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On the solvability of singular boundary value problems on the real line in the critical growth case
Lazer-McKenna conjecture for higher order elliptic problem with critical growth
Department of Mathematical Science, Tsinghua University, Beijing 100084, China |
$ \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)} $ |
$ B_1 $ |
$ \mathbb{R}^{N} $ |
$ s_1 $ |
$ \lambda $ |
$ \varphi_1 > 0 $ |
$ \left((-\Delta )^m, \mathcal{D}_0^{m,2}(B_1) \right) $ |
$ \lambda_1 $ |
$ \hbox{ max }_{y \in B_1} \varphi_1(y) = 1 $ |
$ u_+ = \hbox{ max }(u,0) $ |
$ m^* = \frac{2N}{N-2m} $ |
$ (P) $ |
$ s_1 $ |
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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