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Superlinear elliptic boundary value problems. (English) Zbl 0839.35048
By using a new form of the mountain pass lemma due to the author, he extends results obtained by Ambrosetti and Rabinowitz on this topic.
35J65Nonlinear boundary value problems for linear elliptic equations
58E05Abstract critical point theory
49J27Optimal control problems in abstract spaces (existence)
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[2][AT] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calculus of Variations, to appear
[3][ALP] S. Ahmad, A.C. Lazer, J. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J.25, 933–944 (1976) · Zbl 0351.35036 · doi:10.1512/iumj.1976.25.25074
[4][CM1] D.G. Costa and C.A. Magalhaes, Variational elliptic problems which are nonquadratic at infinity, Trabelhos de Mat.250, 1–14 (1991)
[5][CM2] D.G. Costa and C.A. Magalhaes, Un probleme elliptique nonquadratique a l’infini, C.R. Acad. Sci. Paris,315, 1059–1062 (1992)
[6][dFM] D.G. de Figueiredo and I. Massabo, Semilinear elliptic equations with primitive of the nonlinearity interacting with the first eigenvalue, J. Math. Anal & Appl.,156, 381–394 (1991) · Zbl 0741.35013 · doi:10.1016/0022-247X(91)90404-N
[7][FG] A. Fonda and J.-P. Gossez, semicoercive variational problems at resonance: an abstract approach, Louvain-la-Neuve, Report No.143, 1988
[8][LL] E.A. Landesman and A.C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech.19, 609–623 (1970)
[9][LM] A.C. Lazer and P.J. McKenna, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues, Comm. P.D.E., I,10, 107–150 (1985), II,11, 1653–1676 (1986) · Zbl 0572.35036 · doi:10.1080/03605308508820374
[10][MWW] J. Mawhin, J.R. Ward and M. Willem, Variational methods and semilinear elliptic equations, Arch. Rat. Mech. Anal.95, 269–277 (1986) · Zbl 0656.35044 · doi:10.1007/BF00251362
[11][R] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conf. Board of Math. Sci. Reg. Conf. Ser. in Math., No.65, Amer. Math. Soc., 1986
[12][Sc1] M. Schechter, New saddle point theorems. Proceedings of an International Symposium on Generalized Functions and their Applications, Varanasi, India, December 23–26, 1991. Plenum, N.Y., 1993, pp. 213–219
[13][Sc2] M. Schechter, A generalization of the saddle point method with applications, Annales Polonici Mathematici,57, 269–281 (1992)
[14][Sc3] M. Schechter, Critical points over splitting subspaces, Nonlinearity,6, 417–427 (1993) · Zbl 0779.58010 · doi:10.1088/0951-7715/6/3/005
[15][Sc4] M. Schechter, Splitting subspaces and critical points, Applicable Analysis,49, 33–48 (1993) · Zbl 0802.35049 · doi:10.1080/00036819308840163
[16][Sc5] M. Schechter, The intrinsic mountain pass, to appear
[17][Sc6] M. Schechter, Critical points when there is no saddle point geometry, preprint.
[18][Sc7] M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc.,331, 681–703 (1992) · Zbl 0757.35026 · doi:10.2307/2154135
[19][Si] E.A. de B.e. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Analysis TMA,16, 455–477 (1991) · Zbl 0731.35042 · doi:10.1016/0362-546X(91)90070-H
[20][ST] M. Schechter and K. Tintarev, Pairs of critical points produced by linking subsets with applications to semilinear elliptic problems, Bull. Soc. Math. Belg.44, 249–261 (1992)