Boundary Value Problems
Volume 2009 (2009), Article ID 584203, 17 pages
doi:10.1155/2009/584203
Research Article

Multiple Positive Solutions for Singular Elliptic Equations with Concave-Convex Nonlinearities and Sign-Changing Weights

Tsing-San Hsu and Huei-Li Lin

Center for General Education, Chang Gung University, Kwei-Shan, Tao-Yuan 333, Taiwan

Received 5 December 2008; Accepted 11 March 2009

Academic Editor: Pavel Drabek

Copyright © 2009 Tsing-San Hsu and Huei-Li Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study existence and multiplicity of positive solutions for the following Dirichlet equations: − Δ 𝑢 − ( 𝜇 / | 𝑥 | 2 ) 𝑢 = 𝜆 𝑓 ( 𝑥 ) | 𝑢 | 𝑞 − 2 𝑢 + 𝑔 ( 𝑥 ) | 𝑢 | 2 ∗ − 2 𝑢 in Ω , 𝑢 = 0 on 𝜕 Ω , where 0 ∈ Ω ⊂ ℝ 𝑁 ( 𝑁 ≥ 3 ) is a bounded domain with smooth boundary 𝜕 Ω , 𝜆 > 0 , 0 ≤ 𝜇 < 𝜇 = ( 𝑁 − 2 ) 2 / 4 , 2 ∗ = 2 𝑁 / ( 𝑁 − 2 ) , 1 ≤ 𝑞 < 2 , and 𝑓 , 𝑔 are continuous functions on Ω which are somewhere positive but which may change sign on Ω .

1. Introduction and Main Results

In this paper, we study the existence and multiplicity of positive solutions for the following singular elliptic equation: where 𝑁 ≥ 3 ( 𝜕 Ω ) is a bounded domain with smooth boundary 𝜆 > 0 , 0 ≤ 𝜇 < 𝜇 = ( 𝑁 − 2 ) 2 / 4 , 𝜇 , 1 ≤ 𝑞 < 2 < 𝑝 is the best constant in the Hardy inequality, 𝑓 , 𝑔 ∶ Ω → ℝ , and Ω are continuous functions which are somewhere positive but which may change sign on 𝑝 . We will assume in this paper that 𝑝 = 2 ∗ = 2 𝑁 / ( 𝑁 − 2 ) is a critical Sobolev exponent, that is, 𝜇 = 0 .

When 𝑓 ( 𝑥 ) ≡ 𝑔 ( 𝑥 ) ≡ 1 and weight functions Ω on ( 𝑃 𝜇 , 𝜆 𝑓 , 𝑔 ) , 2 < 𝑝 ≤ 2 ∗ has been studied extensively for 𝑞 > 1 and various 𝜆 0 > 0 . See, for example, [1–3] and the references therein. In [4], Wu has proved that there exists 𝑃 𝜇 , 𝜆 , 𝑓 , 𝑔 such that ( 𝜆 ∈ ( 0 , 𝜆 0 ) ) admits at least two solutions for all 1 ≤ 𝑞 < 2 with 𝑝 ∈ ( 2 , 2 ∗ ) , a subcritical exponent 𝑔 ( 𝑥 ) ≡ 1 , Ω on 𝑓 and Ω is a continuous function which change sign in 𝑃 𝜇 , 𝜆 , 𝑓 , 𝑔 . In a recent work [5], Hsu-Lin have showed the existence and multiplicity of positive solutions of ( 𝑝 = 2 ∗ ) with a critical exponent 𝑓 , 𝑔 and sign-changing weight functions 𝑃 𝜇 , 𝜆 , 𝑓 , 𝑔 .

To proceed, we make some motivations of the present paper. In [6], Chen studied ( 0 ≤ 𝜇 < 𝜇 − 1 ) assuming that 1 ≤ 𝑞 < 2 , 𝑝 = 2 ∗ , 𝑓 ( 𝑥 ) ≡ 𝑔 ( 𝑥 ) ≡ 1 and Ω on Λ > 0 . He proved that there exists 𝑃 𝜇 , 𝜆 , 𝑓 , 𝑔 such that ( 𝐻 1 0 ( Ω ) ) has at least two positive solutions in 𝜆 ∈ ( 0 , Λ ) for any 𝑃 𝜇 , 𝜆 , 𝑓 , 𝑔 . But we do not see any multiplicity results about ( 𝑝 = 2 ∗ ) in the case of the critical exponent 𝑓 , 𝑔 and the weight functions 𝜇 ∈ [ 0 , 𝜇 ) sign-changing. In the present paper, we continue the study of [5] by considering the general case 𝜇 ∈ [ 0 , 𝜇 ) . We will extend the results of [6] to the more general case with 𝑓 , 𝑔 and the weight functions Ω which may change sign on 𝑓 1 . Our assumptions are

( 𝑓 ∈ 𝐶 ( Ω ) ) 𝑓 + = m a x { 𝑓 , 0 } ≢ 0 and Ω in 𝑔 1 ,( 𝑔 ∈ 𝐶 ( Ω ) ) 𝑔 + = m a x { 𝑔 , 0 } ≢ 0 and Ω in Λ 1 = 2 − 𝑞 2 ∗ | | 𝑔 − 𝑞 + | | ∞ ( 2 − 𝑞 ) / ( 2 ∗ − 2 ) 2 ∗ − 2 2 ∗ | | 𝑓 − 𝑞 + | | ∞ | Ω | ( 𝑞 − 2 ∗ ) / 2 ∗ 𝑆 𝜇 ( 𝑁 / 2 ) − ( 𝑁 / 4 ) 𝑞 + ( 𝑞 / 2 ) > 0 , ( 1 . 1 ) . Setwhere Ω is the Lebesgue measure of 𝑆 𝜇 , and 𝑃 𝜇 , 𝜆 , 𝑓 , 𝑔 is the best Sobolev constant (see (2.2). Now, we state the first main result about the existence of positive solution of ( ( 𝑓 1 ) ).

Theorem 1.1. Assume ( 𝑔 1 ) and 𝜆 ∈ ( 0 , Λ 1 ) hold. If 𝑃 𝜇 , 𝜆 , 𝑓 , 𝑔 , then ( ( 𝑃 𝜇 ) ) (simply written as 𝐻 1 0 ( Ω ) from now on) has at least one positive solution in ( 𝑃 𝜇 ) .

In order to get the second positive solution of 𝑓 , we need some additional assumptions about 𝑔 and 𝑓 . We assume the following conditions on 𝑔 and 𝑓 2 :

( 𝛽 0 ) there exist 𝜌 0 > 0 and 𝐵 ( 0 , 2 𝜌 0 ) ⊂ Ω such that 𝑓 ( 𝑥 ) ≥ 𝛽 0 and 𝑥 ∈ 𝐵 ( 0 , 2 𝜌 0 ) for all 𝑔 2 ;( | 𝑔 + | ∞ = 𝑔 ( 0 ) = m a x 𝑥 ∈ Ω 𝑔 ( 𝑥 ) ) 𝑔 ( 𝑥 ) > 0 , 𝑥 ∈ 𝐵 ( 0 , 2 𝜌 0 ) for all √ 𝛽 ∈ ( √ 𝜇 − 𝜇 𝑁 / √ 𝜇 , √ 𝜇 − 𝜇 ( 𝑁 + 1 ) / 𝜇 ) and there exists 𝑔 ( 𝑥 ) = 𝑔 ( 0 ) + 𝑜 | 𝑥 | 𝛽 a s 𝑥 ⟶ 0 . ( 1 . 2 ) such that

Theorem 1.2. Assume that ( 𝑓 2 ) - ( 𝑔 1 ) and ( 𝑔 2 ) - Λ 2 > 0 hold. Then there exists 𝜆 ∈ ( 0 , Λ 2 ) such that for ( 𝑃 𝜇 ) , 𝐻 1 0 ( Ω ) has at least two positive solutions in ( 𝑓 1 ) .

This paper is organized as follows. In Sections 2 and 3, we give some preliminaries and some properties of Nehari manifold. In Sections 4 and 5, we complete proofs of Theorems 1.1 and 1.2.

2. Preliminaries

Throughout this paper, ( 𝑔 1 ) and 𝐸 will be assumed. The dual space of a Banach space 𝐸 − 1 will be denoted by 𝐻 1 0 ( Ω ) . ‖ ⋅ ‖ denotes the standard Sobolev space, whose norm 𝐿 2 ( Ω ) is induced by the standard inner product. We denote the norm in | ⋅ | 2 by 𝐿 2 ( ℝ 𝑁 ) and the norm in | ⋅ | 𝐿 2 ( ℝ 𝑁 ) by 𝒟 1 , 2 ( ℝ 𝑁 ) = { 𝑢 ∈ 𝐿 2 ∗ ( ℝ 𝑁 ) ∶ ∇ 𝑢 ∈ 𝐿 2 ( ℝ 𝑁 ) } . ‖ ⋅ ‖ 2 𝒟 = ∫ ℝ 𝑁 | ∇ ⋅ | 2 𝑑 𝑥 with usual norm | Ω | . Ω is the Lebesgue measure of 𝐵 ( 𝑥 , 𝑟 ) . 𝑥 is a ball centered at 𝑟 with radius 𝑂 ( 𝜀 𝑡 ) . | 𝑂 ( 𝜀 𝑡 ) | / 𝜀 𝑡 ≤ 𝐶 denotes 𝑜 ( 𝜀 𝑡 ) , | 𝑜 ( 𝜀 𝑡 ) | / 𝜀 𝑡 → 0 denotes 𝜀 → 0 as 𝑜 𝑛 ( 1 ) , and 𝑜 𝑛 ( 1 ) → 0 denotes 𝑛 → ∞ as Ω . All integrals are taken over 𝐶 unless stated otherwise. 𝐶 𝑖 , 𝐻 1 0 ( Ω ) will denote various positive constants, the exact values of which are not important. On ‖ 𝑢 ‖ 2 𝜇 = | ∇ 𝑢 | 2 − 𝜇 | 𝑥 | 2 𝑢 2 𝑑 𝑥 . ( 2 . 1 ) , we use the normThanks to the Hardy inequality, the norm ‖ ⋅ ‖ is equivalent to the usual norm 𝐻 1 0 ( Ω ) of 𝐻 1 0 ( Ω ) . ‖ ⋅ ‖ 𝜇 with the norm 𝐻 is simply denoted by 𝜇 ∈ [ 0 , 𝜇 ) . For all 𝑆 𝜇 = i n f 𝑢 ∈ 𝒟 1 , 2 ( ℝ 𝑁 ) ⧵ { 0 } ∫ ℝ 𝑁 | ∇ 𝑢 | 2 − 𝜇 / | 𝑥 | 2 𝑢 2 𝑑 𝑥 ∫ ℝ 𝑁 | 𝑢 | 2 ∗ 𝑑 𝑥 2 / 2 ∗ . ( 2 . 2 ) , we define the constantFrom [7, 8], Ω ⊂ ℝ 𝑁 is independent of 𝑆 𝜇 ( Ω ) = i n f 𝑢 ∈ 𝐻 1 0 ( Ω ) ⧵ { 0 } ∫ Ω | ∇ 𝑢 | 2 − 𝜇 / | 𝑥 | 2 𝑢 2 𝑑 𝑥 ∫ Ω | 𝑢 | 2 ∗ 𝑑 𝑥 2 / 2 ∗ , ( 2 . 3 ) in the sense that if then 𝜇 = ( ( 𝑁 − 2 ) / 2 ) 2 .

Let 𝛾 1 = √ √ 𝜇 − 𝜇 − 𝜇 , 𝛾 2 = √ √ 𝜇 + 𝜇 − 𝜇 , 𝑆 𝜇 ; Catrina and Wang [9], Terracini [10] proved that 1 𝑈 ( 𝑥 ) = | 𝑥 | 𝛾 1 / √ 𝜇 + | 𝑥 | 𝛾 2 / √ 𝜇 √ 𝜇 . ( 2 . 4 ) is attained by the functionMoreover, for 𝑈 𝜀 ( 𝑥 ) = 𝜀 − ( 𝑁 − 2 ) / 2 [ 4 𝑁 ( 𝜇 − 𝜇 ) / ( 𝑁 − 2 ) ] ( 𝑁 − 2 ) / 4 𝑈 ( 𝑥 / 𝜀 ) , 𝜇 − Δ 𝑢 − | 𝑥 | 2 𝑢 = | 𝑢 | 2 ∗ − 2 𝑢 i n 0 x 0 0 0 a 0 0 x 0 0 0 a 0 0 x 0 0 0 a 0 ℝ 𝑁 ⧵ { 0 } , 𝑢 ⟶ 0 a s 0 x 0 0 0 a 0 0 x 0 0 0 a 0 0 x 0 0 0 a 0 | 𝑥 | ⟶ ∞ . ( 2 . 5 ) satisfiesFrom [11, Theorem B], all the positive solutions of problem (2.5) must have the form of 𝑈 𝜀 . Moreover, 𝑆 𝜇 attains 𝑐 ∈ ℝ .

We end these preliminaries by the following definition.

Definition 2.1. Let 𝐸 , 𝐼 ∈ 𝐶 1 ( 𝐸 , ℝ ) be a Banach space and { 𝑢 𝑛 } .
(i) ( P S ) 𝑐 is a 𝐸 -sequence in 𝐼 for 𝐼 ( 𝑢 𝑛 ) = 𝑐 + 𝑜 𝑛 ( 1 ) if 𝐼 ( 𝑢 𝑛 ) = 𝑜 𝑛 ( 1 ) and 𝐸 − 1 strongly in 𝑛 → ∞ . as 𝐼 (ii) We say that ( P S ) 𝑐 satisfies the ( P S ) 𝑐 -condition if any { 𝑢 𝑛 } -sequence 𝐸 in 𝐼 for ( 𝑃 𝜇 ) has a convergent subsequence.

3. Nehari Manifold

Associated with 𝐽 𝜆 , we consider the energy functional 𝐻 in 𝑢 ∈ 𝐻 , for each 𝐽 𝜆 1 ( 𝑢 ) = 2 ‖ 𝑢 ‖ 2 𝜇 − 𝜆 𝑞 𝑓 | 𝑢 | 𝑞 1 𝑑 𝑥 − 2 ∗ 𝑔 | 𝑢 | 2 ∗ 𝑑 𝑥 . ( 3 . 1 ) as follows:It is well known that 𝐶 1 is of 𝐻 in ( 𝑃 𝜇 ) , and the solutions of 𝐽 𝜆 are the critical points of the energy functional 𝐽 𝜆 (see Rabinowitz [12]).

As the energy functional 𝐻 is not bounded below on 𝒩 𝜆 = 𝑢 ∈ 𝐻 ⧵ { 0 } ∶ ⟨ 𝐽 𝜆 ( 𝑢 ) , 𝑢 ⟩ = 0 . ( 3 . 2 ) , it is useful to consider the functional Nehari manifoldThus, ⟨ 𝐽 𝜆 ( 𝑢 ) , 𝑢 ⟩ = ‖ 𝑢 ‖ 2 𝜇 − 𝜆 𝑓 | 𝑢 | 𝑞 𝑑 𝑥 − 𝑔 | 𝑢 | 2 ∗ 𝑑 𝑥 = 0 . ( 3 . 3 ) if and only ifNote that ( 𝑃 𝜇 ) contains every nonzero solution of 𝐽 𝜆 . Moreover, we have the following results.

Lemma 3.1. The energy functional 𝒩 𝜆 is coercive and bounded below on 𝑢 ∈ 𝒩 𝜆 .

Proof. If ( 𝑓 1 ) , then by 𝐽 𝜆 2 ( 𝑢 ) = ∗ − 2 2 ∗ 2 ‖ 𝑢 ‖ 2 𝜇 2 − 𝜆 ∗ − 𝑞 2 ∗ 𝑞 𝑓 | 𝑢 | 𝑞 ≥ 1 𝑑 𝑥 ( 3 . 4 ) 𝑁 ‖ 𝑢 ‖ 2 𝜇 2 − 𝜆 ∗ − 𝑞 2 ∗ 𝑞 𝑆 𝜇 − ( 𝑞 / 2 ) | Ω | 2 ∗ − 𝑞 / 2 ∗ ‖ 𝑢 ‖ 𝑞 𝜇 | | 𝑓 + | | ∞ . ( 3 . 5 ) , (3.3), the Hölder inequality and the Sobolev embedding theoremThus, 𝜓 𝜆 ( 𝑢 ) = ⟨ 𝐽 𝜆 ( 𝑢 ) , 𝑢 ⟩ . ( 3 . 6 ) is coercive and bounded below on 𝑢 ∈ 𝒩 𝜆 .

DefineThen for 𝒩 𝜆 ,Similar to the method used in Tarantello [13], we split 𝑢 𝜆 into three parts:Then, we have the following results.

Lemma 3.2. Assume that 𝒩 𝜆 is a local minimizer for 𝑢 𝜆 ∉ 𝒩 0 𝜆 on 𝐽 𝜆 ( 𝑢 𝜆 ) = 0 and 𝐻 − 1 ( Ω ) . Then 𝜆 ∈ ( 0 , Λ 1 ) in 𝒩 0 𝜆 = ∅ .

Proof. Our proof is almost the same as that in Brown-Zhang [14, Theorem 2.3] (or see Binding-Drábek-Huang [15]).

Lemma 3.3. If Λ 1 , then 𝜆 ∈ ( 0 , Λ 1 ) , where 𝒩 0 𝜆 ≠ ∅ is the same as in (1.1).

Proof. Suppose otherwise, that is there exists 𝑢 ∈ 𝒩 0 𝜆 such that ‖ 𝑢 ‖ 2 𝜇 = 2 ∗ − 𝑞 2 − 𝑞 𝑔 | 𝑢 | 2 ∗ 𝑑 𝑥 , ‖ 𝑢 ‖ 2 𝜇 2 = 𝜆 ∗ − 𝑞 2 ∗ − 2 𝑓 | 𝑢 | 𝑞 𝑑 𝑥 . ( 3 . 9 ) . Then by (3.7), for ( 𝑓 1 ) , we haveMoreover, by ‖ 𝑢 ‖ 𝜇 ≥ 2 − 𝑞 ( 2 ∗ − 𝑞 ) | 𝑔 + | ∞ 𝑆 2 ∗ 𝜇 / 2 1 / ( 2 ∗ − 2 ) , ‖ 𝑢 ‖ 𝜇 ≤ 𝜆 2 ∗ − 𝑞 2 ∗ 𝑆 − 2 𝜇 − ( 𝑞 / 2 ) | Ω | ( 2 ∗ − 𝑞 ) / 2 ∗ | | 𝑓 + | | ∞ 1 / ( 2 − 𝑞 ) . ( 3 . 1 0 ) , 𝜆 ≥ 2 − 𝑞 2 ∗ | | 𝑔 − 𝑞 + | | ∞ ( 2 − 𝑞 ) / ( 2 ∗ − 2 ) 2 ∗ − 2 2 ∗ | | 𝑓 − 𝑞 + | | ∞ | Ω | ( 𝑞 − 2 ∗ ) / 2 ∗ 𝑆 𝜇 ( 𝑁 / 2 ) − ( 𝑁 / 4 ) 𝑞 + ( 𝑞 / 2 ) = Λ 1 , ( 3 . 1 1 ) , the Hölder inequality, and the Sobolev embedding theorem, we haveThis implieswhich is a contradiction. Thus, we can conclude that if 𝒩 𝜆 = 𝒩 + 𝜆 ∪ 𝒩 − 𝜆 , we have 𝛼 𝜆 = i n f 𝑢 ∈ 𝒩 𝜆 𝐽 𝜆 ( 𝑢 ) , 𝛼 + 𝜆 = i n f 𝑢 ∈ 𝒩 + 𝜆 𝐽 𝜆 ( 𝑢 ) , 𝛼 − 𝜆 = i n f 𝑢 ∈ 𝒩 − 𝜆 𝐽 𝜆 ( 𝑢 ) . ( 3 . 1 2 ) .

By Lemma 3.3, we write 𝜆 ∈ ( 0 , Λ 1 ) and defineThen we get the following result.

Lemma 3.4. (i) If 𝜆 ∈ ( 0 , ( 𝑞 / 2 ) Λ 1 ) , then one has 𝛼 − 𝜆 > 𝑑 0 .
(ii) If 𝑑 0 , then 𝜆 , 𝜇 , 𝑞 , 𝑁 , 𝑆 𝜇 , | 𝑓 + | ∞ , | 𝑔 + | ∞ for some positive constant | Ω | depending on 𝑢 ∈ 𝒩 + 𝜆 and 2 − 𝑞 2 ∗ − 𝑞 ‖ 𝑢 ‖ 2 𝜇 > 𝑔 | 𝑢 | 2 ∗ 𝑑 𝑥 , ( 3 . 1 3 ) .

Proof. (i) Let 𝐽 𝜆 1 ( 𝑢 ) = 2 − 1 𝑞 ‖ 𝑢 ‖ 2 𝜇 + 1 𝑞 − 1 2 ∗ 𝑔 | 𝑢 | 2 ∗ < 1 𝑑 𝑥 2 − 1 𝑞 + 1 𝑞 − 1 2 ∗ 2 − 𝑞 2 ∗ − 𝑞 ‖ 𝑢 ‖ 2 𝜇 = − 2 − 𝑞 𝑞 𝑁 ‖ 𝑢 ‖ 2 𝜇 < 0 . ( 3 . 1 4 ) . By (3.7)and so Therefore, from the definitions of 𝛼 𝜆 ≤ 𝛼 + 𝜆 < 0 , 𝑢 ∈ 𝒩 − 𝜆 , we can deduce that 2 − 𝑞 2 ∗ − 𝑞 ‖ 𝑢 ‖ 2 𝜇 < 𝑔 | 𝑢 | 2 ∗ 𝑑 𝑥 . ( 3 . 1 5 ) .
(ii) Let ( 𝑔 1 ) . By (3.7)Moreover, by ‖ 𝑢 ‖ 𝜇 > 2 − 𝑞 2 ∗ − 𝑞 | 𝑔 + | ∞ 1 / ( 2 ∗ − 2 ) 𝑆 𝜇 𝑁 / 4 ∀ 𝑢 ∈ 𝒩 − 𝜆 . ( 3 . 1 7 ) and the Sobolev embedding theorem, This impliesBy (3.5) in the proof of Lemma 3.1Thus, if 𝑑 0 = 𝑑 0 ( 𝜆 , 𝑞 , 𝑁 , 𝑆 𝜇 , | 𝑓 + | ∞ , | 𝑔 + | ∞ , | Ω | ) , thenfor some positive constant ∫ 𝑔 | 𝑢 | 2 ∗ 𝑑 𝑥 > 0 . This completes the proof.

For each 𝑡 m a x = ( 2 − 𝑞 ) ‖ 𝑢 ‖ 2 𝜇 2 ∗ ∫ − 𝑞 𝑔 | 𝑢 | 2 ∗ 𝑑 𝑥 1 / ( 2 ∗ − 2 ) > 0 . ( 3 . 2 0 ) with 𝜆 ∈ ( 0 , Λ 1 ) , we writeThen the following lemma holds.

Lemma 3.5. Let ∫ 𝑔 | 𝑢 | 2 ∗ 𝑑 𝑥 > 0 . For each ∫ 𝑓 | 𝑢 | 𝑞 𝑑 𝑥 ≤ 0 with 𝑡 − > 𝑡 m a x , one has the following:
(i) if 𝑡 − 𝑢 ∈ 𝒩 − 𝜆 , then there exists a unique 𝐽 𝜆 𝑡 − 𝑢 ) = s u p 𝑡 ≥ 0 𝐽 𝜆 ( 𝑡 𝑢 ) , ( 3 . 2 1 ) such that ∫ 𝑓 | 𝑢 | 𝑞 𝑑 𝑥 > 0 and
(ii) if 𝑡 + 𝑢 ∈ 𝒩 + 𝜆 , then there exist unique 𝑡 − 𝑢 ∈ 𝒩 − 𝜆 such that 𝐽 𝜆 𝑡 + 𝑢 = i n f 0 ≤ 𝑡 ≤ 𝑡 m a x 𝐽 𝜆 ( 𝑡 𝑢 ) , 𝐽 𝜆 𝑡 − 𝑢 = s u p 𝑡 ≥ 0 𝐽 𝜆 ( 𝑡 𝑢 ) . ( 3 . 2 2 ) , 𝜆 ∈ ( 0 , Λ 1 ) and

Proof. The proof is almost the same as that in Brown-Wu [16, Lemma 2.6], and is omitted here.

4. Proof of Theorem 1.1

First, we will use the idea of Tarantello [13] to get the following results.

Proposition 4.1. (i) If { 𝑢 𝑛 } ⊂ 𝒩 𝜆 , then there exists a 𝐻 -sequence 𝐽 𝜆 in 𝜆 ∈ ( 0 , ( 𝑞 / 2 ) Λ 1 ) for ( P S ) 𝛼 − 𝜆 .
(ii) If { 𝑢 𝑛 } ⊂ 𝒩 − 𝜆 , then there exists a 𝐻 -sequence 𝐽 𝜆 in 𝐽 𝜆 for 𝒩 + 𝜆 .

Proof. The proof is almost the same as that in Wu [4, Proposition 9] (or see Hsu-Lin [5, Proposition 3.3]).

Now, we establish the existence of a local minimum for 𝜆 ∈ ( 0 , Λ 1 ) on 𝐽 𝜆 .

Theorem 4.2. If 𝑢 𝜆 , then 𝒩 + 𝜆 has a minimizer 𝐽 𝜆 ( 𝑢 𝜆 ) = 𝛼 𝜆 = 𝛼 + 𝜆 in 𝑢 𝜆 and it satisfies
(i) ( 𝑃 𝜇 ) ,(ii) 𝐽 𝜆 ( 𝑢 𝜆 ) → 0 is a positive solution of 𝜆 → 0 + ,(iii) ( 𝑖 ) as { 𝑢 𝑛 } .

Proof. By Proposition 4.1 𝐽 𝜆 , there exists a minimizing sequence 𝒩 𝜆 for 𝐽 𝜆 𝑢 𝑛 = 𝛼 𝜆 + 𝑜 𝑛 ( 1 ) , 𝐽 𝜆 𝑢 𝑛 = 𝑜 𝑛 ( 1 ) i n 𝐻 − 1 . ( 4 . 1 ) on 𝐽 𝜆 such thatSince { 𝑢 𝑛 } is coercive on 𝐻 (see Lemma 3.1), we get that 𝑢 𝜆 ∈ 𝐻 is bounded in 𝑢 𝑛 ⇀ 𝑢 𝜆 𝑢 w e a k l y 0 x 0 0 0 a 0 0 x 0 0 0 a 0 0 x 0 0 0 a 0 i n 0 x 0 0 0 a 0 0 x 0 0 0 a 0 0 x 0 0 0 a 0 𝐻 , 𝑛 ⟶ 𝑢 𝜆 𝑢 a l m o s t e v e r y 0 x 0 0 0 a 0 w h e r e 0 x 0 0 0 a 0 0 x 0 0 0 a 0 0 x 0 0 0 a 0 i n 0 x 0 0 0 a 0 0 x 0 0 0 a 0 0 x 0 0 0 a 0 Ω , 𝑛 ⟶ 𝑢 𝜆 s t r o n g l y i n 𝐿 𝑠 ( Ω ) ∀ 1 ≤ 𝑠 < 2 ∗ . ( 4 . 2 ) . Going if necessary to a subsequence, we can assume that there exists 𝑢 𝜆 such thatFirst, we claim that 𝑢 𝜆 is a nontrivial solution of ( 𝑃 𝜇 ) . By (4.1) and (4.2), it is easy to see that 𝑢 𝑛 ∈ 𝒩 𝜆 is a solution of 𝜆 𝑓 | | 𝑢 𝑛 | | 𝑞 𝑞 2 𝑑 𝑥 = ∗ − 2 2 2 ∗ ‖ ‖ 𝑢 − 𝑞 𝑛 ‖ ‖ 2 𝜇 − 2 ∗ 𝑞 2 ∗ 𝐽 − 𝑞 𝜆 𝑢 𝑛 . ( 4 . 3 ) . From 𝑛 → ∞ and (3.4), we deduce thatLet 𝜆 𝑓 | | 𝑢 𝜆 | | 𝑞 2 𝑑 𝑥 ≥ − ∗ 𝑞 2 ∗ 𝛼 − 𝑞 𝜆 > 0 . ( 4 . 4 ) in (4.3), by (4.1), (4.2), and 𝑢 𝜆 ∈ 𝒩 𝜆 , we getThus, 𝑢 𝑛 → 𝑢 𝜆 is a nontrivial solution of 𝐻 . Now we prove that 𝐽 𝜆 ( 𝑢 𝜆 ) = 𝛼 𝜆 strongly in 𝑢 ∈ 𝒩 𝜆 and 𝐽 𝜆 1 ( 𝑢 ) = 𝑁 ‖ 𝑢 ‖ 2 𝜇 − 2 ∗ − 𝑞 2 ∗ 𝑞 𝜆 𝑓 | 𝑢 | 𝑞 𝑑 𝑥 . ( 4 . 5 ) . By (4.3), if 𝐽 𝜆 ( 𝑢 𝜆 ) = 𝛼 𝜆 , thenIn order to prove that 𝛼 𝜆 ≤ 𝐽 𝜆 𝑢 𝜆 = 1 𝑁 ‖ ‖ 𝑢 𝜆 ‖ ‖ 2 𝜇 − 2 ∗ − 𝑞 2 ∗ 𝑞 𝜆 𝑓 | | 𝑢 𝜆 | | 𝑞 𝑑 𝑥 ≤ l i m i n f 𝑛 → ∞ 1 𝑁 ‖ ‖ 𝑢 𝑛 ‖ ‖ 2 𝜇 − 2 ∗ − 𝑞 2 ∗ 𝑞 𝜆 𝑓 | | 𝑢 𝑛 | | 𝑞 𝑑 𝑥 ≤ l i m i n f 𝑛 → ∞ 𝐽 𝜆 𝑢 𝑛 = 𝛼 𝜆 . ( 4 . 6 ) , it suffices to recall that 𝐽 𝜆 ( 𝑢 𝜆 ) = 𝛼 𝜆 , by (4.5) and applying Fatou's lemma to getThis implies that 𝑣 𝑛 = 𝑢 𝑛 − 𝑢 𝜆 and ‖ ‖ 𝑣 𝑛 ‖ ‖ 2 𝜇 = ‖ ‖ 𝑢 𝑛 ‖ ‖ 2 𝜇 − ‖ ‖ 𝑢 𝜆 ‖ ‖ 2 𝜇 + 𝑜 𝑛 ( 1 ) . ( 4 . 7 ) Let 𝑢 𝑛 → 𝑢 𝜆 , then by Brézis-Lieb lemma [17] implies thatTherefore, 𝑢 𝜆 ∈ 𝒩 + 𝜆 strongly in 𝑢 𝜆 ∈ 𝒩 − 𝜆 . Moreover, we have 𝑡 + 0 . On the contrary, if 𝑡 − 0 , then by Lemma 3.5, there are unique 𝑡 + 0 𝑢 𝜆 ∈ 𝒩 + 𝜆 and 𝑡 − 0 𝑢 𝜆 ∈ 𝒩 − 𝜆 such that 𝑡 + 0 < 𝑡 − 0 = 1 and 𝑑 𝐽 𝑑 𝑡 𝜆 ( 𝑡 + 0 𝑢 𝜆 𝑑 ) = 0 , 2 𝑑 𝑡 2 𝐽 𝜆 𝑡 + 0 𝑢 𝜆 > 0 , ( 4 . 8 ) . In particular, we have 𝑡 + 0 < 𝑡 ≤ 𝑡 − 0 . Sincethere exists 𝐽 𝜆 𝑡 + 0 𝑢 𝜆 < 𝐽 𝜆 𝑡 𝑢 𝜆 ≤ 𝐽 𝜆 𝑡 − 0 𝑢 𝜆 = 𝐽 𝜆 𝑢 𝜆 , ( 4 . 9 ) such that 𝐽 𝜆 ( 𝑢 𝜆 ) = 𝐽 𝜆 ( | 𝑢 𝜆 | ) . By Lemma 3.5,which is a contradiction. Since 𝑢 𝜆 and ( 𝑃 𝜇 ) , by Lemma 3.2 we may assume that 𝑢 𝜆 is a nontrivial nonnegative solution of ( 𝑃 𝜇 ) . Standard arguments implies that 0 > 𝛼 𝜆 2 > − 𝜆 ∗ − 𝑞 2 ∗ 𝑞 𝑆 𝜇 − ( 𝑞 / 2 ) | Ω | ( 2 ∗ − 𝑞 ) / 2 ∗ ‖ ‖ 𝑢 𝜆 ‖ ‖ 𝑞 𝜇 | | 𝑓 + | | ∞ . ( 4 . 1 0 ) is a positive solution of 𝐽 𝜆 ( 𝑢 𝜆 ) → 0 . Moreover, by Lemma 3.4 (i) and (3.5), we haveThis implies that ( 𝑃 𝜇 ) as 𝑢 𝜆 .

Now, we begin the proof of Theorem 1.1: By Theorem 4.2, we obtain ( 𝑃 𝜇 ) has a positive solution 𝐽 𝜆 .

5. Proof of Theorem 1.2

Next, we will establish the existence of the second positive solution of ( P S ) 𝛼 − 𝜆 by proving that ( 𝑓 1 ) satisfies the ( 𝑔 1 ) -condition.

Lemma 5.1. Assume that { 𝑢 𝑛 } and ( P S ) 𝑐 hold. If 𝐽 𝜆 is a 𝑢 𝑛 ⇀ 𝑢 -sequence for 𝐻 with 𝐽 𝜆 ( 𝑢 ) = 0 in 𝐶 0 , then 𝑞 , 𝑁 , 𝑆 𝜇 , | 𝑓 + | ∞ , and there exists a constant | Ω | depending on 𝐽 𝜆 ( 𝑢 ) ≥ − 𝐶 0 𝜆 2 / ( 2 − 𝑞 ) and { 𝑢 𝑛 } , such that ( P S ) 𝑐 .

Proof. If 𝐽 𝜆 is a 𝑢 𝑛 ⇀ 𝑢 -sequence for 𝐻 with 𝐽 𝜆 ( 𝑢 ) = 0 in ⟨ 𝐽 𝜆 ( 𝑢 ) , 𝑢 ⟩ = 0 , it is easy to see that 𝑔 ( 𝑥 ) | 𝑢 | 2 ∗ 𝑑 𝑥 = ‖ 𝑢 ‖ 2 𝜇 − 𝜆 𝑓 ( 𝑥 ) | 𝑢 | 𝑞 𝑑 𝑥 . ( 5 . 1 ) . This implies that 𝐽 𝜆 1 ( 𝑢 ) = 2 − 1 2 ∗ ‖ 𝑢 ‖ 2 𝜇 − 1 𝑞 − 1 2 ∗ 𝜆 𝑓 ( 𝑥 ) | 𝑢 | 𝑞 𝑑 𝑥 . ( 5 . 2 ) , andConsequently,Using the Hölder inequality, the Young inequality, and the Sobolev embedding theorem, we havewhere | Ω | is a positive constant depending on ( 𝑓 1 ) and ( 𝑔 1 ) .

Lemma 5.2. Assume that 𝐽 𝜆 and ( P S ) 𝑐 hold. Then the functional 𝑐 ∈ ( − ∞ , ( 1 / 𝑁 ) | 𝑔 + | ∞ − ( 𝑁 − 2 ) / 2 𝑆 𝜇 𝑁 / 2 − 𝐶 0 𝜆 2 / ( 2 − 𝑞 ) ) satisfies the 𝐶 0 -condition for all { 𝑢 𝑛 } ⊂ 𝐻 where ( P S ) 𝑐 is the positive constant given in Lemma 5.1.

Proof. Let 𝐽 𝜆 ( 𝑢 𝑛 ) = 𝑐 + 𝑜 𝑛 ( 1 ) be a 𝐽 𝜆 ( 𝑢 𝑛 ) = 𝑜 𝑛 ( 1 ) -sequence which satisfies { 𝑢 𝑛 } and 𝐻 . Using standard arguments it follows that { 𝑢 𝑛 } is bounded in 𝑢 ∈ 𝐻 . Thus, there exists a subsequence still denoted by 𝑢 𝑛 𝑢 ⇀ 𝑢 w e a k l y i n 𝐻 , 𝑛 ⟶ 𝑢 s t r o n g l y i n 𝐿 𝑠 ( Ω ) ∀ 1 ≤ 𝑠 < 2 ∗ , 𝑢 𝑛 ⟶ 𝑢 a . e . 0 x 0 0 0 a 0 o n Ω . ( 5 . 4 ) and a function ( 𝑓 1 ) such thatBy 𝐽 𝜆 ( 𝑢 ) = 0 , 𝜆 | | 𝑢 𝑓 ( 𝑥 ) 𝑛 | | 𝑞 𝑑 𝑥 = 𝜆 𝑓 ( 𝑥 ) | 𝑢 | 𝑞 𝑑 𝑥 + 𝑜 𝑛 ( 1 ) , ( 5 . 5 ) , and Lemma 5.1, we have that 𝑣 𝑛 = 𝑢 𝑛 − 𝑢 and
Let Ω . Then by ‖ ‖ 𝑣 𝑛 ‖ ‖ 2 𝜇 = ‖ ‖ 𝑢 𝑛 ‖ ‖ 2 𝜇 − ‖ 𝑢 ‖ 2 𝜇 + 𝑜 𝑛 | | 𝑣 ( 1 ) , ( 5 . 6 ) 𝑔 ( 𝑥 ) 𝑛 | | 2 ∗ | | 𝑢 𝑑 𝑥 = 𝑔 ( 𝑥 ) 𝑛 | | 2 ∗ 𝑑 𝑥 − 𝑔 ( 𝑥 ) | 𝑢 | 2 ∗ 𝑑 𝑥