August  2022, 42(8): 4061-4094. doi: 10.3934/dcds.2022046

Concentrated solutions for a critical elliptic equation

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

2. 

Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China

*Corresponding author: sytian@whut.edu.cn

Received  January 2022 Published  August 2022 Early access  April 2022

Fund Project: 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

In this paper, we are concerned with the following elliptic equation
$ \begin{equation*} \begin{cases} -\Delta u = Q(x)u^{2^*-1 }+\varepsilon u^{s}, \; &{\;{\rm{in}}\;\; \Omega},\\ \ u>0, \; &{\;{\rm{in}}\;\; \Omega}, \\ \ u = 0, &{\;{\rm{on}}\;\; \partial \Omega}, \end{cases} \end{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.
Citation: Lipeng Duan, Shuying Tian. Concentrated solutions for a critical elliptic equation. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4061-4094. doi: 10.3934/dcds.2022046
References:
[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. Yan, Singularly 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.

show all references

References:
[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. Yan, Singularly 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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