Abstract and Applied Analysis

Abstract and Applied Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 652109 | 24 pages | https://doi.org/10.1155/2009/652109

Multiplicity Results for p-Laplacian with Critical Nonlinearity of Concave-Convex Type and Sign-Changing Weight Functions

Academic Editor: Pavel DrΓ‘bek
Received04 Dec 2008
Revised24 Jun 2009
Accepted07 Sep 2009
Published19 Oct 2009

Abstract

The multiple results of positive solutions for the following quasilinear elliptic equation: βˆ’ Ξ” 𝑝 𝑒 = πœ† 𝑓 ( π‘₯ ) | 𝑒 | π‘ž βˆ’ 2 𝑒 + 𝑔 ( π‘₯ ) | 𝑒 | 𝑝 βˆ— βˆ’ 2 𝑒 in Ξ© , 𝑒 = 0 on πœ• Ξ© , are established. Here, 0 ∈ Ξ© is a bounded smooth domain in ℝ 𝑁 , Ξ” 𝑝 denotes the 𝑝 -Laplacian operator, 1 ≀ π‘ž < 𝑝 < 𝑁 , 𝑝 βˆ— = 𝑁 𝑝 / ( 𝑁 βˆ’ 𝑝 ) , πœ† is a positive real parameter, and 𝑓 , 𝑔 are continuous functions on Ξ© which are somewhere positive but which may change sign on Ξ© . The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.

1. Introduction

In this paper, we study the multiple results of positive solutions for the following quasilinear elliptic equation:

βˆ’ Ξ” 𝑝 𝑒 = πœ† 𝑓 ( π‘₯ ) | 𝑒 | π‘ž βˆ’ 2 𝑒 + 𝑔 ( π‘₯ ) | 𝑒 | 𝑝 βˆ— βˆ’ 2 𝑒 i n Ξ© , 𝑒 = 0 o n πœ• Ξ© , ( ξ€· 𝐸 πœ† 𝑓 , 𝑔 ξ€Έ ) where πœ† > 0 , Ξ” 𝑝 𝑒 = d i v ( | βˆ‡ 𝑒 | 𝑝 βˆ’ 2 βˆ‡ 𝑒 ) is the 𝑝 -Laplacian, 0 ∈ Ξ© is a bounded domain in ℝ 𝑁 with smooth boundary πœ• Ξ© , 1 < π‘ž < 𝑝 < 𝑁 , 𝑝 βˆ— = 𝑁 𝑝 / ( 𝑁 βˆ’ 𝑝 ) is the so-called critical Sobolev exponent and the weight functions 𝑓 , 𝑔 are satisfying the following conditions:

( 𝑓 1 ) 𝑓 ∈ 𝐢 ( Ξ© ) and 𝑓 + = m a x { 𝑓 , 0 } β‰’ 0 ;( 𝑓 2 ) there exist 𝛽 0 , 𝜌 0 > 0 and π‘₯ 0 ∈ Ξ© such that 𝐡 ( π‘₯ 0 , 2 𝜌 0 ) βŠ‚ Ξ© and 𝑓 ( π‘₯ ) β‰₯ 𝛽 0 for all π‘₯ ∈ 𝐡 ( π‘₯ 0 , 2 𝜌 0 ) . Without loss of generality, we assume that π‘₯ 0 = 0 ,( 𝑔 1 ) 𝑔 ∈ 𝐢 ( Ξ© ) and 𝑔 + = m a x { 𝑔 , 0 } β‰’ 0 ;( 𝑔 2 ) | 𝑔 + | ∞ = 𝑔 ( 0 ) = m a x π‘₯ ∈ Ξ© 𝑔 ( π‘₯ ) ;( 𝑔 3 ) 𝑔 ( π‘₯ ) > 0 for all π‘₯ ∈ 𝐡 ( 0 , 2 𝜌 0 ) ;( 𝑔 4 ) there exists 𝛽 > 𝑁 / ( 𝑝 βˆ’ 1 ) such that

ξ€· 𝑔 ( π‘₯ ) = 𝑔 ( 0 ) + π‘œ | π‘₯ | 𝛽 ξ€Έ a s π‘₯ ⟢ 0 . ( 1 . 1 )

For the weight functions 𝑓 ≑ 𝑔 ≑ 1 , ( 𝐸 πœ† 𝑓 , 𝑔 ) has been studied extensively. Historically, the role played by such concave-convex nonlinearities in producing multiple solutions was investigated first in the work [1]. They studied the following semilinear elliptic equation:

βˆ’ Ξ” 𝑒 = πœ† 𝑒 π‘ž βˆ’ 1 + 𝑒 2 βˆ— βˆ’ 1 i n Ξ© , 𝑒 > 0 i n Ξ© , 𝑒 = 0 o n πœ• Ξ© , ( 1 . 2 ) for 1 < π‘ž < 2 and showed the existence of πœ† 0 > 0 such that (1.2) admits at least two solutions for all πœ† ∈ ( 0 , πœ† 0 ) and no solution for πœ† > πœ† 0 . Subsequently, in the work [2, 3], the corresponding quasilinear version has been studied

βˆ’ Ξ” 𝑝 𝑒 = πœ† 𝑒 π‘ž βˆ’ 1 + 𝑒 𝑝 βˆ— βˆ’ 1 i n Ξ© , 𝑒 > 0 i n Ξ© , 𝑒 = 0 o n πœ• Ξ© , ( 1 . 3 ) where 1 < 𝑝 < 𝑁 and 1 < π‘ž < 𝑝 . They obtained results similar to the results of [1] above, but only for some ranges of the exponents 𝑝 and π‘ž . We summarize their results in what follows.

Theorem 1.1 (see [2, 3]). Assume that either 2 𝑁 / ( 𝑁 + 2 ) < 𝑝 < 3 or 𝑝 > 3 , 𝑝 > π‘ž > 𝑝 βˆ— βˆ’ 2 / ( 𝑝 βˆ’ 1 ) . Then there exists πœ† 0 > 0 such that (1.3) admits at least two solutions for all πœ† ∈ ( 0 , πœ† 0 ) and no solution for πœ† > πœ† 0 .

It is possible to get complete multiplicity result for problem (1.3) if Ξ© is taken to be a ball in ℝ 𝑁 . Prashanth and Sreenadh [4] have studied (1.3) in the unit ball 𝐡 𝑁 ( 0 ; 1 ) in ℝ 𝑁 and obtained the following results.

Theorem 1.2 (see [4]). Let Ξ© = 𝐡 𝑁 ( 0 ; 1 ) , 1 < 𝑝 < 𝑁 , 1 < π‘ž < 𝑝 . Then there exists πœ† 0 > 0 such that (1.3) admits at least two solutions for all πœ† ∈ ( 0 , πœ† 0 ) and no solution for πœ† > πœ† 0 . Additionally, if 1 < 𝑝 < 2 , then (1.3) admits exactly two solutions for all small πœ† > 0 .

For 𝑝 = 2 , Tang [5] has studied the exact multiplicity about the following semilinear elliptic equation:

βˆ’ Ξ” 𝑒 = πœ† 𝑒 π‘ž βˆ’ 1 + 𝑒 π‘Ÿ βˆ’ 1 i n 𝐡 𝑁 ( 0 ; 1 ) , 𝑒 > 0 i n 𝐡 𝑁 ( 0 ; 1 ) , 𝑒 = 0 o n πœ• 𝐡 𝑁 ( 0 ; 1 ) , ( 1 . 4 ) where 1 < π‘ž < 2 < π‘Ÿ ≀ 2 𝑁 / ( 𝑁 βˆ’ 2 ) and 𝑁 β‰₯ 3 . We also mention his result below.

Theorem 1.3 (see [5]). There exists πœ† 0 > 0 such that (1.4) admits exactly two solutions for πœ† ∈ ( 0 , πœ† 0 ) , exactly one solution for πœ† = πœ† 0 , and no solution for πœ† > πœ† 0 .

To proceed, we make some motivations of the present paper. Recently, in [6] the author has considered (1.2) with subcritical nonlinearity of concave-convex type, 𝑔 ≑ 1 , and 𝑓 is a continuous function which changes sign in Ξ© , and showed the existence of πœ† 0 > 0 such that (1.2) admits at least two solutions for all πœ† ∈ ( 0 , πœ† 0 ) via the extraction of Palais-Smale sequences in the Nehair manifold. In a recent work [7], the author extended the results of [6] to the quasilinear case with the more general weight functions 𝑓 , 𝑔 but also having subcritical nonlinearity of concave-convex type. In the present paper, we continue the study of [7] by considering critical nonlinearity of concave-convex type and sign-changing weight functions 𝑓 , 𝑔 .

In this paper, we use a variational method involving the Nehari manifold to prove the multiplicity of positive solutions. The Nehari method has been used also in [8] to prove the existence of multiple for a singular elliptic problem. The existence of at least one solution can be obtained by using the same arguments as in the subcritical case [7]. The existence of a second solution needs different arguments due to the lack of compactness of the Palais-Smale sequences. For what, we need addtional assumptions ( 𝑓 2 ) and ( 𝑔 2 ) to prove the compactness of the extraction of Palais-Smale sequences in the Nehari manifold (see Theorem 4.4). The multiplicity result is proved only for the parameter πœ† ∈ ( 0 , ( π‘ž / 𝑝 ) Ξ› 1 ) (see Theorem 1.5) but for all 1 < 𝑝 < 𝑁 and 1 ≀ π‘ž < 𝑝 . This is not the case in the papers referred [2, 3] where the multiplicity is global but not with the full range of 𝑝 , π‘ž and with the weight functions 𝑓 ≑ 𝑔 ≑ 1 . Finally, we mention a recent contribution on 𝑝 -Laplacian equation with changing sign nonlinearity by Figuereido et al. [9] which gives the global multiplicity but not with the full range of 𝑝 and π‘ž . The method used in the paper by Figuereido et al. is similar to the method introduced in [1].

In order to represpent our main results, we need to define the following constant Ξ› 1 . Set

Ξ› 1 =  𝑝 βˆ’ π‘ž ( 𝑝 βˆ— | | 𝑔 βˆ’ π‘ž ) + | | ∞ ξƒͺ ( 𝑝 βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 )  𝑝 βˆ— βˆ’ 𝑝 ( 𝑝 βˆ— | | 𝑓 βˆ’ π‘ž ) + | | ∞ ξƒͺ | | Ξ© | | ( π‘ž βˆ’ 𝑝 βˆ— ) / 𝑝 βˆ— 𝑆 ( 𝑁 / 𝑝 ) βˆ’ ( 𝑁 / 𝑝 2 ) π‘ž + ( π‘ž / 𝑝 ) > 0 , ( 1 . 5 ) where | Ξ© | is the Lebesgue measure of Ξ© and 𝑆 is the best Sobolev constant (see (2.2)).

Theorem 1.4. Assume ( 𝑓 1 ) and ( 𝑔 1 ) hold. If πœ† ∈ ( 0 , Ξ› 1 ) , then ( 𝐸 πœ† 𝑓 , 𝑔 ) admits at least one positive solution 𝑒 πœ† ∈ 𝐢 1 , 𝛼 ( Ξ© ) for some 𝛼 ∈ ( 0 , 1 ) .

Theorem 1.5. Assume that ( 𝑓 1 ) - ( 𝑓 2 ) and ( 𝑔 1 ) - ( 𝑔 4 ) hold. If πœ† ∈ ( 0 , ( π‘ž / 𝑝 ) Ξ› 1 ) , then ( 𝐸 πœ† 𝑓 , 𝑔 ) admits at least two positive solutions 𝑒 πœ† , π‘ˆ πœ† ∈ 𝐢 1 , 𝛼 ( Ξ© ) for some 𝛼 ∈ ( 0 , 1 ) .

This paper is organized as follows. In Section 2, we give some preliminaries and some properties of Nehari manifold. In Sections 3 and 4, we complete proofs of Theorems 1.4 and 1.5.

2. Preliminaries and Nehari Manifold

Throughout this paper, ( 𝑓 1 ) and ( 𝑔 1 ) will be assumed. The dual space of a Banach space 𝐸 will be denoted by 𝐸 βˆ’ 1 . π‘Š 0 1 , 𝑝 ( Ξ© ) denotes the standard Sobolev space with the following norm:

β€– 𝑒 β€– 𝑝 = ξ€œ Ξ© | | | | βˆ‡ 𝑒 𝑝 𝑑 π‘₯ . ( 2 . 1 ) π‘Š 0 1 , 𝑝 ( Ξ© ) with the norm β€– β‹… β€– is simply denoted by π‘Š . We denote the norm in 𝐿 𝑝 ( Ξ© ) by | β‹… | 𝑝 and the norm in 𝐿 𝑝 ( ℝ 𝑁 ) by | β‹… | 𝐿 𝑝 ( ℝ 𝑁 ) . | Ξ© | is the Lebesgue measure of Ξ© . 𝐡 ( π‘₯ , π‘Ÿ ) is a ball centered at π‘₯ with radius π‘Ÿ . 𝑂 ( πœ€ 𝑑 ) denotes | 𝑂 ( πœ€ 𝑑 ) | / πœ€ 𝑑 ≀ 𝐢 , π‘œ ( πœ€ 𝑑 ) denotes | π‘œ ( πœ€ 𝑑 ) | / πœ€ 𝑑 β†’ 0 as πœ€ β†’ 0 , and π‘œ 𝑛 ( 1 ) denotes π‘œ 𝑛 ( 1 ) β†’ 0 as 𝑛 β†’ ∞ . 𝐢 , 𝐢 𝑖 will denote various positive constants; the exact values of which are not important. 𝑆 is the best Sobolev embedding constant defined by

𝑆 = i n f 𝑒 ∈ π‘Š ⧡ { 0 } | | | | βˆ‡ 𝑒 𝑝 𝑝 | 𝑒 | 𝑝 𝑝 βˆ— . ( 2 . 2 )

Definition 2.1. Let 𝑐 ∈ ℝ , 𝐸 be a Banach space and 𝐼 ∈ 𝐢 1 ( 𝐸 , ℝ ) .
(i) { 𝑒 𝑛 } is a (PS) 𝑐 -sequence in 𝐸 for 𝐼 if 𝐼 ( 𝑒 𝑛 ) = 𝑐 + π‘œ 𝑛 ( 1 ) and 𝐼 ξ…ž ( 𝑒 𝑛 ) = π‘œ 𝑛 ( 1 ) strongly in 𝐸 βˆ’ 1 as 𝑛 β†’ ∞ . (ii) We say that 𝐼 satisfies the (PS) 𝑐 condition if any (PS) 𝑐 -sequence { 𝑒 𝑛 } in 𝐸 for 𝐼 has a convergent subsequence.

Associated with ( 𝐸 πœ† 𝑓 , 𝑔 ), we consider the energy functional 𝐽 πœ† in π‘Š , for each 𝑒 ∈ π‘Š ,

𝐽 πœ† ( 1 𝑒 ) = 𝑝 β€– 𝑒 β€– 𝑝 βˆ’ πœ† π‘ž ξ€œ Ξ© 𝑓 | 𝑒 | π‘ž 1 𝑑 π‘₯ βˆ’ 𝑝 βˆ— ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ . ( 2 . 3 ) It is well known that 𝐽 πœ† is of 𝐢 1 in π‘Š and the solutions of ( 𝐸 πœ† 𝑓 , 𝑔 ) are the critical points of the energy functional 𝐽 πœ† (see Rabinowitz [10]).

As the energy functional 𝐽 πœ† is not bounded below on π‘Š , it is useful to consider the functional on the Nehari manifold

𝒩 πœ† = ξ€½  𝐽 𝑒 ∈ π‘Š ⧡ { 0 } ∢ ξ…ž πœ†  ξ€Ύ . ( 𝑒 ) , 𝑒 = 0 ( 2 . 4 ) Thus, 𝑒 ∈ 𝒩 πœ† if and only if

 𝐽 ξ…ž πœ† (  𝑒 ) , 𝑒 = β€– 𝑒 β€– 𝑝 ξ€œ βˆ’ πœ† Ξ© 𝑓 | 𝑒 | π‘ž ξ€œ 𝑑 π‘₯ βˆ’ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ = 0 . ( 2 . 5 ) Note that 𝒩 πœ† contains every nonzero solution of ( 𝐸 πœ† 𝑓 , 𝑔 ). Moreover, we have the following results.

Lemma 2.2. The energy functional 𝐽 πœ† is coercive and bounded below on 𝒩 πœ† .

Proof. If 𝑒 ∈ 𝒩 πœ† , then by ( 𝑓 1 ) , (2.5), and the HΓΆlder inequality and the Sobolev embedding theorem we have 𝐽 πœ† 𝑝 ( 𝑒 ) = βˆ— βˆ’ 𝑝 𝑝 βˆ— 𝑝 β€– 𝑒 β€– 𝑝 ξ‚΅ 𝑝 βˆ’ πœ† βˆ— βˆ’ π‘ž 𝑝 βˆ— π‘ž ξ‚Ά ξ€œ Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ ( 2 . 6 ) β‰₯ 1 𝑁 β€– 𝑒 β€– 𝑝 ξ‚΅ 𝑝 βˆ’ πœ† βˆ— βˆ’ π‘ž 𝑝 βˆ— π‘ž ξ‚Ά 𝑆 βˆ’ π‘ž / 𝑝 | | Ξ© | | ( 𝑝 βˆ— βˆ’ π‘ž ) / 𝑝 βˆ— β€– 𝑒 β€– π‘ž | | 𝑓 + | | ∞ . ( 2 . 7 ) Thus, 𝐽 πœ† is coercive and bounded below on 𝒩 πœ† .

Define

πœ“ πœ†  𝐽 ( 𝑒 ) = ξ…ž πœ†  . ( 𝑒 ) , 𝑒 ( 2 . 8 ) Then for 𝑒 ∈ 𝒩 πœ† ,

 πœ“ ξ…ž πœ† (  𝑒 ) , 𝑒 = 𝑝 β€– 𝑒 β€– 𝑝 ξ€œ βˆ’ πœ† π‘ž Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ βˆ’ 𝑝 βˆ— ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ ( 2 . 9 ) = ( 𝑝 βˆ’ π‘ž ) β€– 𝑒 β€– 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ ξ€œ βˆ’ π‘ž Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ ( 2 . 1 0 ) ξ€· 𝑝 = πœ† βˆ— ξ€Έ ξ€œ βˆ’ π‘ž Ξ© 𝑓 | 𝑒 | π‘ž ξ€· 𝑝 𝑑 π‘₯ βˆ’ βˆ— ξ€Έ βˆ’ 𝑝 β€– 𝑒 β€– 𝑝 . ( 2 . 1 1 ) Similar to the method used in Tarantello [11], we split 𝒩 πœ† into three parts:

𝒩 + πœ† = ξ€½ 𝑒 ∈ 𝒩 πœ† ∢  πœ“ ξ…ž πœ†  ξ€Ύ , 𝒩 ( 𝑒 ) , 𝑒 > 0 0 πœ† = ξ€½ 𝑒 ∈ 𝒩 πœ† ∢  πœ“ ξ…ž πœ†  ξ€Ύ , 𝒩 ( 𝑒 ) , 𝑒 = 0 βˆ’ πœ† = ξ€½ 𝑒 ∈ 𝒩 πœ† ∢  πœ“ ξ…ž πœ†  ξ€Ύ . ( 𝑒 ) , 𝑒 < 0 ( 2 . 1 2 ) Then, we have the following results.

Lemma 2.3. Assume that 𝑒 πœ† is a local minimizer for 𝐽 πœ† on 𝒩 πœ† and 𝑒 πœ† βˆ‰ 𝒩 0 πœ† . Then 𝐽 ξ…ž πœ† ( 𝑒 πœ† ) = 0 in π‘Š βˆ’ 1 .

Proof. Our proof is almost the same as that in Brown and Zhang [12, Theorem 2 . 3 ] (or see Binding et al. [13]).

Lemma 2.4. One has the following.
(i) If 𝑒 ∈ 𝒩 + πœ† , then ∫ Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ > 0 .
(ii) If 𝑒 ∈ 𝒩 0 πœ† , then ∫ Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ > 0 and ∫ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ > 0 .
(iii) If 𝑒 ∈ 𝒩 βˆ’ πœ† , then ∫ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ > 0 .

Proof. The proof is immediate from (2.10) and (2.11).

Moreover, we have the following result.

Lemma 2.5. If πœ† ∈ ( 0 , Ξ› 1 ) , then 𝒩 0 πœ† = βˆ… where Ξ› 1 is the same as in (1.5).

Proof. Suppose otherwise that there exists πœ† ∈ ( 0 , Ξ› 1 ) such that 𝒩 0 πœ† β‰  βˆ… . Then by (2.10) and (2.11), for 𝑒 ∈ 𝒩 0 πœ† , we have β€– 𝑒 β€– 𝑝 = 𝑝 βˆ— βˆ’ π‘ž ξ€œ 𝑝 βˆ’ π‘ž Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ , β€– 𝑒 β€– 𝑝 𝑝 = πœ† βˆ— βˆ’ π‘ž 𝑝 βˆ— ξ€œ βˆ’ 𝑝 Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ . ( 2 . 1 3 ) Moreover, by ( 𝑓 1 ) , ( 𝑔 1 ) , and the HΓΆlder inequality and the Sobolev embedding theorem, we have  β€– 𝑒 β€– β‰₯ 𝑝 βˆ’ π‘ž ( 𝑝 βˆ— | | 𝑔 βˆ’ π‘ž ) + | | ∞ 𝑆 𝑝 βˆ— / 𝑝 ξƒͺ 1 / ( 𝑝 βˆ— βˆ’ 𝑝 ) , ξ‚Έ πœ† 𝑝 β€– 𝑒 β€– ≀ βˆ— βˆ’ π‘ž 𝑝 βˆ— 𝑆 βˆ’ 𝑝 βˆ’ π‘ž / 𝑝 | | Ξ© | | ( 𝑝 βˆ— βˆ’ π‘ž ) / 𝑝 βˆ— | | 𝑓 + | | ∞ ξ‚Ή 1 / ( 𝑝 βˆ’ π‘ž ) . ( 2 . 1 4 ) This implies  πœ† β‰₯ 𝑝 βˆ’ π‘ž ( 𝑝 βˆ— | | 𝑔 βˆ’ π‘ž ) + | | ∞ ξƒͺ ( 𝑝 βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 )  𝑝 βˆ— βˆ’ 𝑝 ( 𝑝 βˆ— | | 𝑓 βˆ’ π‘ž ) + | | ∞ ξƒͺ | | Ξ© | | ( π‘ž βˆ’ 𝑝 βˆ— ) / 𝑝 βˆ— 𝑆 ( 𝑁 / 𝑝 ) βˆ’ ( 𝑁 / 𝑝 2 ) π‘ž + ( π‘ž / 𝑝 ) = Ξ› 1 , ( 2 . 1 5 ) which is a contradiction. Thus, we can conclude that if πœ† ∈ ( 0 , Ξ› 1 ) , we have 𝒩 0 πœ† = βˆ… .

By Lemma 2.5, we write 𝒩 πœ† = 𝒩 + πœ† βˆͺ 𝒩 βˆ’ πœ† and define

𝛼 πœ† = i n f 𝑒 ∈ 𝒩 πœ† 𝐽 πœ† ( 𝑒 ) , 𝛼 + πœ† = i n f 𝑒 ∈ 𝒩 + πœ† 𝐽 πœ† ( 𝑒 ) , 𝛼 βˆ’ πœ† = i n f 𝑒 ∈ 𝒩 βˆ’ πœ† 𝐽 πœ† ( 𝑒 ) . ( 2 . 1 6 ) Then we get the following result.

Theorem 2.6. (i) If πœ† ∈ ( 0 , Ξ› 1 ) and 𝑒 ∈ 𝒩 + πœ† , then one has 𝐽 πœ† ( 𝑒 ) < 0 and 𝛼 πœ† ≀ 𝛼 + πœ† < 0 .
(ii) If πœ† ∈ ( 0 , ( π‘ž / 𝑝 ) Ξ› 1 ) , then 𝛼 βˆ’ πœ† > 𝑑 0 for some positive constant 𝑑 0 depending on πœ† , 𝑝 , π‘ž , 𝑁 , 𝑆 , | 𝑓 + | ∞ , | 𝑔 + | ∞ , and | Ξ© | .

Proof. (i) Let 𝑒 ∈ 𝒩 + πœ† . By (2.10), we have 𝑝 βˆ’ π‘ž 𝑝 βˆ— βˆ’ π‘ž β€– 𝑒 β€– 𝑝 > ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ , ( 2 . 1 7 ) and so 𝐽 πœ† ξ‚΅ 1 ( 𝑒 ) = 𝑝 βˆ’ 1 π‘ž ξ‚Ά β€– 𝑒 β€– 𝑝 + ξ‚΅ 1 π‘ž βˆ’ 1 𝑝 βˆ— ξ‚Ά ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— < 1 𝑑 π‘₯ ξ‚Έ ξ‚΅ 𝑝 βˆ’ 1 π‘ž ξ‚Ά + ξ‚΅ 1 π‘ž βˆ’ 1 𝑝 βˆ— ξ‚Ά 𝑝 βˆ’ π‘ž 𝑝 βˆ— ξ‚Ή βˆ’ π‘ž β€– 𝑒 β€– 𝑝 = βˆ’ 𝑝 βˆ’ π‘ž π‘ž 𝑁 β€– 𝑒 β€– 𝑝 < 0 . ( 2 . 1 8 ) Therefore, from the definition of 𝛼 πœ† , 𝛼 + πœ† , we can deduce that 𝛼 πœ† ≀ 𝛼 + πœ† < 0 .
(ii) Let 𝑒 ∈ 𝒩 βˆ’ πœ† . By (2.10), we have 𝑝 βˆ’ π‘ž 𝑝 βˆ— βˆ’ π‘ž β€– 𝑒 β€– 𝑝 < ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ . ( 2 . 1 9 ) Moreover, by ( 𝑔 1 ) and the Sobolev embedding theorem, we have ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ ≀ 𝑆 βˆ’ 𝑝 βˆ— / 𝑝 β€– 𝑒 β€– 𝑝 βˆ— | | 𝑔 + | | ∞ . ( 2 . 2 0 ) This implies  β€– 𝑒 β€– > 𝑝 βˆ’ π‘ž ( 𝑝 βˆ— | | 𝑔 βˆ’ π‘ž ) + | | ∞ ξƒͺ 1 / ( 𝑝 βˆ— βˆ’ 𝑝 ) 𝑆 𝑁 / 𝑝 2 , βˆ€ 𝑒 ∈ 𝒩 βˆ’ πœ† . ( 2 . 2 1 ) By(2.7) in the proof of Lemma 2.2, we have 𝐽 πœ† ( 𝑒 ) β‰₯ β€– 𝑒 β€– π‘ž ξ‚Έ 𝑝 βˆ— βˆ’ 𝑝 𝑝 βˆ— 𝑝 β€– 𝑒 β€– 𝑝 βˆ’ π‘ž βˆ’ πœ† 𝑆 βˆ’ π‘ž / 𝑝 𝑝 βˆ— βˆ’ π‘ž 𝑝 βˆ— π‘ž | | Ξ© | | ( 𝑝 βˆ— βˆ’ π‘ž ) / 𝑝 βˆ— | | 𝑓 + | | ∞ ξ‚Ή >  𝑝 βˆ’ π‘ž ( 𝑝 βˆ— | | 𝑔 βˆ’ π‘ž ) + | | ∞ ξƒͺ π‘ž / ( 𝑝 βˆ— βˆ’ 𝑝 ) 𝑆 π‘ž 𝑁 / 𝑝 2 Γ— ⎑ ⎒ ⎒ ⎣ 𝑝 βˆ— βˆ’ 𝑝 𝑝 βˆ— 𝑝 𝑆 ( 𝑝 βˆ’ π‘ž ) 𝑁 / 𝑝 2  𝑝 βˆ’ π‘ž ( 𝑝 βˆ— | | 𝑔 βˆ’ π‘ž ) + | | ∞ ξƒͺ ( 𝑝 βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 ) βˆ’ πœ† 𝑆 βˆ’ π‘ž / 𝑝 𝑝 βˆ— βˆ’ π‘ž 𝑝 βˆ— π‘ž | | Ξ© | | ( 𝑝 βˆ— βˆ’ π‘ž ) / 𝑝 βˆ— | | 𝑓 + | | ∞ ⎀ βŽ₯ βŽ₯ ⎦ . ( 2 . 2 2 ) Thus, if πœ† ∈ ( 0 , ( π‘ž / 𝑝 ) Ξ› 1 ) , then 𝐽 πœ† ( 𝑒 ) > 𝑑 0 , βˆ€ 𝑒 ∈ 𝒩 βˆ’ πœ† , ( 2 . 2 3 ) for some positive constant 𝑑 0 = 𝑑 0 ( πœ† , 𝑝 , π‘ž , 𝑁 , 𝑆 , | 𝑓 + | ∞ , | 𝑔 + | ∞ , | Ξ© | ) . This completes the proof.

For each 𝑒 ∈ π‘Š with ∫ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ > 0 , we write

𝑑 m a x =  β€– ( 𝑝 βˆ’ π‘ž ) 𝑒 β€– 𝑝 ( 𝑝 βˆ— ∫ βˆ’ π‘ž ) Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— ξƒͺ 𝑑 π‘₯ 1 / ( 𝑝 βˆ— βˆ’ 𝑝 ) > 0 . ( 2 . 2 4 ) Then the following lemma holds.

Lemma 2.7. Let πœ† ∈ ( 0 , Ξ› 1 ) . For each 𝑒 ∈ π‘Š with ∫ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ > 0 , one has the following:
(i) if ∫ Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ ≀ 0 , then there exists a unique 𝑑 βˆ’ > 𝑑 m a x such that 𝑑 βˆ’ 𝑒 ∈ 𝒩 βˆ’ πœ† and 𝐽 πœ† ( 𝑑 βˆ’ 𝑒 ) = s u p 𝑑 β‰₯ 0 𝐽 πœ† ( 𝑑 𝑒 ) , ( 2 . 2 5 )
(ii) if ∫ Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ > 0 , then there exists unique 0 < 𝑑 + < 𝑑 m a x < 𝑑 βˆ’ such that 𝑑 + 𝑒 ∈ 𝒩 + πœ† , 𝑑 βˆ’ 𝑒 ∈ 𝒩 βˆ’ πœ† , and 𝐽 πœ† ξ€· 𝑑 + 𝑒 ξ€Έ = i n f 0 ≀ 𝑑 ≀ 𝑑 m a x 𝐽 πœ† ( 𝑑 𝑒 ) ; 𝐽 πœ† ( 𝑑 βˆ’ 𝑒 ) = s u p 𝑑 β‰₯ 0 𝐽 πœ† ( 𝑑 𝑒 ) . ( 2 . 2 6 )

Proof. Fix 𝑒 ∈ π‘Š with ∫ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ > 0 . Let π‘˜ ( 𝑑 ) = 𝑑 𝑝 βˆ’ π‘ž β€– 𝑒 β€– 𝑝 βˆ’ 𝑑 𝑝 βˆ— βˆ’ π‘ž ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ f o r 𝑑 β‰₯ 0 . ( 2 . 2 7 ) It is clear that π‘˜ ( 0 ) = 0 , π‘˜ ( 𝑑 ) β†’ βˆ’ ∞ as 𝑑 β†’ ∞ . From π‘˜ ξ…ž ( 𝑑 ) = ( 𝑝 βˆ’ π‘ž ) 𝑑 𝑝 βˆ’ π‘ž βˆ’ 1 β€– 𝑒 β€– 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ 𝑑 βˆ’ π‘ž 𝑝 βˆ— βˆ’ π‘ž βˆ’ 1 ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ , ( 2 . 2 8 ) we can deduce that π‘˜ ξ…ž ( 𝑑 ) = 0 at 𝑑 = 𝑑 m a x , π‘˜ ξ…ž ( 𝑑 ) > 0 for 𝑑 ∈ ( 0 , 𝑑 m a x ) and π‘˜ ξ…ž ( 𝑑 ) < 0 for 𝑑 ∈ ( 𝑑 m a x , ∞ ) . Then π‘˜ ( 𝑑 ) that achieves its maximum at 𝑑 m a x is increasing for 𝑑 ∈ [ 0 , 𝑑 m a x ) and decreasing for 𝑑 ∈ ( 𝑑 m a x , ∞ ) . Moreover, π‘˜ ξ€· 𝑑 m a x ξ€Έ =  β€– ( 𝑝 βˆ’ π‘ž ) 𝑒 β€– 𝑝 ( 𝑝 βˆ— ∫ βˆ’ π‘ž ) Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— ξƒͺ 𝑑 π‘₯ ( 𝑝 βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 ) β€– 𝑒 β€– 𝑝 βˆ’  ( 𝑝 βˆ’ π‘ž ) β€– 𝑒 β€– 𝑝 ( 𝑝 βˆ— ∫ βˆ’ π‘ž ) Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— ξƒͺ 𝑑 π‘₯ ( 𝑝 βˆ— βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 ) ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ = β€– 𝑒 β€– π‘ž  ξ‚΅ 𝑝 βˆ’ π‘ž 𝑝 βˆ— ξ‚Ά βˆ’ π‘ž ( 𝑝 βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 ) βˆ’ ξ‚΅ 𝑝 βˆ’ π‘ž 𝑝 βˆ— ξ‚Ά βˆ’ π‘ž ( 𝑝 βˆ— βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 ) ξƒ­  β€– 𝑒 β€– 𝑝 βˆ— ∫ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— ξƒͺ 𝑑 π‘₯ ( 𝑝 βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 ) β‰₯ β€– 𝑒 β€– π‘ž ξ‚΅ 𝑝 βˆ— βˆ’ 𝑝 𝑝 βˆ— ξ‚Ά  βˆ’ π‘ž 𝑝 βˆ’ π‘ž ( 𝑝 βˆ— ) | | 𝑔 βˆ’ π‘ž + | | ∞ 𝑆 𝑝 βˆ— / 𝑝 ξƒͺ ( 𝑝 βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 ) . ( 2 . 2 9 ) We have ∫ Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ ≀ 0 . There exists a unique 𝑑 βˆ’ > 𝑑 m a x such that π‘˜ ( 𝑑 βˆ’ ∫ ) = πœ† Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ and π‘˜ ξ…ž ( 𝑑 βˆ’ ) < 0 . Now, ( 𝑝 βˆ’ π‘ž ) ( 𝑑 βˆ’ ) 𝑝 β€– 𝑒 β€– 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ ( 𝑑 βˆ’ π‘ž βˆ’ ) 𝑝 ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ = ( 𝑑 βˆ’ ) 1 + π‘ž ξ‚Έ ( 𝑝 βˆ’ π‘ž ) ( 𝑑 βˆ’ ) 𝑝 βˆ’ π‘ž βˆ’ 1 β€– 𝑒 β€– 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ βˆ’ π‘ž ( 𝑑 βˆ’ ) 𝑝 βˆ— βˆ’ π‘ž βˆ’ 1 ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— ξ‚Ή 𝑑 π‘₯ = ( 𝑑 βˆ’ ) 1 + π‘ž π‘˜ ξ…ž ( 𝑑 βˆ’ )  𝐽 < 0 , ξ…ž πœ† ( 𝑑 βˆ’ 𝑒 ) , 𝑑 βˆ’ 𝑒  = ( 𝑑 βˆ’ ) 𝑝 β€– 𝑒 β€– 𝑝 βˆ’ ( 𝑑 βˆ’ ) 𝑝 βˆ— ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ βˆ’ ( 𝑑 βˆ’ ) π‘ž πœ† ξ€œ Ξ© 𝑓 | 𝑒 | π‘ž = 𝑑 π‘₯ ( 𝑑 βˆ’ ) π‘ž ξ‚Έ π‘˜ ( 𝑑 βˆ’ ξ€œ ) βˆ’ πœ† Ξ© 𝑓 | 𝑒 | π‘ž ξ‚Ή 𝑑 π‘₯ = 0 . ( 2 . 3 0 ) Then we have that 𝑑 βˆ’ 𝑒 ∈ 𝒩 βˆ’ πœ† . For 𝑑 > 𝑑 m a x , we have ( 𝑝 βˆ’ π‘ž ) β€– 𝑑 𝑒 β€– 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ ξ€œ βˆ’ π‘ž Ξ© 𝑔 | 𝑑 𝑒 | 𝑝 βˆ— 𝑑 < 0 , 2 𝑑 𝑑 2 𝐽 πœ† 𝑑 ( 𝑑 𝑒 ) < 0 , 𝐽 𝑑 𝑑 πœ† ( 𝑑 𝑒 ) = 𝑑 𝑝 βˆ’ 1 β€– 𝑒 β€– 𝑝 βˆ’ 𝑑 𝑝 βˆ— βˆ’ 1 ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ βˆ’ 𝑑 π‘ž βˆ’ 1 πœ† ξ€œ Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ = 0 f o r 𝑑 = 𝑑 βˆ’ . ( 2 . 3 1 ) Thus, 𝐽 πœ† ( 𝑑 βˆ’ 𝑒 ) = s u p 𝑑 β‰₯ 0 𝐽 πœ† ( 𝑑 𝑒 ) .
(ii) We have ∫ Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ > 0 . By (2.29) and ξ€œ π‘˜ ( 0 ) = 0 < πœ† Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ ≀ πœ† 𝑆 βˆ’ π‘ž / 𝑝 | | Ξ© | | ( 𝑝 βˆ— βˆ’ π‘ž ) / 𝑝 βˆ— β€– 𝑒 β€– π‘ž | | 𝑓 + | | ∞ < β€– 𝑒 β€– π‘ž ξ‚΅ 𝑝 βˆ— βˆ’ 𝑝 𝑝 βˆ— ξ‚Ά  βˆ’ π‘ž 𝑝 βˆ’ π‘ž ( 𝑝 βˆ— | | 𝑔 βˆ’ π‘ž ) + | | ∞ 𝑆 𝑝 βˆ— / 𝑝 ξƒͺ ( 𝑝 βˆ’ π‘ž ) / ( 𝑝 βˆ— βˆ’ 𝑝 ) ξ€· 𝑑 ≀ π‘˜ m a x ξ€Έ f o r ξ€· πœ† ∈ 0 , Ξ› 1 ξ€Έ , ( 2 . 3 2 ) there are unique 𝑑 + and 𝑑 βˆ’ such that 0 < 𝑑 + < 𝑑 m a x < 𝑑 βˆ’ , π‘˜ ξ€· 𝑑 + ξ€Έ ξ€œ = πœ† Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ = π‘˜ ( 𝑑 βˆ’ ) , π‘˜ ξ…ž ξ€· 𝑑 + ξ€Έ > 0 > π‘˜ ξ…ž ( 𝑑 βˆ’ ) . ( 2 . 3 3 ) We have 𝑑 + 𝑒 ∈ 𝒩 + πœ† , 𝑑 βˆ’ 𝑒 ∈ 𝒩 βˆ’ πœ† , and 𝐽 πœ† ( 𝑑 βˆ’ 𝑒 ) β‰₯ 𝐽 πœ† ( 𝑑 𝑒 ) β‰₯ 𝐽 πœ† ( 𝑑 + 𝑒 ) for each 𝑑 ∈ [ 𝑑 + , 𝑑 βˆ’ ] and 𝐽 πœ† ( 𝑑 + 𝑒 ) ≀ 𝐽 πœ† ( 𝑑 𝑒 ) for each 𝑑 ∈ [ 0 , 𝑑 + ] . Thus, 𝐽 πœ† ξ€· 𝑑 + 𝑒 ξ€Έ = i n f 0 ≀ 𝑑 ≀ 𝑑 m a x 𝐽 πœ† ( 𝑑 𝑒 ) , 𝐽 πœ† ( 𝑑 βˆ’ 𝑒 ) = s u p 𝑑 β‰₯ 0 𝐽 πœ† ( 𝑑 𝑒 ) . ( 2 . 3 4 ) This completes the proof.

3. Proof of Theorem 1.4

First, we will use the idea of Tarantello [11] to get the following results.

Lemma 3.1. If πœ† ∈ ( 0 , Ξ› 1 ) , then for each 𝑒 ∈ 𝒩 πœ† , there exist πœ– > 0 and a differentiable function πœ‰ ∢ 𝐡 ( 0 ; πœ– ) βŠ‚ π‘Š β†’ ℝ + such that πœ‰ ( 0 ) = 1 , the function πœ‰ ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) ∈ 𝒩 πœ† , and  πœ‰ ξ…ž  = 𝑝 ∫ ( 0 ) , 𝑣 Ξ© | | | | βˆ‡ 𝑒 𝑝 βˆ’ 2 ∫ βˆ‡ 𝑒 βˆ‡ 𝑣 𝑑 π‘₯ βˆ’ πœ† π‘ž Ξ© 𝑓 | 𝑒 | π‘ž βˆ’ 2 𝑒 𝑣 𝑑 π‘₯ βˆ’ 𝑝 βˆ— ∫ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— βˆ’ 2 𝑒 𝑣 𝑑 π‘₯ ( 𝑝 βˆ’ π‘ž ) β€– 𝑒 β€– 𝑝 βˆ’ ( 𝑝 βˆ— ∫ βˆ’ π‘ž ) Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ ( 3 . 1 ) for all 𝑣 ∈ π‘Š .

Proof. For 𝑒 ∈ 𝒩 πœ† , define a function 𝐹 ∢ ℝ Γ— π‘Š β†’ ℝ by 𝐹 𝑒  𝐽 ( πœ‰ , 𝑀 ) = ξ…ž πœ†  ( πœ‰ ( 𝑒 βˆ’ 𝑀 ) ) , πœ‰ ( 𝑒 βˆ’ 𝑀 ) = πœ‰ 𝑝 ξ€œ Ξ© | | | | βˆ‡ ( 𝑒 βˆ’ 𝑀 ) 𝑝 𝑑 π‘₯ βˆ’ πœ‰ π‘ž πœ† ξ€œ Ξ© 𝑓 | 𝑒 βˆ’ 𝑀 | π‘ž 𝑑 π‘₯ βˆ’ πœ‰ 𝑝 βˆ— ξ€œ Ξ© 𝑔 | 𝑒 βˆ’ 𝑀 | 𝑝 βˆ— 𝑑 π‘₯ . ( 3 . 2 ) Then 𝐹 𝑒 ( 1 , 0 ) = ⟨ 𝐽 ξ…ž πœ† ( 𝑒 ) , 𝑒 ⟩ = 0 and 𝑑 𝐹 𝑑 πœ‰ 𝑒 ( 1 , 0 ) = 𝑝 β€– 𝑒 β€– 𝑝 ξ€œ βˆ’ πœ† π‘ž Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ βˆ’ 𝑝 βˆ— ξ€œ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ = ( 𝑝 βˆ’ π‘ž ) β€– 𝑒 β€– 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ ξ€œ βˆ’ π‘ž Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ β‰  0 . ( 3 . 3 ) According to the implicit function theorem, there exist πœ– > 0 and a differentiable function πœ‰ ∢ 𝐡 ( 0 ; πœ– ) βŠ‚ π‘Š β†’ ℝ such that πœ‰ ( 0 ) = 1 ,  πœ‰ ξ…ž  = 𝑝 ∫ ( 0 ) , 𝑣 Ξ© | | | | βˆ‡ 𝑒 𝑝 βˆ’ 2 ∫ βˆ‡ 𝑒 βˆ‡ 𝑣 𝑑 π‘₯ βˆ’ πœ† π‘ž Ξ© 𝑓 | 𝑒 | π‘ž βˆ’ 2 𝑒 𝑣 𝑑 π‘₯ βˆ’ 𝑝 βˆ— ∫ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— βˆ’ 2 𝑒 𝑣 𝑑 π‘₯ ( 𝑝 βˆ’ π‘ž ) β€– 𝑒 β€– 𝑝 βˆ’ ( 𝑝 βˆ— ∫ βˆ’ π‘ž ) Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— , 𝐹 𝑑 π‘₯ 𝑒 ( πœ‰ ( 𝑣 ) , 𝑣 ) = 0 , βˆ€ 𝑣 ∈ 𝐡 ( 0 ; πœ– ) , ( 3 . 4 ) which is equivalent to  𝐽 ξ…ž πœ†  ( πœ‰ ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) ) , πœ‰ ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) = 0 , βˆ€ 𝑣 ∈ 𝐡 ( 0 ; πœ– ) , ( 3 . 5 ) that is, πœ‰ ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) ∈ 𝒩 πœ† .

Lemma 3.2. Let πœ† ∈ ( 0 , Ξ› 1 ) , then for each 𝑒 ∈ 𝒩 βˆ’ πœ† , there exist πœ– > 0 and a differentiable function πœ‰ βˆ’ ∢ 𝐡 ( 0 ; πœ– ) βŠ‚ π‘Š β†’ ℝ + such that πœ‰ βˆ’ ( 0 ) = 1 , the function πœ‰ βˆ’ ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) ∈ 𝒩 βˆ’ πœ† , and  ( πœ‰ βˆ’ ) ξ…ž  = 𝑝 ∫ ( 0 ) , 𝑣 Ξ© | | | | βˆ‡ 𝑒 𝑝 βˆ’ 2 ∫ βˆ‡ 𝑒 βˆ‡ 𝑣 𝑑 π‘₯ βˆ’ πœ† π‘ž Ξ© 𝑓 | 𝑒 | π‘ž βˆ’ 2 𝑒 𝑣 𝑑 π‘₯ βˆ’ 𝑝 βˆ— ∫ Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— βˆ’ 2 𝑒 𝑣 𝑑 π‘₯ ( 𝑝 βˆ’ π‘ž ) β€– 𝑒 β€– 𝑝 βˆ’ ( 𝑝 βˆ— ∫ βˆ’ π‘ž ) Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ ( 3 . 6 ) for all 𝑣 ∈ π‘Š .

Proof. Similar to the argument in Lemma 3.1, there exist πœ– > 0 and a differentiable function πœ‰ βˆ’ ∢ 𝐡 ( 0 ; πœ– ) βŠ‚ π‘Š β†’ ℝ such that πœ‰ βˆ’ ( 0 ) = 1 and πœ‰ βˆ’ ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) ∈ 𝒩 πœ† for all 𝑣 ∈ 𝐡 ( 0 ; πœ– ) . Since  πœ“ ξ…ž πœ† (  𝑒 ) , 𝑒 = ( 𝑝 βˆ’ π‘ž ) β€– 𝑒 β€– 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ ξ€œ βˆ’ π‘ž Ξ© 𝑔 | 𝑒 | 𝑝 βˆ— 𝑑 π‘₯ < 0 . ( 3 . 7 ) Thus, by the continuity of the function πœ‰ βˆ’ , we have  πœ“ ξ…ž πœ† ( πœ‰ βˆ’ ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) ) , πœ‰ βˆ’  = ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) ( 𝑝 βˆ’ π‘ž ) β€– πœ‰ βˆ’ β€– ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ ξ€œ βˆ’ π‘ž Ξ© 𝑔 | | πœ‰ βˆ’ | | ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) 𝑝 𝑑 π‘₯ < 0 , ( 3 . 8 ) if πœ– sufficiently small, this implies that πœ‰ βˆ’ ( 𝑣 ) ( 𝑒 βˆ’ 𝑣 ) ∈ 𝒩 βˆ’ πœ† .

Proposition 3.3. (i) If πœ† ∈ ( 0 , Ξ› 1 ) , then there exists a (PS) 𝛼 πœ† -sequence { 𝑒 𝑛 } βŠ‚ 𝒩 πœ† in π‘Š for 𝐽 πœ† .
(ii) If πœ† ∈ ( 0 , ( π‘ž / 𝑝 ) Ξ› 1 ) , then there exists a (PS) 𝛼 βˆ’ πœ† -sequence { 𝑒 𝑛 } βŠ‚ 𝒩 βˆ’ πœ† in π‘Š for 𝐽 πœ† .

Proof. (i) By Lemma 2.2 and the Ekeland variational principle [14], there exists a minimizing sequence { 𝑒 𝑛 } βŠ‚ 𝒩 πœ† such that 𝐽 πœ† ξ€· 𝑒 𝑛 ξ€Έ < 𝛼 πœ† + 1 𝑛 , 𝐽 πœ† ξ€· 𝑒 𝑛 ξ€Έ < 𝐽 πœ† 1 ( 𝑀 ) + 𝑛 β€– β€– 𝑀 βˆ’ 𝑒 𝑛 β€– β€– f o r e a c h 𝑀 ∈ 𝒩 πœ† . ( 3 . 9 ) By 𝛼 πœ† < 0 and taking 𝑛 large, we have 𝐽 πœ† ξ€· 𝑒 𝑛 ξ€Έ = ξ‚΅ 1 𝑝 βˆ’ 1 𝑝 βˆ— ξ‚Ά β€– β€– 𝑒 𝑛 β€– β€– 𝑝 βˆ’ ξ‚΅ 1 π‘ž βˆ’ 1 𝑝 βˆ— ξ‚Ά πœ† ξ€œ Ξ© 𝑓 | | 𝑒 𝑛 | | π‘ž 𝑑 π‘₯ < 𝛼 πœ† + 1 𝑛 < 𝛼 πœ† 𝑝 . ( 3 . 1 0 ) From (2.7), (3.10), 𝛼 πœ† < 0 , and the HΓΆlder inequality, we deduce that | | 𝑓 + | | ∞ πœ† 𝑆 βˆ’ π‘ž / 𝑝 | | Ξ© | | ( 𝑝 βˆ— βˆ’ π‘ž ) / 𝑝 βˆ— β€– β€– 𝑒 𝑛 β€– β€– π‘ž ξ€œ β‰₯ πœ† Ξ© 𝑓 | | 𝑒 𝑛 | | π‘ž 𝑑 π‘₯ > βˆ’ 𝑝 βˆ— π‘ž 𝑝 ( 𝑝 βˆ— 𝛼 βˆ’ π‘ž ) πœ† > 0 . ( 3 . 1 1 ) Consequently, 𝑒 𝑛 β‰  0 and putting together (3.10), (3.11), and the HΓΆlder inequality, we obtain β€– β€– 𝑒 𝑛 β€– β€– >  βˆ’ 𝑝 βˆ— π‘ž 𝑝 πœ† ( 𝑝 βˆ— ) | | 𝑓 βˆ’ π‘ž + | | ∞ 𝛼 πœ† 𝑆 π‘ž / 𝑝 | | Ξ© | | ( π‘ž βˆ’ 𝑝 βˆ— ) / 𝑝 βˆ— ξƒ­ 1 / π‘ž , β€– β€– 𝑒 𝑛 β€– β€– <  𝑝 ξ€· 𝑝 βˆ— ξ€Έ βˆ’ π‘ž π‘ž ( 𝑝 βˆ— βˆ’ 𝑝 ) πœ† 𝑆 βˆ’ π‘ž / 𝑝 | | Ξ© | | ( 𝑝 βˆ— βˆ’ π‘ž ) / 𝑝 βˆ— | | 𝑓 + | | ∞ ξƒ­ 1 / ( 𝑝 βˆ’ π‘ž ) . ( 3 . 1 2 ) Now, we show that β€– β€– 𝐽 ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ β€– β€– π‘Š βˆ’ 1 ⟢ 0 , a s 𝑛 ⟢ ∞ . ( 3 . 1 3 ) Apply Lemma 3.1 with 𝑒 𝑛 to obtain the functions πœ‰ 𝑛 ∢ 𝐡 ( 0 ; πœ– 𝑛 ) β†’ ℝ + for some πœ– 𝑛 > 0 , such that πœ‰ 𝑛 ( 𝑀 ) ( 𝑒 𝑛 βˆ’ 𝑀 ) ∈ 𝒩 πœ† . Choose 0 < 𝜌 < πœ– 𝑛 . Let 𝑒 ∈ π‘Š with 𝑒 β‰’ 0 and let 𝑀 𝜌 = 𝜌 𝑒 / β€– 𝑒 β€– . We set πœ‚ 𝜌 = πœ‰ 𝑛 ( 𝑀 𝜌 ) ( 𝑒 𝑛 βˆ’ 𝑀 𝜌 ) . Since πœ‚ 𝜌 ∈ 𝒩 πœ† , we deduce from (3.9) that 𝐽 πœ† ξ€· πœ‚ 𝜌 ξ€Έ βˆ’ 𝐽 πœ† ξ€· 𝑒 𝑛 ξ€Έ 1 β‰₯ βˆ’ 𝑛 β€– β€– πœ‚ 𝜌 βˆ’ 𝑒 𝑛 β€– β€– , ( 3 . 1 4 ) and by the mean value theorem, we have  𝐽 ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ , πœ‚ 𝜌 βˆ’ 𝑒 𝑛  ξ€· β€– β€– πœ‚ + π‘œ 𝜌 βˆ’ 𝑒 𝑛 β€– β€– ξ€Έ 1 β‰₯ βˆ’ 𝑛 β€– β€– πœ‚ 𝜌 βˆ’ 𝑒 𝑛 β€– β€– . ( 3 . 1 5 ) Thus,  𝐽 ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ , βˆ’ 𝑀 𝜌  + ξ€· πœ‰ 𝑛 ξ€· 𝑀 𝜌 ξ€Έ 𝐽 βˆ’ 1 ξ€Έ  ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ , ξ€· 𝑒 𝑛 βˆ’ 𝑀 𝜌 1 ξ€Έ  β‰₯ βˆ’ 𝑛 β€– β€– πœ‚ 𝜌 βˆ’ 𝑒 𝑛 β€– β€– ξ€· β€– β€– πœ‚ + π‘œ 𝜌 βˆ’ 𝑒 𝑛 β€– β€– ξ€Έ . ( 3 . 1 6 ) Since πœ‰ 𝑛 ( 𝑀 𝜌 ) ( 𝑒 𝑛 βˆ’ 𝑀 𝜌 ) ∈ 𝒩 πœ† and (3.16) it follows that  𝐽 βˆ’ 𝜌 ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ , 𝑒 ξƒ’ + ξ€· πœ‰ β€– 𝑒 β€– 𝑛 ξ€· 𝑀 𝜌 ξ€Έ 𝐽 βˆ’ 1 ξ€Έ  ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ βˆ’ 𝐽 ξ…ž πœ† ξ€· πœ‚ 𝜌 ξ€Έ , ξ€· 𝑒 𝑛 βˆ’ 𝑀 𝜌 1 ξ€Έ  β‰₯ βˆ’ 𝑛 β€– β€– πœ‚ 𝜌 βˆ’ 𝑒 𝑛 β€– β€– ξ€· β€– β€– πœ‚ + π‘œ 𝜌 βˆ’ 𝑒 𝑛 β€– β€– ξ€Έ . ( 3 . 1 7 ) Thus,  𝐽 ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ , 𝑒 β€– ξƒ’ ≀ β€– β€– πœ‚ 𝑒 β€– 𝜌 βˆ’ 𝑒 𝑛 β€– β€– + π‘œ ξ€· β€– β€– πœ‚ 𝑛 𝜌 𝜌 βˆ’ 𝑒 𝑛 β€– β€– ξ€Έ 𝜌 + ξ€· πœ‰ 𝑛 ξ€· 𝑀 𝜌 ξ€Έ ξ€Έ βˆ’ 1 𝜌  𝐽 ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ βˆ’ 𝐽 ξ…ž πœ† ξ€· πœ‚ 𝜌 ξ€Έ , ξ€· 𝑒 𝑛 βˆ’ 𝑀 𝜌 . ξ€Έ  ( 3 . 1 8 ) Since β€– πœ‚ 𝜌 βˆ’ 𝑒 𝑛 β€– ≀ 𝜌 πœ‰ 𝑛 ( 𝑀 𝜌 ) + | πœ‰ 𝑛 ( 𝑀 𝜌 ) βˆ’ 1 | β€– 𝑒 𝑛 β€– and l i m 𝜌 β†’ 0 | | πœ‰ 𝑛 ξ€· 𝑀 𝜌 ξ€Έ | | βˆ’ 1 𝜌 ≀ β€– β€– πœ‰ ξ…ž 𝑛 β€– β€– , ( 0 ) ( 3 . 1 9 ) if we let 𝜌 β†’ 0 in (3.18) for a fixed 𝑛 , then by (3.12) we can find a constant 𝐢 > 0 , independent of 𝜌 , such that  𝐽 ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ , 𝑒 ξƒ’ ≀ 𝐢 β€– 𝑒 β€– 𝑛 ξ€· β€– β€– πœ‰ 1 + ξ…ž 𝑛 β€– β€– ξ€Έ . ( 0 ) ( 3 . 2 0 ) The proof will be complete once we show that β€– πœ‰ ξ…ž 𝑛 ( 0 ) β€– is uniformly bounded in 𝑛 . By (3.1), (3.12), ( 𝑓 1 ) , ( 𝑔 1 ) , and the HΓΆlder inequality and the Sobolev embedding theorem, we have  πœ‰ ξ…ž 𝑛  ≀ ( 0 ) , 𝑣 𝑏 β€– 𝑣 β€– | | β€– β€– 𝑒 ( 𝑝 βˆ’ π‘ž ) 𝑛 β€– β€– 𝑝 βˆ’ ( 𝑝 βˆ— ) ∫ βˆ’ π‘ž Ξ© 𝑔 | | 𝑒 𝑛 | | 𝑝 βˆ— | | 𝑑 π‘₯ f o r s o m e 𝑏 > 0 . ( 3 . 2 1 ) We only need to show that | | | | β€– β€– 𝑒 ( 𝑝 βˆ’ π‘ž ) 𝑛 β€– β€– 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ ξ€œ βˆ’ π‘ž Ξ© 𝑔 | | 𝑒 𝑛 | | 𝑝 βˆ— | | | | 𝑑 π‘₯ > 𝐢 ( 3 . 2 2 ) for some 𝐢 > 0 and 𝑛 large enough. We argue by contradiction. Assume that there exists a subsequence { 𝑒 𝑛 } such that ( β€– β€– 𝑒 𝑝 βˆ’ π‘ž ) 𝑛 β€– β€– 𝑝 βˆ’ ξ€· 𝑝 βˆ— ξ€Έ ξ€œ βˆ’ π‘ž Ξ© 𝑔 | | 𝑒 𝑛 | | 𝑝 βˆ— 𝑑 π‘₯ = π‘œ 𝑛 ( 1 ) . ( 3 . 2 3 ) By (3.23) and the fact that 𝑒 𝑛 ∈ 𝒩 πœ† , we get β€– β€– 𝑒 𝑛 β€– β€– 𝑝 = 𝑝 βˆ— βˆ’ π‘ž ξ€œ 𝑝 βˆ’ π‘ž Ξ© 𝑔 | | 𝑒 𝑛 | | 𝑝 βˆ— 𝑑 π‘₯ + π‘œ 𝑛 β€– β€– 𝑒 ( 1 ) , 𝑛 β€– β€– 𝑝 𝑝 = πœ† βˆ— βˆ’ π‘ž 𝑝 βˆ— ξ€œ βˆ’ 𝑝 Ξ© 𝑓 | | 𝑒 𝑛 | | π‘ž 𝑑 π‘₯ + π‘œ 𝑛 ( 1 ) . ( 3 . 2 4 ) Moreover, by ( 𝑓 1 ) , ( 𝑔 1 ) , and the HΓΆlder inequality and the Sobolev embedding theorem, we have β€– β€– 𝑒 𝑛 β€– β€– β‰₯  𝑝 βˆ’ π‘ž ( 𝑝 βˆ— | | 𝑔 βˆ’ π‘ž ) + | | ∞ 𝑆 𝑝 βˆ— / 𝑝 ξƒ­ 1 / ( 𝑝 βˆ— βˆ’ 𝑝 ) + π‘œ 𝑛 β€– β€– 𝑒 ( 1 ) , 𝑛 β€– β€– ≀  πœ† ξ€· 𝑝 βˆ— ξ€Έ | | 𝑓 βˆ’ π‘ž + | | ∞ 𝑝 βˆ— 𝑆 βˆ’ 𝑝 βˆ’ π‘ž / 𝑝 | | Ξ© | | ( 𝑝 βˆ— βˆ’ π‘ž ) / 𝑝 βˆ— ξƒ­ 1 / ( 𝑝 βˆ’ π‘ž ) + π‘œ 𝑛 ( 1 ) . ( 3 . 2 5 ) This implies πœ† β‰₯ Ξ› 1 which is a contradiction. We obtain  𝐽 ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ , 𝑒 ξƒ’ ≀ 𝐢 β€– 𝑒 β€– 𝑛 . ( 3 . 2 6 ) This completes the proof of (i).
(ii) Similarly, by using Lemma 3.2, we can prove (ii). We will omit detailed proof here.

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

Theorem 3.4. If πœ† ∈ ( 0 , Ξ› 1 ) , then 𝐽 πœ† has a minimizer 𝑒 πœ† in 𝒩 + πœ† and it satisfies that
(i) 𝐽 πœ† ( 𝑒 πœ† ) = 𝛼 πœ† = 𝛼 + πœ† ;
(ii) 𝑒 πœ† is a positive solution of ( 𝐸 πœ† 𝑓 , 𝑔 ) in 𝐢 1 , 𝛼 ( Ξ© ) for some 𝛼 ∈ ( 0 , 1 ) .

Proof. By Proposition 3.3(i), there exists a minimizing sequence { 𝑒 𝑛 } for 𝐽 πœ† on 𝒩 πœ† such that 𝐽 πœ† ξ€· 𝑒 𝑛 ξ€Έ = 𝛼 πœ† + π‘œ 𝑛 ( 1 ) , 𝐽 ξ…ž πœ† ξ€· 𝑒 𝑛 ξ€Έ = π‘œ 𝑛 ( 1 ) i n π‘Š βˆ’ 1 . ( 3 . 2 7 ) Since 𝐽 πœ† is coercive on 𝒩 πœ† (see Lemma 2.2), we get that { 𝑒 𝑛 } is bounded in π‘Š . Going if necessary to a subsequence, we can assume that there exists 𝑒 πœ† ∈ π‘Š such that 𝑒 𝑛 ⇀ 𝑒 πœ† w e a k l y i n 𝑒 π‘Š , 𝑛 ⟢ 𝑒 πœ† a l m o s t e v e r y w h e r e i n 𝑒 Ξ© , 𝑛 ⟢ 𝑒 πœ† s t r o n g l y i n 𝐿 𝑠 ( Ξ© ) βˆ€ 1 ≀ 𝑠 < 𝑝 βˆ— . ( 3 . 2 8 ) First, we claim that 𝑒 πœ† is a nontrivial solution of ( 𝐸 πœ† 𝑓 , 𝑔 ). By (3.27) and (3.28), it is easy to see that 𝑒 πœ† is a solution of ( 𝐸 πœ† 𝑓 , 𝑔 ). From 𝑒 𝑛 ∈ 𝒩 πœ† and (2.6), we deduce that πœ† ξ€œ Ξ© 𝑓 | | 𝑒 𝑛 | | π‘ž π‘ž ξ€· 𝑝 𝑑 π‘₯ = βˆ— ξ€Έ βˆ’ 𝑝 𝑝 ( 𝑝 βˆ— β€– β€– 𝑒 βˆ’ π‘ž ) 𝑛 β€– β€– 𝑝 βˆ’ 𝑝 βˆ— π‘ž 𝑝 βˆ— 𝐽 βˆ’ π‘ž πœ† ξ€· 𝑒 𝑛 ξ€Έ . ( 3 . 2 9 ) Let 𝑛 β†’ ∞ in (3.29), by (3.27), (3.28), and 𝛼 πœ† < 0 , we get ξ€œ Ξ© 𝑓 | | 𝑒 πœ† | | π‘ž 𝑝 𝑑 π‘₯ β‰₯ βˆ’ βˆ— π‘ž 𝑝 βˆ— 𝛼 βˆ’ π‘ž πœ† > 0 . ( 3 . 3 0 ) Thus, 𝑒 πœ† ∈ 𝒩 πœ† is a nontrivial solution of ( 𝐸 πœ† 𝑓 , 𝑔 ). Now we prove that 𝑒 𝑛 β†’ 𝑒 πœ† strongly in π‘Š and 𝐽 πœ† ( 𝑒 πœ† ) = 𝛼 πœ† . By (3.29), if 𝑒 ∈ 𝒩 πœ† , then 𝐽 πœ† 𝑝 ( 𝑒 ) = βˆ— βˆ’ 𝑝 𝑝 βˆ— 𝑝 β€– 𝑒 β€– 𝑝 βˆ’ 𝑝 βˆ— βˆ’ π‘ž 𝑝 βˆ— π‘ž πœ† ξ€œ Ξ© 𝑓 | 𝑒 | π‘ž 𝑑 π‘₯ . ( 3 . 3 1 ) In order to prove that 𝐽 πœ† ( 𝑒 πœ† ) = 𝛼 πœ† , it suffices to recall that 𝑒 πœ† ∈ 𝒩 πœ† , by (3.31), and applying Fatou's lemma to get 𝛼 πœ† ≀ 𝐽 πœ† ξ€· 𝑒 πœ† ξ€Έ = 𝑝 βˆ— βˆ’ 𝑝 𝑝 βˆ— 𝑝 β€– β€– 𝑒 πœ† β€– β€– 𝑝 βˆ’ 𝑝 βˆ— βˆ’ π‘ž 𝑝 βˆ— π‘ž πœ† ξ€œ Ξ© 𝑓 | | 𝑒 πœ† | | π‘ž 𝑑 π‘₯ ≀ l i m i n f 𝑛 β†’ ∞ ξ‚΅ 𝑝 βˆ— βˆ’ 𝑝 𝑝 βˆ— 𝑝 β€– β€– 𝑒 𝑛 β€– β€– 𝑝 βˆ’ 𝑝 βˆ— βˆ’ π‘ž 𝑝 βˆ— π‘ž πœ† ξ€œ Ξ© 𝑓 | | 𝑒 𝑛 | | π‘ž ξ‚Ά 𝑑 π‘₯ ≀ l i m i n f 𝑛 β†’ ∞ 𝐽 πœ† ξ€· 𝑒 𝑛 ξ€Έ = 𝛼 πœ† . ( 3 . 3 2 ) This implies that 𝐽 πœ† ( 𝑒 πœ† ) = 𝛼 πœ† and l i m 𝑛 β†’ ∞ β€– 𝑒 𝑛 β€– 𝑝 = β€– 𝑒 πœ† β€– 𝑝 . Let 𝑣 𝑛 = 𝑒 𝑛 βˆ’ 𝑒 πœ† , then BrΓ©zis and Lieb lemma [15] implies that β€– β€– 𝑣 𝑛 β€– β€– 𝑝 = β€– β€– 𝑒 𝑛 β€– β€– 𝑝 βˆ’ β€– β€– 𝑒 πœ† β€– β€– 𝑝 + π‘œ 𝑛 ( 1 ) . ( 3 . 3 3 ) Therefore, 𝑒 𝑛 β†’ 𝑒 πœ† strongly in π‘Š . Moreover, we have 𝑒 πœ† ∈ 𝒩 + πœ† . On the contrary, if 𝑒 πœ† ∈ 𝒩 βˆ’ πœ† , then by Lemma 2.7, there are unique 𝑑 + 0 and 𝑑 βˆ’ 0 such that 𝑑 + 0 𝑒 πœ† ∈ 𝒩 + πœ† and 𝑑 βˆ’ 0 𝑒 πœ† ∈ 𝒩 βˆ’ πœ† . In particular, we have 𝑑 + 0 < 𝑑 βˆ’ 0 = 1 . Since 𝑑 𝐽 𝑑 𝑑 πœ† ξ€· 𝑑 + 0 𝑒 πœ† ξ€Έ 𝑑 = 0 , 2 𝑑 𝑑 2 𝐽 πœ† ξ€· 𝑑 + 0 𝑒 πœ† ξ€Έ > 0 , ( 3 . 3 4 ) there exists 𝑑 + 0 < 𝑑 ≀ 𝑑 βˆ’ 0 such that 𝐽 πœ† ( 𝑑 + 0 𝑒 πœ† ) < 𝐽 πœ† ( 𝑑 𝑒 πœ† ) . By Lemma 2.7, 𝐽 πœ† ξ€· 𝑑 + 0 𝑒 πœ† ξ€Έ < 𝐽 πœ† ξ€· 𝑑 𝑒 πœ† ξ€Έ ≀ 𝐽 πœ† ξ€· 𝑑 βˆ’ 0 𝑒 πœ† ξ€Έ = 𝐽 πœ† ξ€· 𝑒 πœ† ξ€Έ , ( 3 . 3 5 ) which is a contradiction. Since 𝐽 πœ† ( 𝑒 πœ† ) = 𝐽 πœ† ( | 𝑒 πœ† | ) and | 𝑒 πœ† | ∈ 𝒩 + πœ† , by Lemma 2.3 we may assume that 𝑒 πœ† is a nontrivial nonnegative solution of ( 𝐸 πœ† 𝑓 , 𝑔 ). Moreover, from 𝑓 , 𝑔 ∈ 𝐿 ∞ ( Ξ© ) , then using the standard bootstrap argument (see, e.g., [16]) we obtain 𝑒 πœ† ∈ 𝐿 ∞ ( Ξ© ) ; hence by applying regularity results [17, 18] we derive that 𝑒 πœ† ∈ C 1 , 𝛼 ( Ξ© ) for some 𝛼 ∈ ( 0 , 1 ) and finally, by the Harnack inequality [19] we deduce that 𝑒 πœ† > 0 . This completes the proof.

Now, we begin the proof of Theorem 1.4. By Theorem 3.4, we obtain ( 𝐸 πœ† 𝑓 , 𝑔 ) that has a positive solution 𝑒 πœ† in 𝐢 1 , 𝛼 ( Ξ© ) for some 𝛼 ∈ ( 0 , 1 ) .

4. Proof of Theorem 1.5

Next, we will establish the existence of the second positive solution of ( 𝐸 πœ† 𝑓 , 𝑔 ) by proving that 𝐽 πœ† satisfies the (PS) 𝛼 βˆ’ πœ† condition.

Lemma 4.1. Assume that ( 𝑓 1 ) and ( 𝑔 1 ) hold. If { 𝑒 𝑛 } βŠ‚ π‘Š is a (PS) 𝑐 -sequence for 𝐽 πœ† , then { 𝑒 𝑛 } is bounded in π‘Š .

Proof. We argue by contradiction. Assume that β€– 𝑒 𝑛 β€– β†’ ∞ . Let Μ‚ 𝑒 𝑛 = 𝑒 𝑛 / β€– 𝑒 𝑛 β€– . We may assume that Μ‚ 𝑒 𝑛 ⇀ Μ‚ 𝑒 in π‘Š . This implies that Μ‚ 𝑒 𝑛 β†’ Μ‚ 𝑒 strongly in 𝐿 𝑠 ( Ξ© ) for all 1 ≀ 𝑠 < 𝑝 βˆ— and πœ† π‘ž ξ€œ Ξ© 𝑓 | | Μ‚ 𝑒 𝑛 | | π‘ž πœ† 𝑑 π‘₯ = π‘ž ξ€œ Ξ© 𝑓 | | | | Μ‚ 𝑒 π‘ž 𝑑 π‘₯ + π‘œ 𝑛 ( 1 ) . ( 4 . 1 ) Since { 𝑒 𝑛 } is a (PS) 𝑐 -sequence for 𝐽 πœ† and β€– 𝑒 𝑛 β€– β†’ ∞ , there hold 1 𝑝 ξ€œ Ξ© | | βˆ‡ Μ‚ 𝑒 𝑛 | | 𝑝 πœ† β€– β€– 𝑒 𝑑 π‘₯ βˆ’ 𝑛 β€– β€– π‘ž βˆ’ 𝑝 π‘ž ξ€œ Ξ© 𝑓 | | Μ‚ 𝑒 𝑛 | | π‘ž β€– β€– 𝑒 𝑑 π‘₯ βˆ’ 𝑛 β€– β€– 𝑝 βˆ— βˆ’ 𝑝 𝑝 βˆ— ξ€œ Ξ© 𝑔 | | Μ‚ 𝑒 𝑛 | | 𝑝 βˆ— 𝑑 π‘₯ = π‘œ 𝑛 ξ€œ ( 1 ) , Ξ© | | βˆ‡ Μ‚ 𝑒 𝑛 | | 𝑝 β€– β€– 𝑒 𝑑 π‘₯ βˆ’ πœ† 𝑛 β€– β€– π‘ž βˆ’ 𝑝 ξ€œ Ξ© 𝑓 | | Μ‚ 𝑒 𝑛 | | π‘ž β€– β€– 𝑒 𝑑 π‘₯ βˆ’ 𝑛 β€– β€– 𝑝 βˆ— βˆ’ 𝑝 ξ€œ Ξ© 𝑔 | | Μ‚ 𝑒 𝑛 | | 𝑝 βˆ— 𝑑 π‘₯ = π‘œ 𝑛 ( 1 ) . ( 4 . 2 ) From (4.1)-(4.2), we can deduce that ξ€œ Ξ© | | βˆ‡ Μ‚ 𝑒 𝑛 | | 𝑝 𝑝 ξ€· 𝑝 𝑑 π‘₯ = βˆ— ξ€Έ βˆ’ π‘ž π‘ž ( 𝑝 βˆ— β€– β€– 𝑒 βˆ’ 𝑝 ) 𝑛 β€– β€– π‘ž βˆ’ 𝑝 πœ† ξ€œ Ξ© 𝑓 | | | | Μ‚ 𝑒 π‘ž 𝑑 π‘₯ + π‘œ 𝑛 ( 1 ) . ( 4 . 3 ) Since 1 ≀ π‘ž < 2 and β€– 𝑒 𝑛 β€– β†’ ∞ , (4.3) implies ξ€œ Ξ© | | βˆ‡ Μ‚ 𝑒 𝑛 | | 𝑝 𝑑 π‘₯ ⟢ 0 , a s 𝑛 ⟢ ∞ , ( 4 . 4 ) which is contrary to the fact β€– Μ‚ 𝑒 𝑛 β€– = 1 for all 𝑛 .

Lemma 4.2. Assume that ( 𝑓 1 ) and ( 𝑔 1 ) hold. If { 𝑒 𝑛 } βŠ‚ π‘Š is a (PS) 𝑐 -sequence for 𝐽 πœ† with 𝑐 ∈ ( 0 , ( 1 / 𝑁 ) | 𝑔 + | ∞ βˆ’ ( 𝑁 βˆ’ 𝑝 ) / 𝑝 𝑆 𝑁 / 𝑝 ) , then there exists a subsequence of { 𝑒 𝑛 } converging weakly to a nontrivial solution of ( 𝐸 πœ† 𝑓 , 𝑔 ).

Proof. Let { 𝑒 𝑛 } βŠ‚ π‘Š be a (PS) 𝑐 -sequence for 𝐽 πœ† with 𝑐 ∈ ( 0 , ( 1 / 𝑁 ) | 𝑔 + | ∞ βˆ’ ( 𝑁 βˆ’ 𝑝 ) / 𝑝 𝑆 𝑁 / 𝑝 ) . We know from Lemma 4.1 that { 𝑒 𝑛 } is bounded in π‘Š , and then there exists a subsequence of { 𝑒 𝑛 } (still denoted by { 𝑒 𝑛 } and 𝑒 0 ∈ π‘Š such that 𝑒 𝑛 ⇀ 𝑒 0 w e a k l y i n 𝑒 π‘Š , 𝑛 ⟢ 𝑒 0 a l m o s t e v e r y w h e r e i n 𝑒 Ξ© , 𝑛 ⟢ 𝑒 0 s t r o n g l y i n 𝐿 𝑠 ( Ξ© )