Abstract
In this paper, we consider a class of quasilinear elliptic systems with weights and the nonlinearity involving the critical Hardy–Sobolev exponent and one sign-changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.
Highlights
The aim of this paper is to establish the existence and multiplicity of nontrivial non-negative solutions to the quasilinear elliptic system ⎧⎪⎪⎪⎪⎪⎪⎪⎨−div −div|x|−ap|∇u|p−2∇u |x|−ap|∇v|p−2∇v⎪⎪⎪⎪⎪⎪⎪⎩u > 0, v > 0, =1 p∗|x|bp∗ Fu(x, u, v) + λf (x) 1 |x|β |u|q−2u
Exploiting the relationship between the Nehari manifold and fibering maps (i.e., maps of the form t → Jλ(tu) where Jλ is the Euler functional associated with the equation), they gave an interesting explanation of the well-known bifurcation result
By the Lemma 2, we get the existence of a positive constant M such that
Summary
∂Ω, F ∈ C1(Ω × (R+), R+) is positively homogeneous of degree p∗. Here p∗ = p(a, b) pN N −p(1+a−b) is the Hardy–Sobolev critical exponent. Several authors have used the Nehari manifold to solve semilinear and quasilinear problems (see [1, 2, 6, 7, 8, 9, 10, 16, 20] and references therein). Exploiting the relationship between the Nehari manifold and fibering maps (i.e., maps of the form t → Jλ(tu) where Jλ is the Euler functional associated with the equation), they gave an interesting explanation of the well-known bifurcation result. Some authors studied the singular problems with Hardy–Sobolev critical exponents ([3, 17, 18] the references therein).
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