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January  2019, 18(1): 301-322. doi: 10.3934/cpaa.2019016

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

 1 Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto Sumiyoshi-ku, Osaka-shi, Osaka 558-8585 Japan 2 Department of Mathematics, Institute of Applied Mathematical Sciences, National Center for Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan 3 National Center for Theoretical Sciences, No. 1 Sec. 4 Roosevelt Rd., National Taiwan University, Taipei, 10617, Taiwan

* Corresponding author

Received  December 2017 Revised  March 2018 Published  August 2018

Fund Project: The first author is supported by Grant-in-Aid for JSPS Research Fellow (JSPS KAKENHI Grant Number JP16J08945)

Let
 $N ≥ 3$
and
 $Ω \subset \mathbb{R}^N$
be a
 $C^2$
bounded domain. We study the existence of positive solution
 $u ∈ H^1(Ω)$
of
 \begin{align*}\left\{\begin{array}{l}-\Delta u + \lambda u = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} + \tau \frac{|u|^{2^*(s)-2}u}{|x-x_2|^s}\text{ in }\Omega\\\frac{\partial u}{\partial \nu} = 0 \text{ on }\partial\Omega,\end{array}\right.\end{align*}
where
 $τ = 1$
or
 $-1$
,
 $0 < s <2$
,
 $2^*(s) = \frac{2(N-s)}{N-2}$
and
 $x_1, x_2 ∈ \overline{Ω}$
with
 $x_1 ≠ x_2$
. First, we show the existence of positive solutions to the equation provided the positive
 $λ$
is small enough. In case that one of the singularities locates on the boundary and the mean curvature of the boundary at this singularity is positive, the existence of positive solutions is obtained for any
 $λ > 0$
and some
 $s$
depending on
 $τ$
and
 $N$
. Furthermore, we extend the existence theory of solutions to the equations for the case of the multiple singularities.
Citation: Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016
##### References:
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##### References:
 [1] T. Bartsch, S. Peng and Z. Zhang, Existence and non-existence of solutions to elliptic equations related to the Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. Partial Diff. Equ., 30 (2007), 113-136.  doi: 10.1007/s00526-006-0086-1.  Google Scholar [2] G. Cerami, X. Zhong and W. Zou, On some nonlinear elliptic PDEs with Sobolev-Hardy critical exponents and a Li-Lin open problem, Calc. Var. Partial Diff. Equ., 54 (2015), 1793-1829.  doi: 10.1007/s00526-015-0844-z.  Google Scholar [3] J. Chabrowski, On the Neumann problem with the Hardy-Sobolev potential, Annali di Matematica, 186 (2007), 703-719.  doi: 10.1007/s10231-006-0027-9.  Google Scholar [4] N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 767-793.  doi: 10.1016/j.anihpc.2003.07.002.  Google Scholar [5] N. Ghoussoub and F. Robert, Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth, IMRP Int. Math. Res. Pap., 21867 (2006), 1-85.   Google Scholar [6] N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities, Geom. Funct. Anal. 16 (2006), 1201-1245. doi: 10.1007/s00039-006-0579-2.  Google Scholar [7] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.  doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar [8] M. Hashizume, Asymptotic behavior of the least-energy solutions of a semilinear elliptic equation with the Hardy-Sobolev critical exponent, J. Differential Equations, 262 (2017), 3107-3131.  doi: 10.1016/j.jde.2016.11.005.  Google Scholar [9] C. Hsia, C. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal., 259 (2010), 1816-1849.  doi: 10.1016/j.jfa.2010.05.004.  Google Scholar [10] Y. Li and C.-S. Lin, A nonlinear elliptic pde with two Sobolev-Hardy critical exponents, Arch. Ration. Mech. Anal, 203 (2012), 943-968.  doi: 10.1007/s00205-011-0467-2.  Google Scholar [11] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequal ities, Ann. of Math, 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar [12] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/6.  Google Scholar [13] R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal, 68 (2008), 3972-3986.  doi: 10.1016/j.na.2007.04.034.  Google Scholar [14] M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar [15] J.-L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar [16] X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.  doi: 10.1016/0022-0396(91)90014-Z.  Google Scholar [17] X.-X. Zhong and W.-M. Zou, A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in $\mathbb{R}^N$, arXiv: 1504.01133. Google Scholar
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