Abstract
This paper is concerned with a quasilinear elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. The existence and multiplicity results of positive solutions are obtained by variational methods.
Highlights
The aim of this paper is to establish the existence of nontrivial solutions to the following quasilinear elliptic system ⎧ ⎪⎪⎨ − pu − μ |u | p−2 u |x|p =pα |u|α−2|v|β u α+β |x|t + λ |u |q −2 u |x |s
Existence of nontrivial non-negative solutions for elliptic equations with singular potentials were recently studied by several authors, but, essentially, only with a solely critical exponent
Exploiting the relationship between the Nehari manifold and fibering maps (i.e., maps of the form t → Jλ(tu) where Jλ is the Euler function associated with the equation), they gave an interesting explanation of the well-known bifurcation result
Summary
Note that Aμ,0 is the best constant in the Sobolev inequality, namely, Aμ,0( ) = inf u∈D01,p( )\{0}. Existence of nontrivial non-negative solutions for elliptic equations with singular potentials were recently studied by several authors, but, essentially, only with a solely critical exponent. Kang in [25] studied the following elliptic equation via the generalized Mountain-Pass theorem [29]:. Brown and Zhang [12] have studied a subcritical semi-linear elliptic equation with a sign-changing weight function and a bifurcation real parameter in the case p = 2 and Dirichlet boundary conditions. Exploiting the relationship between the Nehari manifold and fibering maps (i.e., maps of the form t → Jλ(tu) where Jλ is the Euler function associated with the equation), they gave an interesting explanation of the well-known bifurcation result.
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