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February  2020, 40(2): 1159-1189. doi: 10.3934/dcds.2020074

Lazer-McKenna conjecture for higher order elliptic problem with critical growth

Yuxia Guo , and  Ting Liu

Department of Mathematical Science, Tsinghua University, Beijing 100084, China

* Corresponding author: Yuxia Guo

Received  June 2019 Revised  June 2019 Published  November 2019

Fund Project: The first author is supported by NSFC(11771235).

This paper is concerned with the following problem involving critical Sobolev exponent and polyharmonic operator:
$ \begin{cases} (-\Delta )^mu = u_{+}^{m^*-1}+ \lambda u-s_1 \varphi_1, \hbox{ in } B_1, \\ u \in \mathcal{D}_0^{m,2}(B_1), \end{cases}\;\;\;\;\;\;\;\;\;\;\;\;\;{(P)} $
where
$ B_1 $
 is the unit ball in
$ \mathbb{R}^{N} $
,
$ s_1 $
 and
$ \lambda $
 are two positive parameters,
$ \varphi_1 > 0 $
 is the eigenfunction of
$ \left((-\Delta )^m, \mathcal{D}_0^{m,2}(B_1) \right) $
 corresponding to the first eigenvalue
$ \lambda_1 $
 with
$ \hbox{ max }_{y \in B_1} \varphi_1(y) = 1 $
,
$ u_+ = \hbox{ max }(u,0) $
 and
$ m^* = \frac{2N}{N-2m} $
. By using the Lyapunov-Schmits reduction method, we prove that the number of solutions for
$ (P) $
 is unbounded as the parameter
$ s_1 $
 tends to infinity, therefore proving the Lazer-McKenna conjecture for the higher order operator equation with critical growth.
Keywords: Peak solutions, finite-dimensional reduction, higher order, elliptic problem, critical growth.
Mathematics Subject Classification: Primary: 35J65; Secondary: 35B38.
Citation: Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074
References:
[1]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Math. Pura Appl., 93 (1973), 231-246.  doi: 10.1007/BF02412022.  Google Scholar

[2]

A. Bahri, Critical Points at Infinity in some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182, Longman Scientific & Technical, 1989.  Google Scholar

[3]

B. Breuer, P. McKenna and M. Plum, Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof, J. Differ. Equ., 195 (2003), 243-269.  doi: 10.1016/S0022-0396(03)00186-4.  Google Scholar

[4]

M. Calanchi and B. Ruf, Elliptic equations with one-sided critical growth, Electronic J.D.E, (2002), 1–21.  Google Scholar

[5]

E. Dancer, A counter example to the Lazer-McKenna conjecture, Nonlinear Anal., 13 (1989), 19-21.  doi: 10.1016/0362-546X(89)90030-8.  Google Scholar

[6]

E. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differ. Equ., 210 (2005), 317-351.  doi: 10.1016/j.jde.2004.07.017.  Google Scholar

[7]

E. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, part 2, Comm. PDE., 30 (2005), 1331-1358.  doi: 10.1080/03605300500258865.  Google Scholar

[8]

D. De Figueiredo, On the superlinear Ambrosetti-Prodi problem, Nonlinear Anal., 8 (1984), 655-665.  doi: 10.1016/0362-546X(84)90010-5.  Google Scholar

[9]

D. De Figueiredo, P. Shrikanth and S. Santra, Non-radially symmetric solutions for a superlinear Ambrosetti-Prodi type problem in a ball, Nonlinear Anal., 7 (2005), 849-866.  doi: 10.1142/S0219199705001982.  Google Scholar

[10]

D. De Figueiredo and S. Solimini, A variational approach to superlinear elliptic problems, Comm. PDE., 9 (1984), 699-717.  doi: 10.1080/03605308408820345.  Google Scholar

[11]

D. De Figueiredo and J. Yang, Critical superlinear Ambrosetti-Prodi problems. Topol. Methods, Nonlinear Anal., 14 (1999), 59-80.  doi: 10.12775/TMNA.1999.022.  Google Scholar

[12]

F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.  Google Scholar

[13]

H. Hofer, Variational and topological methods in partial ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514.  doi: 10.1007/BF01457453.  Google Scholar

[14]

A. Lazer and P. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294.  doi: 10.1016/0022-247X(81)90166-9.  Google Scholar

[15]

A. Lazer and P. McKenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Royal Soc. Edinburgh, 95 (1983), 275-283.  doi: 10.1017/S0308210500012993.  Google Scholar

[16]

A. Lazer and P. McKenna, A symmetric theorem and application to nonlinear partial differential equaitons, J. Differ. Equ., 72 (1988), 95-106.  doi: 10.1016/0022-0396(88)90150-7.  Google Scholar

[17]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var. Partial. Differ. Equ., 28 (2007), 471-508.  doi: 10.1007/s00526-006-0051-z.  Google Scholar

[18]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, part 2, J. Differ. Equ., 227 (2006), 301-332.  doi: 10.1016/j.jde.2006.02.011.  Google Scholar

[19]

O. Rey, The role of the Green's function in a nonlinear elliptic problem involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52.  doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[20]

B. Ruf and S. Solimini, On a calss of superlinear Sturm-Liouville problems with acbitrarily many solutions, J. Math. Anal., 17 (1986), 761-771.  doi: 10.1137/0517055.  Google Scholar

[21]

B. Ruf and P. N. Srikanth, Multiplicity results for superlinear elliptic problems with partial interference with spectrum, J. Math. Anal. Appl., 118 (1986), 15-23.  doi: 10.1016/0022-247X(86)90286-6.  Google Scholar

[22]

B. Ruf and P. N. Srikanth, Multiplicity results for ODEs with nonlinearities crossing all but a finite number of eigenvalues, Nonlinear Anal., 10 (1986), 157-163.  doi: 10.1016/0362-546X(86)90043-X.  Google Scholar

[23]

S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 143-156.  doi: 10.1016/S0294-1449(16)30407-3.  Google Scholar

[24]

J. Wei and S. Yan, Lazer-McKenna conjecture: The critical case, J. Funct. Anal., 244 (2007), 639-667.  doi: 10.1016/j.jfa.2006.11.002.  Google Scholar

[25]

S. Yan, Multipeak solutions for a nonlinear Neumann problem in exterior domains, Adv. Diff. Equ., 7 (2002), 919-950.   Google Scholar

show all references

References:
[1]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Math. Pura Appl., 93 (1973), 231-246.  doi: 10.1007/BF02412022.  Google Scholar

[2]

A. Bahri, Critical Points at Infinity in some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182, Longman Scientific & Technical, 1989.  Google Scholar

[3]

B. Breuer, P. McKenna and M. Plum, Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof, J. Differ. Equ., 195 (2003), 243-269.  doi: 10.1016/S0022-0396(03)00186-4.  Google Scholar

[4]

M. Calanchi and B. Ruf, Elliptic equations with one-sided critical growth, Electronic J.D.E, (2002), 1–21.  Google Scholar

[5]

E. Dancer, A counter example to the Lazer-McKenna conjecture, Nonlinear Anal., 13 (1989), 19-21.  doi: 10.1016/0362-546X(89)90030-8.  Google Scholar

[6]

E. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differ. Equ., 210 (2005), 317-351.  doi: 10.1016/j.jde.2004.07.017.  Google Scholar

[7]

E. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, part 2, Comm. PDE., 30 (2005), 1331-1358.  doi: 10.1080/03605300500258865.  Google Scholar

[8]

D. De Figueiredo, On the superlinear Ambrosetti-Prodi problem, Nonlinear Anal., 8 (1984), 655-665.  doi: 10.1016/0362-546X(84)90010-5.  Google Scholar

[9]

D. De Figueiredo, P. Shrikanth and S. Santra, Non-radially symmetric solutions for a superlinear Ambrosetti-Prodi type problem in a ball, Nonlinear Anal., 7 (2005), 849-866.  doi: 10.1142/S0219199705001982.  Google Scholar

[10]

D. De Figueiredo and S. Solimini, A variational approach to superlinear elliptic problems, Comm. PDE., 9 (1984), 699-717.  doi: 10.1080/03605308408820345.  Google Scholar

[11]

D. De Figueiredo and J. Yang, Critical superlinear Ambrosetti-Prodi problems. Topol. Methods, Nonlinear Anal., 14 (1999), 59-80.  doi: 10.12775/TMNA.1999.022.  Google Scholar

[12]

F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.  Google Scholar

[13]

H. Hofer, Variational and topological methods in partial ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514.  doi: 10.1007/BF01457453.  Google Scholar

[14]

A. Lazer and P. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294.  doi: 10.1016/0022-247X(81)90166-9.  Google Scholar

[15]

A. Lazer and P. McKenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Royal Soc. Edinburgh, 95 (1983), 275-283.  doi: 10.1017/S0308210500012993.  Google Scholar

[16]

A. Lazer and P. McKenna, A symmetric theorem and application to nonlinear partial differential equaitons, J. Differ. Equ., 72 (1988), 95-106.  doi: 10.1016/0022-0396(88)90150-7.  Google Scholar

[17]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var. Partial. Differ. Equ., 28 (2007), 471-508.  doi: 10.1007/s00526-006-0051-z.  Google Scholar

[18]

G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, part 2, J. Differ. Equ., 227 (2006), 301-332.  doi: 10.1016/j.jde.2006.02.011.  Google Scholar

[19]

O. Rey, The role of the Green's function in a nonlinear elliptic problem involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52.  doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[20]

B. Ruf and S. Solimini, On a calss of superlinear Sturm-Liouville problems with acbitrarily many solutions, J. Math. Anal., 17 (1986), 761-771.  doi: 10.1137/0517055.  Google Scholar

[21]

B. Ruf and P. N. Srikanth, Multiplicity results for superlinear elliptic problems with partial interference with spectrum, J. Math. Anal. Appl., 118 (1986), 15-23.  doi: 10.1016/0022-247X(86)90286-6.  Google Scholar

[22]

B. Ruf and P. N. Srikanth, Multiplicity results for ODEs with nonlinearities crossing all but a finite number of eigenvalues, Nonlinear Anal., 10 (1986), 157-163.  doi: 10.1016/0362-546X(86)90043-X.  Google Scholar

[23]

S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 143-156.  doi: 10.1016/S0294-1449(16)30407-3.  Google Scholar

[24]

J. Wei and S. Yan, Lazer-McKenna conjecture: The critical case, J. Funct. Anal., 244 (2007), 639-667.  doi: 10.1016/j.jfa.2006.11.002.  Google Scholar

[25]

S. Yan, Multipeak solutions for a nonlinear Neumann problem in exterior domains, Adv. Diff. Equ., 7 (2002), 919-950.   Google Scholar

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