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Concentrated solutions for a critical elliptic equation

The authors were supported by NSFC grants (No.12071364, 11871387, 11771167), The Science and Technology Foundation of Guizhou Province ([2001]ZK008), the Fundamental Research Funds for the Central Universities(No. WUT: 2020IA003) and the China Scholarship Council

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  • In this paper, we are concerned with the following elliptic equation

    where $ N\geq 3 $, $ s\in [1, 2^*-1) $ with $ 2^* = \frac{2N}{N-2} $, $ \varepsilon>0 $, $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^N $. Under some conditions on $ Q(x) $, Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997,461–483) proved that there exists a single-peak solution for small $ \varepsilon $ if $ N\geq 4 $ and $ s\in (1, 2^*-1) $. And they proposed in Remark 1.7 of their paper that

    $ \begin{array}{c} ''it\; is\; interesting\; to\; know\; the \;existence\; of\; single-peak \;solutions \; for \;small \;\varepsilon\;\\ and \;s = 1''.\end{array} $

    Also it was addressed in Remark 1.8 of their paper that

    $ \begin{array}{c} ''the\; question\; of \;solutions \;concentrated \;at \;several\; points \;at \;the \;same \;time \;is \\ still \;open''. \end{array}$

    Here we give some confirmative answers to the above two questions. Furthermore, we prove the local uniqueness of the multi-peak solutions. And our results show that the concentration of the solutions to above problem is delicate whether $ s = 1 $ or $ s>1 $.


    Correction: “Technology Foundation of Guizhou Province” is corrected to “The Science and Technology Foundation of Guizhou Province" under Fund Project.

    Mathematics Subject Classification: Primary: 35A01, 35B25; Secondary: 35J20, 35J60.


    \begin{equation} \\ \end{equation}
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  • [1] T. BartschA. Micheletti and A. Pistoia, The Morse property for functions of Kirchhoff-Routh path type, Discrete Contin. Dyn. Syst. Series. S, 12 (2019), 1867-1877.  doi: 10.3934/dcdss.2019123.
    [2] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), 437-478.  doi: 10.1002/cpa.3160360405.
    [3] D. CaoS. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 4037-4063.  doi: 10.1007/s00526-015-0930-2.
    [4] D. Cao and E. Noussair, Multiple positive and nodal solutions for semilinear elliptic problems with critical exponents, Indiana Univ. Math. J., 44 (1995), 1249-1271.  doi: 10.1512/iumj.1995.44.2027.
    [5] D. CaoS. Peng and  S. YanSingularly Perturbed Methods for Nonlinear Elliptic Problems, Cambridge University Press, 2021.  doi: 10.1017/9781108872638.
    [6] D. Cao and X. Zhong, Multiplicity of positive solutions for semilinear elliptic equations involving the critical Sobolev exponents, Nonlin. Anal. TMA, 29 (1997), 461-483.  doi: 10.1016/S0362-546X(96)00051-X.
    [7] J. Chabrowski and S. Yan, Concentration of solutions for a nonlinear elliptic problem with near critical exponent, Topol. Methods Nonlinear Anal., 13 (1999), 199-233.  doi: 10.12775/TMNA.1999.011.
    [8] Y. DengC. Lin and S. Yan, On the prescribed scalar curvature problem in $ \mathbb R^N$, local uniqueness and periodicity, J. Math. Pures Appl., 104 (2015), 1013-1044.  doi: 10.1016/j.matpur.2015.07.003.
    [9] J. Escobar, Positive solutions for some semilinear elliptic equations with critical Sobolev exponents, Comm. Pure Appl. Math., 40 (1987), 623-657.  doi: 10.1002/cpa.3160400507.
    [10] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.
    [11] L. Glangetas, Uniqueness of positive solutions of a nonlinear elliptic equation involving the critical exponent, Nonlin. Anal. TMA, 20 (1993), 571-603.  doi: 10.1016/0362-546X(93)90039-U.
    [12] Y. Guo, M. Musso, S. Peng and S. Yan, Non-degeneracy of multi-bubbling solutions for the prescribed scalar curvature equations and applications, J. Funct. Anal., 279 (2020), 108553, 29 pp. doi: 10.1016/j.jfa.2020.108553.
    [13] Z. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 159-174.  doi: 10.1016/s0294-1449(16)30270-0.
    [14] W. Kulpa, The Poincaré-Miranda theorem, Am. Math. Mon., 104 (1997), 545-550.  doi: 10.2307/2975081.
    [15] C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7. 
    [16] M. Musso and A. Pistoia, Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent, Indiana Univ. Math. J., 51 (2002), 541-579.  doi: 10.1512/iumj.2002.51.2199.
    [17] S. PengC. Wang and S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal., 274 (2018), 2606-2633.  doi: 10.1016/j.jfa.2017.12.008.
    [18] O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52.  doi: 10.1016/0022-1236(90)90002-3.
    [19] S. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda u=0$, Dokl. Akad. Nauk, 165 (1965), 36-39. 
    [20] J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $S^N$, J. Funct. Anal., 258 (2010), 3048-3081.  doi: 10.1016/j.jfa.2009.12.008.
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