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

Abstract

The multiple results of positive solutions for the following quasilinear elliptic equation: in on , are established. Here, is a bounded smooth domain in denotes the -Laplacian operator, is a positive real parameter, and are continuous functions on which are somewhere positive but which may change sign on . The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.

Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms

“…In this paper, we study (1.1) and extend the results of [11,18,19] to the more general case 1 < q < p < N, −∞ < μ <μ, f, g are sign changing and Ω is a smooth domain (not necessarily bounded) in ℝ N (N ≥ 3). By extracting the Palais-Smale sequence in the Nehari manifold, the existence of at least two positive solutions of (1.1) is verified.…”

“…When p = 2, 1 < q <2, μ ∈ [0,μ), f, g are sign changing and Ω is bounded, [18] studied (1.1) and obtained that there exists Λ >0 such that (1.1) has at least two positive solutions for all l (0, Λ). For the case p ≠ 2, [19] studied (1.1) and obtained the multiplicity of positive solutions when 1 < q < p < N, μ = 0, f, g are sign changing and Ω is bounded. However, little has been done for this type of problem (1.1).…”

“…The proof is similar to [19,25] and the details are omitted. □ Now, we establish the existence of a local minimum for J l on N λ .…”

THREE NONTRIVIAL NONNEGATIVE SOLUTIONS FOR SOME CRITICAL p-LAPLACIAN SYSTEMS WITH LOWER-ORDER NEGATIVE PERTURBATIONS

“…In recent years, more and more attention have been paid to the existence and multiplicity of nonnegative or positive solutions for the elliptic problems involving concave terms and critical Sobolev exponent. Results relating to these problems can be found in [1], [2], [4,5,12,13], [7,8,9], [11,14,15,16,17,18,19,20,21], and the references therein. By the results of the above papers we know that the number of nontrivial solutions for problem (1) is affected by the concave-convex nonlinearities.…”

“…By (11), (12), (14), (15) and (16), there exist µ 1 > 0 and η 2 satisfies that 0 < η 2 ≤ η 1 such that…”