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},
pages = {423--457},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {2},
year = {2010},
mrnumber = {2731162},
zbl = {1204.35094},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_2_423_0/}
}
TY - JOUR AU - Wei, Juncheng AU - Yan, Shusen TI - On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 423 EP - 457 VL - 9 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2010_5_9_2_423_0/ LA - en ID - ASNSP_2010_5_9_2_423_0 ER -
%0 Journal Article %A Wei, Juncheng %A Yan, Shusen %T On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 423-457 %V 9 %N 2 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2010_5_9_2_423_0/ %G en %F 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, Série 5, Tome 9 (2010) no. 2, pp. 423-457. http://www.numdam.org/item/ASNSP_2010_5_9_2_423_0/
[1] , and , Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations 13 (2001), 311–337. | MR | Zbl
[2] Adimurthi and S. Prashanth, Critical exponent problem in -border-line between existence and non-existence of positive solutions for Dirichlet problem, Adv. Differential Equations 5 (2000), 67–95. | MR
[3] and , On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1973), 231–247. | MR | Zbl
[4] , “Critical Points at Infinity in Some Variational Problems”, Research Notes in Mathematics, Vol. 182, Longman-Pitman, 1989. | MR | Zbl
[5] , and , 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 | Zbl
[6] , and , Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof, J. Differential Equations 195 (2003), 243–269. | MR | Zbl
[7] and , Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477. | MR | Zbl
[8] and , Elliptic equations with one-sided critical growth, Electron. J. Differential Equations (2002), 1–21. | EuDML | MR | Zbl
[9] , A Counter example to the Lazer-McKenna conjecture, Nonlinear Anal. 13 (1989), 19–21. | MR | Zbl
[10] and , On the superlinear Lazer-McKenna conjecture: the non-homogeneous case, Adv. Differential Equations 12(2007), 961–993. | MR | Zbl
[11] and , On the superlinear Lazer-McKenna conjecture, J. Differential Equations 210 (2005), 317–351. | MR | Zbl
[12] and , On the superlinear Lazer-McKenna conjecture, part two, Comm. Partial Differential Equations 30 (2005), 1331–1358. | MR | Zbl
[13] and , The Lazer-McKenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc. 78 (2008), 639-662. | MR | Zbl
[14] , and , Two-bubble solutions in the super-critical Bahri-Coron’s problem, Calc. Var. Partial Differential Equations 16 (2003), 113–145. | MR | Zbl
[15] , and , Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 45–82. | EuDML | Numdam | MR | Zbl
[16] , and , The Brezis-Nirenberg problem near criticality in dimension 3, J. Math. Pures Appl. (9) 83 (2004), 1405–1456. | MR | Zbl
[17] and , The two dimensional Lazer-McKenna conjecture for an exponential nonlinearity, J. Differential Equations 231 (2006), 108–134. | MR | Zbl
[18] , On the superlinear Ambrosetti-Prodi problem, Nonlinear Anal. 8 (1984), 655–665. | MR | Zbl
[19] and , A variational approach to superlinear elliptic problems, Comm. Partial Differential Equations 9 (1984), 699–717. | MR | Zbl
[20] and , Critical superlinear Ambrosetti-Prodi problems, Topol. Methods Nonlinear Anal. 14 (1999), 59–80. | MR | Zbl
[21] , The critical Lazer-McKenna conjecture in low dimensions, J. Differential Equations 231 (2008), 108–134. | MR | Zbl
[22] , and , Bubble towers for supercritical semilinear elliptic equations, J. Funct. Anal. 221 (2005), 251–302. | MR | Zbl
[23] , Variational and topological methods in partial ordered Hilbert spaces, Math. Ann. 261 (1982), 493–514. | EuDML | MR | Zbl
[24] and , On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981), 282–294. | MR | Zbl
[25] and , On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Roy. Soc. Edinburgh 95A (1983), 275–283. | MR | Zbl
[26] and , A symmetric theorem and application to nonlinear partial differential equations, J. Differential Equations 72 (1988), 95–106. | MR | Zbl
[27] , and , The superlinear Lazer-McKenna onjecture for an elliptic problem with critical growth, Calc. Var. Partial Differential Equations 28 (2007), 471–508. | MR | Zbl
[28] , and , The superlinear Lazer-McKenna onjecture for an elliptic problem with critical growth, part II, J. Differential Equations 227 (2006), 301–332. | MR | Zbl
[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 | Zbl
[30] and , Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity, J. Eur. Math. Soc. (JEMS) 7 (2005), 449–476. | EuDML | MR | Zbl
[31] and , On a class of superlinear Sturm-Liouville problems with arbitrarily many solutions, SIAM J. Math. Anal. 17 (1986), 761–771. | MR | Zbl
[32] and , Multiplicity results for superlinear elliptic problems with partial interference with the spectrum, J. Math. Anal. Appl. 118 (1986), 15–23. | MR | Zbl
[33] and , Multiplicity results for ODEs with nonlinearities crossing all but a finite number of eigenvalues, Nonlinear Anal. 10 (1986), 174–163. | MR | Zbl
[34] , Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Lineáire 2 (1985), 143–156. | EuDML | Numdam | MR | Zbl
[35] , and , A Neumann problem with critical exponent in non-convex domains and Lin-Ni’s conjecture, Trans. Amer. Math. Soc., to appear. | Zbl
[36] and , Lazer-McKenna conjecture: the critical case, J. Funct. Anal. 244 (2007), 639–667. | MR | Zbl
[37] and , Infinitely many solutions for the prescribed scalar curvature problem, J. Funct. Anal. 258 (2010), 3048–3081. | MR | Zbl
[38] , Multipeak solutions for a nonlinear Neumann problem in exterior domains, Adv. Differential Equations 7 (2002), 919–950. | MR | Zbl
