Abstract
Abstract In this article, we consider a class of Kirchhoff equations with critical Hardy-Sobolev exponent and indefinite nonlinearity, which has not been studied in the literature. We prove very nicely that this equation has at least two solutions in ℝ3. And some known results in the literature are improved.
Highlights
The equation − a + b|∇u| dx ∆u + V(x)u = f (x, u), x ∈ R (1.1)R u>, x∈R is closely related to the stationary analogue of the equation L ρ∂ u ∂t p h E L ∂u ∂t dx ∂u ∂x
In this article, we consider a class of Kirchho equations with critical Hardy-Sobolev exponent and inde nite nonlinearity, which has not been studied in the literature
The main result in this paper is the following theorem
Summary
Which was presented by Kirchho [16] in 1883. Since an abstract functional framework to the following equation utt − a + b |∇u| dx u = f (x, u), Ω was rst introduced by Lions [20] in 1978, Equation (1.1) has received much more attention, there are many results about the existence of ground state solutions, sign-changing solutions, multiplicity of solutions and concentration of solutions, we may refer to [2, 10, 12,13,14,15, 19, 23,24,25,26,27] and the references therein. The commonest method used in the existing literature is to use the mountain pass theorem, we refer to [11]. |∇u| dx is homogeneous of degree 4, one usually assume that f is 4-superlinear at in nity or satis es
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