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The borderline case of Bahri-Lions result and related problems. (English) Zbl 07496944

Summary: In this paper, we analyse the borderline case of Bahri-Lions result in [Zbl 0645.58013; Zbl 0923.34025; Zbl 0963.35001; Zbl 0669.34035]. Combining a Gidas-Spruck type a priori estimate, this method also leads us to partial affirmative answers of the open problems in [X. Yue and W. Zou, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 94, 171–184 (2014; Zbl 1285.35026), Section 7].

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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[1] Ackermann, N.; Cano, A.; Hernández-Martínez, E., Spectral density estimates with partial symmetries and an application to Bahri-Lions-type results, Calc. Var. Partial Differ. Equ., 56, Article 6 pp. (2017) · Zbl 1372.35078
[2] Bahri, A., Topological results on a certain class of functionals and application, J. Funct. Anal., 41, 3, 397-427 (1981) · Zbl 0499.35050
[3] Bahri, A.; Berestycki, H., A perturbation method in critical point theory and applications, Trans. Am. Math. Soc., 267, 1, 1-32 (1981) · Zbl 0476.35030
[4] Bahri, A.; Lions, P. L., Morse index of some min-max critical points. I. Application to multiplicity results, Commun. Pure Appl. Math., 41, 8, 1027-1037 (1988) · Zbl 0645.58013
[5] Bolle, P., On the Bolza problem, J. Differ. Equ., 152, 2, 274-288 (1999) · Zbl 0923.34025
[6] Bolle, P.; Ghoussoub, N.; Tehrani, H., The multiplicity of solutions in non-homogeneous boundary value problems, Manuscr. Math., 101, 3, 325-350 (2000) · Zbl 0963.35001
[7] Candela, A. M.; Palmieri, G.; Salvatore, A., Radial solutions of semilinear elliptic equations with broken symmetry, Topol. Methods Nonlinear Anal., 27, 1, 117-132 (2006) · Zbl 1135.35339
[8] Chang, K.-C., Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications, vol. 6 (1993), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA
[9] Chen, Z.; Lin, C.-S.; Zou, W., Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrödinger system, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 15, 859-897 (2016) · Zbl 1343.35101
[10] Cingolani, S.; Vannella, G., Marino-Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces, Ann. Mat. Pura Appl. (4), 186, 1, 157-185 (2007) · Zbl 1232.58006
[11] Dancer, E. N.; Wei, J.; Weth, T., A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 27, 3, 953-969 (2010) · Zbl 1191.35121
[12] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ., 6, 8, 883-901 (1981) · Zbl 0462.35041
[13] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Classics in Mathematics (2001), Springer-Verlag: Springer-Verlag Berlin · Zbl 1042.35002
[14] Hirano, N.; Zou, W., A perturbation method for multiple sign-changing solutions, Calc. Var. Partial Differ. Equ., 37, 1-2, 87-98 (2010) · Zbl 1186.35041
[15] H. Li, Multiple positive solutions for coupled Schrödinger equations with perturbations. II, in preparation. · Zbl 1460.35129
[16] Li, H.; Wang, Z.-Q., Multiple positive solutions for coupled Schrödinger equations with perturbations, Commun. Pure Appl. Anal., 20, 2, 867-884 (2021) · Zbl 1460.35129
[17] Li, P.; Yau, S.-T., On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88, 3, 309-318 (1983) · Zbl 0554.35029
[18] Li, Y.; Liu, Z.; Zhao, C., Nodal solutions of a perturbed elliptic problem, Topol. Methods Nonlinear Anal., 32, 1, 49-68 (2008) · Zbl 1173.35497
[19] Liu, J.; Liu, X.; Wang, Z.-Q., Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differ. Equ., 261, 12, 7194-7236 (2016) · Zbl 1352.35162
[20] Marino, A.; Prodi, G., Metodi perturbativi nella teoria di Morse, Univ. Genova Pubbl. Ist. Mat. (2), vol. 99 (1974), (Italian), i+42 pp · Zbl 0311.58006
[21] Qi, Z.; Zhang, Z., Existence of multiple solutions to a class of nonlinear Schrödinger system with external sources terms, J. Math. Anal. Appl., 420, 2, 972-986 (2014) · Zbl 1300.35133
[22] Rabinowitz, P. H., Multiple critical points of perturbed symmetric functionals, Trans. Am. Math. Soc., 272, 2, 753-769 (1982) · Zbl 0589.35004
[23] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65 (1986), Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI · Zbl 0609.58002
[24] Struwe, M., Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscr. Math., 32, 3-4, 335-364 (1980) · Zbl 0456.35031
[25] Struwe, M., Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 34 (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0864.49001
[26] Tanaka, K., Morse indices at critical points related to the symmetric mountain pass theorem and applications, Commun. Partial Differ. Equ., 14, 1, 99-128 (1989) · Zbl 0669.34035
[27] Yue, X.; Zou, W., Infinitely many solutions for the perturbed Bose-Einstein condensates system, Nonlinear Anal., Theory Methods Appl., 94, 1, 171-184 (2014) · Zbl 1285.35026
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