(4.20)

Lemma 4.4. Assume that (π‘Ž1) and (𝑏1) hold. If {𝑒𝑛}βŠ‚π» is a (PS)𝑐-sequence for π½πœ† with π‘βˆˆ(0,π‘†βˆž), then there exists a subsequence of {𝑒𝑛} converging weakly to a nonzero solution of (πΈπ‘Ž,πœ†π‘) in ℝ𝑁.

Proof. Let {𝑒𝑛}βŠ‚π» be a (PS)𝑐-sequence for π½πœ† with π‘βˆˆ(0,π‘†βˆž). We know from Lemma 4.1 that {𝑒𝑛} is bounded in 𝐻, and then there exist a subsequence of {𝑒𝑛} (still denoted by {𝑒𝑛}) and 𝑒0∈𝐻 such that 𝑒𝑛⇀𝑒0weaklyin𝑒𝐻,𝑛→𝑒0almosteverywhereinℝ𝑁,𝑒𝑛→𝑒0stronglyin𝐿𝑠locξ€·β„π‘ξ€Έβˆ€1≀𝑠<2βˆ—.(4.21) It is easy to see that π½ξ…žπœ†(𝑒0)=0 and by (𝑏1), Egorov theorem and HΓΆlder inequality, we have πœ†ξ€œβ„π‘||𝑒𝑏(π‘₯)𝑛||π‘žξ€œπ‘‘π‘₯=πœ†β„π‘||𝑒𝑏(π‘₯)0||π‘žπ‘‘π‘₯+π‘œπ‘›(1).(4.22)
Next we verify that 𝑒0β‰’0. Arguing by contradiction, we assume 𝑒0≑0. By (π‘Ž1), for any πœ€>0, there exists 𝑅0>0 such that |π‘Ž(π‘₯)βˆ’1|<πœ€ for all π‘₯∈[𝐡𝑁(0;𝑅0)]𝐢. Since 𝑒𝑛→0 strongly in 𝐿𝑠loc(ℝ𝑁) for 1≀𝑠<2βˆ—, {𝑒𝑛} is a bounded sequence in 𝐻, therefore βˆ«β„π‘(π‘Ž(π‘₯)βˆ’1)|𝑒𝑛|π‘βˆ«β‰€πΆπ΅π‘(0;𝑅0)|𝑒𝑛|𝑝+πœ€πΆ. Setting π‘›β†’βˆž, then πœ€β†’0, we havelimπ‘›β†’βˆžξ€œβ„π‘||π‘’π‘Ž(π‘₯)𝑛||𝑝𝑑π‘₯=limπ‘›β†’βˆžξ€œβ„π‘||𝑒𝑛||𝑝𝑑π‘₯.(4.23) We set 𝑙=limπ‘›β†’βˆžξ€œβ„π‘||π‘’π‘Ž(π‘₯)𝑛||𝑝𝑑π‘₯=limπ‘›β†’βˆžξ€œβ„π‘||𝑒𝑛||𝑝𝑑π‘₯.(4.24) Since π½ξ…žπœ†(𝑒𝑛)=π‘œπ‘›(1) and {𝑒𝑛} is bounded, then by (4.22), we can deduce that 0=limπ‘›β†’βˆžξ«π½ξ…žπœ†ξ€·π‘’π‘›ξ€Έ,𝑒𝑛=limπ‘›β†’βˆžξ‚΅β€–β€–π‘’π‘›β€–β€–2βˆ’ξ€œβ„π‘||π‘’π‘Ž(π‘₯)𝑛||𝑝𝑑π‘₯=limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–2βˆ’π‘™,(4.25) that is, limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–2=𝑙.(4.26)
If 𝑙=0, then we get 𝑐=limπ‘›β†’βˆžπ½πœ†(𝑒𝑛)=0, which contradicts to 𝑐>0. Thus we conclude that 𝑙>0. Furthermore, by the definition of 𝑆𝑝 we obtain‖‖𝑒𝑛‖‖2β‰₯π‘†π‘ξ‚΅ξ€œβ„π‘||𝑒𝑛||𝑝𝑑π‘₯2/𝑝.(4.27) Then, as π‘›β†’βˆž, we have 𝑙=limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–2β‰₯𝑆𝑝𝑙2/𝑝,(4.28) which implies that 𝑙β‰₯𝑆𝑝𝑝/(π‘βˆ’2).(4.29) Hence, from (4.2) and (4.22)–(4.29), we get 𝑐=limπ‘›β†’βˆžπ½πœ†ξ€·π‘’π‘›ξ€Έ=12limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–2βˆ’1𝑝limπ‘›β†’βˆžξ€œβ„π‘||π‘’π‘Ž(π‘₯)𝑛||π‘πœ†π‘‘π‘₯βˆ’π‘žlimπ‘›β†’βˆžξ€œβ„π‘||𝑒𝑏(π‘₯)𝑛||π‘ž=ξ‚΅1𝑑π‘₯2βˆ’1𝑝𝑙β‰₯π‘βˆ’2𝑆2𝑝𝑝𝑝/(π‘βˆ’2)=π‘†βˆž.(4.30) This is a contradiction to 𝑐<π‘†βˆž. Therefore, 𝑒0 is a nonzero solution of (πΈπ‘Ž,πœ†π‘).

Now, we establish the existence of a local minimum of π½πœ† on π’©βˆ’πœ†.

Theorem 4.5. Assume that (π‘Ž1) and (𝑏1)-(𝑏2) hold. If πœ†βˆˆ(0,(π‘ž/2)Ξ›0), then there exists π‘ˆπœ†βˆˆπ’©βˆ’πœ† such that (i)π½πœ†(π‘ˆπœ†)=π›Όβˆ’πœ†,(ii)π‘ˆπœ† is a positive solution of (πΈπ‘Ž,πœ†π‘).

Proof. If πœ†βˆˆ(0,(π‘ž/2)Ξ›0), then by Theorem 2.5(ii), Proposition 3.2(ii) and Lemma 4.3(ii), there exists a (PS)π›Όβˆ’πœ†-sequence {𝑒𝑛}βŠ‚π’©βˆ’πœ† in 𝐻 for π½πœ† with π›Όβˆ’πœ†βˆˆ(0,π‘†βˆž). From Lemma 4.4, there exist a subsequence still denoted by {𝑒𝑛} and a nonzero solution π‘ˆπœ†βˆˆπ» of (πΈπ‘Ž,πœ†π‘) such that π‘’π‘›β‡€π‘ˆπœ† weakly in 𝐻.
First, we prove that π‘ˆπœ†βˆˆπ’©βˆ’πœ†. On the contrary, if π‘ˆπœ†βˆˆπ’©+πœ†, then by π’©βˆ’πœ† is closed in 𝐻, we have β€–π‘ˆπœ†β€–2<liminfπ‘›β†’βˆžβ€–π‘’π‘›β€–2. From (2.9) and π‘Ž(π‘₯)>0 for all π‘₯βˆˆβ„π‘, we getξ€œβ„π‘||π‘ˆπ‘(π‘₯)πœ†||π‘žξ€œπ‘‘π‘₯>0,ℝ𝑁||π‘ˆπ‘Ž(π‘₯)πœ†||𝑝𝑑π‘₯>0.(4.31) By Lemma 2.4(ii), there exists a unique π‘‘βˆ’πœ† such that π‘‘βˆ’πœ†π‘ˆπœ†βˆˆπ’©βˆ’πœ†. If π‘’βˆˆπ’©πœ†, then it is easy to see that π½πœ†(𝑒)=π‘βˆ’22𝑝‖𝑒‖2βˆ’π‘βˆ’π‘žπœ†ξ€œπ‘π‘žβ„π‘π‘(π‘₯)|𝑒|π‘žπ‘‘π‘₯.(4.32) From (3.1), π‘’π‘›βˆˆπ’©βˆ’πœ† and (4.32), we can deduce that π›Όβˆ’πœ†β‰€π½πœ†ξ€·π‘‘βˆ’πœ†π‘ˆπœ†ξ€Έ<limπ‘›β†’βˆžπ½πœ†ξ€·π‘‘βˆ’πœ†π‘’π‘›ξ€Έβ‰€limπ‘›β†’βˆžπ½πœ†ξ€·π‘’π‘›ξ€Έ=π›Όβˆ’πœ†(4.33) which is a contradiction. Thus, π‘ˆπœ†βˆˆπ’©βˆ’πœ†.
Next, by the same argument as that in Theorem 3.3, we get that π‘’π‘›β†’π‘ˆπœ† strongly in 𝐻 and π½πœ†(π‘ˆπœ†)=π›Όβˆ’πœ†>0 for all πœ†βˆˆ(0,(π‘ž/2)Ξ›0). Since π½πœ†(π‘ˆπœ†)=π½πœ†(|π‘ˆπœ†|) and |π‘ˆπœ†|βˆˆπ’©βˆ’πœ†, by Lemma 2.2 we may assume that π‘ˆπœ† is a nonzero nonnegative solution of (πΈπ‘Ž,πœ†π‘). Finally, by the Harnack inequality [22] we deduce that π‘ˆπœ†>0 in ℝ𝑁.

Now, we complete the proof of Theorem 1.1. By Theorems 3.3, 4.5, we obtain (πΈπ‘Ž,πœ†π‘) has two positive solutions π‘’πœ† and π‘ˆπœ† such that π‘’πœ†βˆˆπ’©+πœ†, π‘ˆπœ†βˆˆπ’©βˆ’πœ†. Since 𝒩+πœ†βˆ©π’©βˆ’πœ†=βˆ…, this implies that π‘’πœ† and π‘ˆπœ† are distinct. It completes the proof of Theorem 1.1.