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

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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 and Continuous Dynamical Systems, 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.
[2]	A. Bahri, Critical Points at Infinity in some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182, Longman Scientific & Technical, 1989.
[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.
[4]	M. Calanchi and B. Ruf, Elliptic equations with one-sided critical growth, Electronic J.D.E, (2002), 1–21.
[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.
[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.
[7]	E. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, part 2, Comm. PDE., 30 (2005), 1331-1358. doi: 10.1080/03605300500258865.
[8]	D. De Figueiredo, On the superlinear Ambrosetti-Prodi problem, Nonlinear Anal., 8 (1984), 655-665. doi: 10.1016/0362-546X(84)90010-5.
[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.
[10]	D. De Figueiredo and S. Solimini, A variational approach to superlinear elliptic problems, Comm. PDE., 9 (1984), 699-717. doi: 10.1080/03605308408820345.
[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.
[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.
[13]	H. Hofer, Variational and topological methods in partial ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514. doi: 10.1007/BF01457453.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	S. Yan, Multipeak solutions for a nonlinear Neumann problem in exterior domains, Adv. Diff. Equ., 7 (2002), 919-950.

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.
[2]	A. Bahri, Critical Points at Infinity in some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182, Longman Scientific & Technical, 1989.
[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.
[4]	M. Calanchi and B. Ruf, Elliptic equations with one-sided critical growth, Electronic J.D.E, (2002), 1–21.
[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.
[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.
[7]	E. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, part 2, Comm. PDE., 30 (2005), 1331-1358. doi: 10.1080/03605300500258865.
[8]	D. De Figueiredo, On the superlinear Ambrosetti-Prodi problem, Nonlinear Anal., 8 (1984), 655-665. doi: 10.1016/0362-546X(84)90010-5.
[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.
[10]	D. De Figueiredo and S. Solimini, A variational approach to superlinear elliptic problems, Comm. PDE., 9 (1984), 699-717. doi: 10.1080/03605308408820345.
[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.
[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.
[13]	H. Hofer, Variational and topological methods in partial ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514. doi: 10.1007/BF01457453.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	S. Yan, Multipeak solutions for a nonlinear Neumann problem in exterior domains, Adv. Diff. Equ., 7 (2002), 919-950.

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