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

Abstract

In this paper, we are concerned with the following quasilinear elliptic equationwhere Ω ⊂ ℝ N is a smooth domain with smooth boundary ∂Ω such that 0 Ω,

Solutions for the quasi-linear elliptic problems involving the critical Sobolev exponent

“…The property of the Nehari manifold was used to prove the existence of multiple positive solutions for ( 1.4 ). Furthermore, Hsu [ 11 , 12 ] improved and complemented the main results obtained in [ 10 ]. Recently, Goyal and Sreenadh [ 13 ] investigated a class of singular N -Laplacian problems with exponential nonlinearities in .…”

Multiple positive solutions of degenerate nonlocal problems on unbounded domain

“…They showed that there exists such that ( E λ ) admits at least two positive solutions for , a positive solution for , and no positive solution for . For a more detailed reference on this subject, we refer the interested reader to .…”

“…Motivated by the findings mentioned earlier, we are interested in the existence of multiple positive solutions of without condition . Moreover, we will suppose f ( x , u ) a nonhomogenous case in , and thus, the method of the decomposition of Nehari manifold, which is used in , is not applicable for . Our problem is not only nonlocal and degenerate but also is studied in an unbounded domain.…”