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.

@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} }

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] Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations 13 (2001), 311–337. | MR 1865001 | Zbl 1086.35035

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

[3] 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

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

,[5] 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

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

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

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

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

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

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

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

and ,[13] 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

and ,[14] 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

, and ,[15] 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

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

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

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

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

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

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

,[22] Bubble towers for supercritical semilinear elliptic equations, J. Funct. Anal. 221 (2005), 251–302. | MR 2124865 | Zbl 1129.35379

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

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

and ,[25] 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

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

and ,[27] 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

, and ,[28] 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

, and ,[29] 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] 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

and ,[31] 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

and ,[32] 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

and ,[33] 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

and ,[34] 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] A Neumann problem with critical exponent in non-convex domains and Lin-Ni’s conjecture, Trans. Amer. Math. Soc., to appear. | Zbl 1204.35093

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

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

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

,