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

and Applied Analysis 3 Theorem 1.3 see 5 . There exists λ0 > 0 such that 1.4 admits exactly two solutions for λ ∈ 0, λ0 , exactly one solution for λ λ0, and no solution for λ > λ0. To proceed, wemake somemotivations 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 Ω, and showed the existence of λ0 > 0 such that 1.2 admits at least two solutions for all λ ∈ 0, λ0 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 f2 and g2 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 λ ∈ 0, q/p Λ1 see Theorem 1.5 but for all 1 < p < N and 1 ≤ q < 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, q 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 q. 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 Λ1. Set Λ1 ( p − q ( p∗ − q)∣∣g ∣∞ ) p−q / p∗−p ( p∗ − p ( p∗ − q)∣∣f ∣∞ ) |Ω| q−p∗ /p∗ S N/p − N/p2 q q/p > 0, 1.5 where |Ω| is the Lebesgue measure of Ω and S is the best Sobolev constant see 2.2 . Theorem 1.4. Assume f1 and g1 hold. If λ ∈ 0,Λ1 , then Eλf,g admits at least one positive solution uλ ∈ C1,α Ω for some α ∈ 0, 1 . Theorem 1.5. Assume that f1 f2 and g1 – g4 hold. If λ ∈ 0, q/p Λ1 , then Eλf,g admits at least two positive solutions uλ,Uλ ∈ C1,α Ω 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, f1 and g1 will be assumed. The dual space of a Banach space E will be denoted by E−1. W 0 Ω denotes the standard Sobolev space with the following 4 Abstract and Applied Analysis norm: ‖u‖ ∫