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Keywords:
semilinear elliptic equations; multiple solutions; shooting method; variational methods
semilinear elliptic equations; multiple solutions; shooting method; variational methods
Summary:
Given a semilinear elliptic boundary value problem having the zero solution and where the nonlinearity crosses the first eigenvalue, we perturb it by a positive forcing term; we show the existence of two solutions under certain conditions that can be weakened in the onedimensional case.
References:
[1] A. Ambrosetti G. Prodi: On the inversion of some differential mappings with singularities between Banach spaces. Ann. Mat. Pura Appl. (4) 93 (1972), 231-247. DOI 10.1007/BF02412022 | MR 0320844
[2] H. Berestycki D. G. Figueiredo: Double resonance in semilinear elliptic problems. Comm. Partial Differential Equations, 6 (1) (1981), 91-120. DOI 10.1080/03605308108820172 | MR 0597753
[3] M. S. Berger E. Podolak: On the solutions of a nonlinear Dirichlet problem. Indiana Univ. Math. J. 24 (1975), 837-846. DOI 10.1512/iumj.1975.24.24066 | MR 0377274
[4] H. Brezis L. Nirenberg: $H^1$ versus $C^1$ local minimizers. C. R. Acad. Sci. Paris 317, Serie I (1993), 465-472. MR 1239032
[5] D. De Figueiredo: On the superlinear Ambrosetti-Prodi problem. Nonlinear Anal. 8 (1984), no. 6, 351-366. MR 0746723 | Zbl 0554.35045
[6] G. Dinca L. Sanchez: Multiple solutions of boundary value problems: an elementary approach via the shooting method. Nonlinear Differential Equations Appl. 1 (1994), 163-178. DOI 10.1007/BF01193950 | MR 1273348
[7] J. V. A. Gongalves: On multiple solutions for a semilinear Dirichlet problem. Houston J. Math. 12 (1986), 43-53. MR 0855792
[8] J. P. Gossez P. Omari: Nonresonance with respect to the Fučík spectrum for the periodic solutions of second order ordinary differential equations. Nonlinear Anal. 14 (1990), no. 12, 1079-1104. DOI 10.1016/0362-546X(90)90069-S | MR 1059615
[9] H. Kaper M. K. Kwong: On two conjectures concerning the multiplicity result of solutions of a Dirichlet problem. Siam J. Math. Anal. 23 (1992), 571-578. DOI 10.1137/0523029 | MR 1158822
[10] J. L. Kazdan F. Warner: Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. 28 (1975), 567-597. DOI 10.1002/cpa.3160280502 | MR 0477445
[11] A. C. Lazer P. J. McKenna: On a conjecture related to the number of solutions of a nonlinear Dirichlet problem. Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 275-283. MR 0726879
[12] A. Marino A. M. Micheletti A. Pistoia: Some variational results on semilinear problems with asymptotically nonsymmetric behaviour. Quaderno Sc. Normale Superiore Nonlinear Analysis, A Tribute in honour of G. Prodi. 1991, pp. 243-256. MR 1205387
[13] J. Mawhin: Point fixes, points critiques et problèmes aux limites. Sem. Math. Sup. 92. Presses Univ. Montreal, 1985. MR 0789982
[14] P. Rabinowitz: Minimax methods in critical point theory and applications to differential equations. CBMS Reg. Conf. 65. Amer. Math. Soc, Providence, R.I., 1986. MR 0845785
[15] F. Zanolin: Continuation theorems for the periodic problem via the translation operator. Preprint, 1993. MR 1490010
[16] E. Zeidler: Nonlinear Functional Analysis and its Applications I. Springer-Verlag, New York, 1985. MR 0816732
[17] B. Zinner: Multiplicity of solutions for two point boundary value problems with jumping nonlinearities. J. Math. Anal. Appl. 176 (1993), 282-291. MR 1222169
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