Multiple Positive Solutions for Semilinear Elliptic Equations with Sign - Changing Weight Functions

and Applied Analysis 3 In order to describe our main result, we need to define Λ0 ( 2 − q ( p − q‖a‖L∞ ) 2−q / p−2 ( p − 2 ( p − q)‖b ‖Lq∗ ) S p 2−q /2 p−2 q/2 p > 0, 1.3 where ‖a‖L∞ supx∈RNa x , ‖b ‖Lq∗ ∫ RN |b x |qdx 1/q∗ and Sp is the best Sobolev constant for the imbedding of H1 R into L R . Theorem 1.1. Assume that a1 , b1 b2 hold. If λ ∈ 0, q/2 Λ0 , Ea,λb admits at least two positive solutions in H1 R . 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 Ea,λb . At the end of this section, we explain some notations employed. In the following discussions, we will consider H H1 R with the norm ‖u‖ ∫ RN |∇u|2 u2 dx . We denote by Sp the best constant which is given by Sp inf u∈H\{0} ‖u‖ (∫ RN |u|pdx)2/p . 1.4 The dual space ofH will be denoted byH∗. 〈·, ·〉 denote the dual pair betweenH∗ andH. We denote the norm in L R by ‖ · ‖Ls for 1 ≤ s ≤ ∞. B x; r is a ball in R centered at x with radius r. on 1 denotes on 1 → 0 as n → ∞. C, Ci will denote various positive constants, the exact values of which are not important. 2. Preliminary Results Associated with 1.3 , the energy functional Jλ : H → R defined by Jλ u 1 2 ‖u‖ − 1 p ∫ RN a x |u|dx − λ q ∫ RN b x |u|dx, 2.1 for all u ∈ H is considered. It is well-known that Jλ ∈ C1 H,R and the solutions of Ea,λb are the critical points of Jλ. Since Jλ is not bounded from below on H, we will work on the Nehari manifold. For λ > 0 we define Nλ { u ∈ H \ {0} : 〈J ′ λ u , u 〉 0 } . 2.2 Note that Nλ contains all nonzero solutions of Ea,λb and u ∈ Nλ if and only if 〈 J ′ λ u , u 〉 ‖u‖ − ∫ RN a x |u|dx − λ ∫ RN b x |u|dx 0. 2.3 Lemma 2.1. Jλ is coercive and bounded from below on Nλ. 4 Abstract and Applied Analysis Proof. If u ∈ Nλ, then by b1 , 2.3 , and the Hölder and Sobolev inequalities, one has Jλ u p − 2 2p ‖u‖ − λ ( p − q pq )∫ RN b x |u|dx 2.4 ≥ p − 2 2p ‖u‖ − λ ( p − q pq ) S −q/2 p ‖b ‖Lq∗ ‖u‖. 2.5 Since q < 2 < p, it follows that Jλ is coercive and bounded from below on Nλ. The Nehari manifold is closely linked to the behavior of the function of the form φu : t → Jλ tu for t > 0. Such maps are known as fibering maps and were introduced by Drábek and Pohozaev in 20 and are also discussed by Brown and Zhang in 10 . If u ∈ H, we have φu t t2 2 ‖u‖ − t p p ∫ RN a x |u|dx − t q q λ ∫ RN b x |u|dx, φu t t‖u‖ − tp−1 ∫ RN a x |u|dx − tq−1λ ∫ RN b x |u|dx, φ′′ u t ‖u‖ − ( p − 1)tp−2 ∫ RN a x |u|dx − (q − 1)tq−2λ ∫ RN b x |u|dx. 2.6 It is easy to see that tφu t ‖tu‖ − ∫ RN a x |tu|dx − λ ∫ RN b x |tu|dx, 2.7 and so, for u ∈ H \ {0} and t > 0, φu t 0 if and only if tu ∈ Nλ that is, the critical points of φu correspond to the points on the Nehari manifold. In particular, φu 1 0 if and only if u ∈ Nλ. Thus, it is natural to split Nλ into three parts corresponding to local minima, local maxima, and points of inflection. Accordingly, we define N λ { u ∈ Nλ : φ′′ u 1 > 0 } , Nλ { u ∈ Nλ : φ′′ u 1 0 } , Nλ { u ∈ Nλ : φ′′ u 1 < 0 } , 2.8 and note that if u ∈ Nλ, that is, φu 1 0, then φ′′ u 1 ( 2 − q)‖u‖2 − (p − q) ∫ RN a x |u|dx, 2.9 ( 2 − p)‖u‖2 − (q − p)λ ∫ RN b x |u|dx. 2.10 We now derive some basic properties of N λ , N0 λ , and N− λ . Abstract and Applied Analysis 5 Lemma 2.2. Suppose that u0 is a local minimizer for Jλ on Nλ and u0 / ∈ Nλ, then J ′ λ u0 0 in H∗. Proof. See the work of Brown and Zhang in 10, Theorem 2.3 . Lemma 2.3. If λ ∈ 0,Λ0 , thenNλ ∅. Proof. We argue by contradiction. Suppose that there exists λ ∈ 0,Λ0 such thatNλ / ∅. Then for u ∈ N0 λ by 2.9 and the Sobolev inequality, we have 2 − q p − q‖u‖ 2 ∫ RN a x |u|dx ≤ ‖a‖L∞S p ‖u‖, 2.11and Applied Analysis 5 Lemma 2.2. Suppose that u0 is a local minimizer for Jλ on Nλ and u0 / ∈ Nλ, then J ′ λ u0 0 in H∗. Proof. See the work of Brown and Zhang in 10, Theorem 2.3 . Lemma 2.3. If λ ∈ 0,Λ0 , thenNλ ∅. Proof. We argue by contradiction. Suppose that there exists λ ∈ 0,Λ0 such thatNλ / ∅. Then for u ∈ N0 λ by 2.9 and the Sobolev inequality, we have 2 − q p − q‖u‖ 2 ∫ RN a x |u|dx ≤ ‖a‖L∞S p ‖u‖, 2.11