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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.
MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
58E05Abstract critical point theory
49J27Optimal control problems in abstract spaces (existence)
References:
[1][AR] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal.14 349–381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[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)