On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 2, p. 423-457

We consider an elliptic problem of Ambrosetti-Prodi type involving critical Sobolev exponent on a bounded smooth domain. We show that if the domain has some symmetry, the problem has infinitely many (distinct) solutions whose energy approach to infinity even for a fixed parameter, thereby obtaining a stronger result than the Lazer-McKenna conjecture.

Classification:  35J65,  35B38,  47H15
@article{ASNSP_2010_5_9_2_423_0,
author = {Wei, Juncheng and Yan, Shusen},
title = {On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {2},
year = {2010},
pages = {423-457},
zbl = {1204.35094},
mrnumber = {2731162},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_2_423_0}
}

Wei, Juncheng; Yan, Shusen. On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 2, pp. 423-457. http://www.numdam.org/item/ASNSP_2010_5_9_2_423_0/

[1] J. Ai, K. S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations 13 (2001), 311–337. | MR 1865001 | Zbl 1086.35035

[2] Adimurthi and S. Prashanth, Critical exponent problem in ${ℝ}^{2}$-border-line between existence and non-existence of positive solutions for Dirichlet problem, Adv. Differential Equations 5 (2000), 67–95. | MR 1734537

[3] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1973), 231–247. | MR 320844 | Zbl 0288.35020

[4] A. Bahri, “Critical Points at Infinity in Some Variational Problems”, Research Notes in Mathematics, Vol. 182, Longman-Pitman, 1989. | MR 1019828 | Zbl 0676.58021

[5] A. Bahri, Y. Y. Li and O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations 3 (1995), 67–93. | MR 1384837 | Zbl 0814.35032

[6] B. Breuer, P. J. Mckenna and M. Plum, Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof, J. Differential Equations 195 (2003), 243–269. | MR 2019251 | Zbl 1156.35359

[7] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477. | MR 709644 | Zbl 0541.35029

[8] M. Calanchi and B. Ruf, Elliptic equations with one-sided critical growth, Electron. J. Differential Equations (2002), 1–21. | MR 1938385 | Zbl 1022.35009

[9] E. N. Dancer, A Counter example to the Lazer-McKenna conjecture, Nonlinear Anal. 13 (1989), 19–21. | MR 973364 | Zbl 0691.35039

[10] E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: the non-homogeneous case, Adv. Differential Equations 12(2007), 961–993. | MR 2351835 | Zbl 1162.35037

[11] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations 210 (2005), 317–351. | MR 2119987 | Zbl 1190.35082

[12] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, part two, Comm. Partial Differential Equations 30 (2005), 1331–1358. | MR 2180307 | Zbl 1330.35147

[13] E. N. Dancer and S. Yan, The Lazer-McKenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc. 78 (2008), 639-662. | MR 2456896 | Zbl 1202.35088

[14] M. Del Pino, P. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron’s problem, Calc. Var. Partial Differential Equations 16 (2003), 113–145. | MR 1956850 | Zbl 1142.35421

[15] M. Del Pino, M. Musso and A. Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 45–82. | Numdam | MR 2114411 | Zbl 1130.35064

[16] M. Del Pino, J. Dolbeault and M. Musso, The Brezis-Nirenberg problem near criticality in dimension 3, J. Math. Pures Appl. (9) 83 (2004), 1405–1456. | MR 2103187 | Zbl 1130.35040

[17] M. Del Pino and C. Munoz, The two dimensional Lazer-McKenna conjecture for an exponential nonlinearity, J. Differential Equations 231 (2006), 108–134. | MR 2287880 | Zbl 1159.35372

[18] D. G. De Figueiredo, On the superlinear Ambrosetti-Prodi problem, Nonlinear Anal. 8 (1984), 655–665. | MR 746723 | Zbl 0554.35045

[19] D. G. De Figueiredo and S. Solimini, A variational approach to superlinear elliptic problems, Comm. Partial Differential Equations 9 (1984), 699–717. | MR 745022 | Zbl 0552.35030

[20] D. G. De Figueiredo and J. Yang, Critical superlinear Ambrosetti-Prodi problems, Topol. Methods Nonlinear Anal. 14 (1999), 59–80. | MR 1758880 | Zbl 0958.35055

[21] O. Druet, The critical Lazer-McKenna conjecture in low dimensions, J. Differential Equations 231 (2008), 108–134. | MR 2446190 | Zbl 1155.35030

[22] Y. Ge, R. Jing and F. Pacard, Bubble towers for supercritical semilinear elliptic equations, J. Funct. Anal. 221 (2005), 251–302. | MR 2124865 | Zbl 1129.35379

[23] H. Hofer, Variational and topological methods in partial ordered Hilbert spaces, Math. Ann. 261 (1982), 493–514. | MR 682663 | Zbl 0488.47034

[24] A. C. Lazer and P. J. Mckenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981), 282–294. | MR 639539 | Zbl 0496.35039

[25] A. C. Lazer and P. J. Mckenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Roy. Soc. Edinburgh 95A (1983), 275–283. | MR 726879 | Zbl 0533.35037

[26] A. C. Lazer and P. J. Mckenna, A symmetric theorem and application to nonlinear partial differential equations, J. Differential Equations 72 (1988), 95–106. | MR 929199 | Zbl 0666.47038

[27] G. Li, S. Yan and J. Yang, The superlinear Lazer-McKenna onjecture for an elliptic problem with critical growth, Calc. Var. Partial Differential Equations 28 (2007), 471–508. | MR 2293982 | Zbl 1194.35169

[28] G. Li, S. Yan and J. Yang, The superlinear Lazer-McKenna onjecture for an elliptic problem with critical growth, part II, J. Differential Equations 227 (2006), 301–332. | MR 2233963 | Zbl 1254.35095

[29] O. Rey, The role of the Green’s function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1–52. | MR 1040954 | Zbl 0786.35059

[30] O. Rey and J. Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity, J. Eur. Math. Soc. (JEMS) 7 (2005), 449–476. | MR 2159223 | Zbl 1129.35406

[31] B. Ruf and S. Solimini, On a class of superlinear Sturm-Liouville problems with arbitrarily many solutions, SIAM J. Math. Anal. 17 (1986), 761–771. | MR 846387 | Zbl 0608.34019

[32] B. Ruf and P. N. Srikanth, Multiplicity results for superlinear elliptic problems with partial interference with the spectrum, J. Math. Anal. Appl. 118 (1986), 15–23. | MR 849438 | Zbl 0601.35042

[33] B. Ruf and P. N. Srikanth, Multiplicity results for ODEs with nonlinearities crossing all but a finite number of eigenvalues, Nonlinear Anal. 10 (1986), 174–163. | MR 825214 | Zbl 0586.34017

[34] S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Lineáire 2 (1985), 143–156. | Numdam | MR 794004 | Zbl 0583.35044

[35] L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in non-convex domains and Lin-Ni’s conjecture, Trans. Amer. Math. Soc., to appear. | Zbl 1204.35093

[36] J. Wei and S. Yan, Lazer-McKenna conjecture: the critical case, J. Funct. Anal. 244 (2007), 639–667. | MR 2297039 | Zbl 1231.35072

[37] J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem, J. Funct. Anal. 258 (2010), 3048–3081. | MR 2595734 | Zbl 1209.53028

[38] S. Yan, Multipeak solutions for a nonlinear Neumann problem in exterior domains, Adv. Differential Equations 7 (2002), 919–950. | MR 1895112 | Zbl 1208.35064