On the Neumann problem with the Hardy–Sobolev potential

Abstract

In this paper we investigate the solvability of the nonlinear Neumann problem (1.1) with indefinite weight functions and a critical Hardy-Sobolev nonlinearity. We examine the common effect of the shape of the graph of a weight function and the mean curvature of the boundary on the existence of solutions of problem (1.1). We also investigate the regularity of solutions.

On a Neumann problem with critical exponents and Hardy potentials

“…For some t > 0, Han and Liu [5] considered the multiplicity of solutions of (1.5) with λ < 0. For relevant papers on problem (1.5) for t ≥ 0, see [6][7][8][9][10][11] and the references cited therein. Note that the classical Hardy inequality does not hold any more in H 1 (Ω).…”

On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities

“…Hence, there does not exist a positive solution to (3). So, only the case where λ > 0 are adderessed in literature.…”

“…In this case, Ghossoub-Kang [4] showed that (3) has a positive solution if the mean curvature of ∂Ω at 0, H(0) is positive. Furthermore, Chabrowski [3] investigated the solvability of the nonlinear Neumann problem with indefinite weight functions −∆u + λu = Q(x)|u| 2 * (s)−2 u |x| s , u > 0 in Ω and gives some sufficient condition on Q(x) provided the mean curvature of ∂Ω at 0, H(0) > 0. Recently, concerning the equation (3) the first author investigated the case when H(0) ≤ 0 in [7].…”

Asymptotic behavior of the least-energy solutions of a semilinear elliptic equation with the Hardy–Sobolev critical exponent

“…In the interior singularity case, the existence and non-existence results of the minimizer for µ N s,λ (Ω) are obtained by [13]. In the boundary singularity case, some results are due to [5], [9] and [13]. Due to these results, the attainability for µ N s,λ (Ω) is different for each situation.…”

“…Our main purpose of this paper is to investigate the asymptotic behavior of the least-energy solutions of (1) as λ → ∞. In [5] and [9], the existence of the least energy solutions of (1) is guaranteed for any λ > 0 if the mean curvature of ∂Ω at 0 is positive. Thus it is natural that we investigate the asymptotic behavior of the least-energy solutions of (1).…”