Strongly indefinite systems with critical Sobolev exponents

Hulshof, Josephus; Mitidieri, Enzo; vanderVorst, Robertus

doi:10.1090/S0002-9947-98-02159-X

Strongly indefinite systems with critical Sobolev exponents
HTML articles powered by AMS MathViewer

by Josephus Hulshof, Enzo Mitidieri and Robertus vanderVorst PDF

Trans. Amer. Math. Soc. 350 (1998), 2349-2365 Request permission

Abstract:

We consider an elliptic system of Hamiltonian type on a bounded domain. In the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms. Our proof is based on an adaptation of the dual variational method as applied before to the scalar case.

References

Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
A. Ambrosetti and M. Struwe, A note on the problem $-\Delta u=\lambda u+u|u|^{2^\ast -2}$, Manuscripta Math. 54 (1986), no. 4, 373–379. MR 829403, DOI 10.1007/BF01168482
Vieri Benci and Donato Fortunato, The dual method in critical point theory. Multiplicity results for indefinite functionals, Ann. Mat. Pura Appl. (4) 132 (1982), 215–242 (1983) (English, with Italian summary). MR 696044, DOI 10.1007/BF01760982
Vieri Benci and Paul H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. MR 537061, DOI 10.1007/BF01389883
Haïm Brézis and Elliott Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. MR 699419, DOI 10.1090/S0002-9939-1983-0699419-3
Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405
Frank H. Clarke and Ivar Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), no. 2, 103–116. MR 562546, DOI 10.1002/cpa.3160330202
Ph. Clément, D. G. de Figueiredo, and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), no. 5-6, 923–940. MR 1177298, DOI 10.1080/03605309208820869
Ph. Clément, D.G. de Figueiredo & E. Mitidieri, A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities, Nonlinear Partial Differential Equations (Fès, 1994), Pitman Res. Notes in Math. Ser., vol. 343, Longman Sci. Tech., Harlow, 1996, pp. 73–91.
Ph. Clément and Robert C. A. M. Van der Vorst, Interpolation spaces for $\partial _T$-systems and applications to critical point theory, PanAmer. Math. J. 4 (1994), no. 4, 1–45. MR 1310320
Ph. Clément and R. C. A. M. Van der Vorst, On a semilinear elliptic system, Differential Integral Equations 8 (1995), no. 6, 1317–1329. MR 1329843
Djairo G. de Figueiredo and Patricio L. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343 (1994), no. 1, 99–116. MR 1214781, DOI 10.1090/S0002-9947-1994-1214781-2
Djairo G. de Figueiredo and Enzo Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal. 17 (1986), no. 4, 836–849. MR 846392, DOI 10.1137/0517060
Djairo G. de Figueiredo and Enzo Mitidieri, Maximum principles for cooperative elliptic systems, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 2, 49–52 (English, with French summary). MR 1044413
Helmut Hofer, On strongly indefinite functionals with applications, Trans. Amer. Math. Soc. 275 (1983), no. 1, 185–214. MR 678344, DOI 10.1090/S0002-9947-1983-0678344-2
Josephus Hulshof and Robertus van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal. 114 (1993), no. 1, 32–58. MR 1220982, DOI 10.1006/jfan.1993.1062
Josephus Hulshof and Robertus C. A. M. van der Vorst, Asymptotic behaviour of ground states, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2423–2431. MR 1363170, DOI 10.1090/S0002-9939-96-03669-6
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. MR 850686, DOI 10.4171/RMI/12
P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441–467. MR 678562, DOI 10.1137/1024101
Giovanni Mancini and Enzo Mitidieri, Positive solutions of some coercive-anticoercive elliptic systems, Ann. Fac. Sci. Toulouse Math. (5) 8 (1986/87), no. 3, 257–292 (English, with French summary). MR 948755
Enzo Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993), no. 1-2, 125–151. MR 1211727, DOI 10.1080/03605309308820923
S. I. Pohožaev, The eigenfunctions of quasilinear elliptic problems, Mat. Sb. (N.S.) 82 (124) (1970), 192–212 (Russian). MR 0273200
Patrizia Pucci and James Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), no. 3, 681–703. MR 855181, DOI 10.1512/iumj.1986.35.35036
L. A. Peletier and R. C. A. M. Van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations 5 (1992), no. 4, 747–767. MR 1167492
R. C. A. M. Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal. 116 (1992), no. 4, 375–398. MR 1132768, DOI 10.1007/BF00375674
Robert C. A. M. Van der Vorst, Fourth-order elliptic equations with critical growth, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 295–299 (English, with English and French summaries). MR 1320374