Topological Methods in Nonlinear Analysis

Critical superlinear Ambrosetti-Prodi problems

Djairo G. de Figueiredo and Yang Jianfu

Full-text: Open access

Abstract

We consider the existence of multiple solutions for problem (1.1) below with either $\lambda\neq \lambda$ or $\lambda=\lambda_1$, where $\lambda_k$, $k=1,2,\ldots$ are eigenvalues of $(-\Delta, H^1_0(\Omega))$. The local bifurcation from $\lambda=\lambda_k$ is also investigated.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 14, Number 1 (1999), 59-80.

Dates
First available in Project Euclid: 29 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475179422

Mathematical Reviews number (MathSciNet)
MR1758880

Zentralblatt MATH identifier
0958.35055

Citation

de Figueiredo, Djairo G.; Jianfu, Yang. Critical superlinear Ambrosetti-Prodi problems. Topol. Methods Nonlinear Anal. 14 (1999), no. 1, 59--80. https://projecteuclid.org/euclid.tmna/1475179422


Export citation

References

  • H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems , Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 145–151 \ref\key 2
  • A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces , Ann. Math. Pura Appl., 93 (1972), 231–247 \ref\key 3
  • H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaire elliptiques , J. Funct. Anal., 40 (1981), 1–29 \ref\key 4
  • M. S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem , Indiana Univ. Math. J., 24 (1975), 837–846 \ref\key 5
  • R. Böhme, Die Lösung der Verzwergungsgleichungen für nichtlineare Eigenwertprobleme , Math. Z., 127 (1972), 105–126 \ref\key 6
  • H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials , J. Math. Pures Appl., 58 (1978), 137–151 \ref\key 7
  • H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of integrals , Proc. Amer. Math. Soc., 88 (1983), 486–490 \ref\key 8
  • H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents , Comm. Pure Appl. Math., 24(1983), 437–477 \ref\key 9
  • A. Castro, Metodos variacionales y analisis funcional no lineal , X Coloquio Colombiano de Matematicos 1980 \ref\key 10
  • A. Castro, Reduction via minimax , Lecture Notes in Math., 957 , Springer \ref\key 11
  • J. Chabrowski and Yang Jianfu, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent , Z. Angev. Math. Phys., 48(1998), 276–293 \ref\key 12
  • E. N. Dancer, On the ranges of certain weakly nonlinear elliptic partial differential equations , J. Math. Pures Appl., 57 (1978), 351–366 \ref\key 13
  • Deng Yinbin, On the superlinear Ambrosetti–Prodi problem involving critical Sobolev exponents , Nonlinear Anal., 17 (1991), 1111–1124 \ref\key 14
  • D. G. de Figueiredo, The Ekeland variational principle with applications and detours , Tata Inst. Fund. Res. Lectures on Math. and Phys., 81 (1989) \ref\key 15
  • D. G. de Figueiredo, On superlinear elliptic problems with nonlinearities interacting only with higher eigenvalues , Rocky Mountain J. Math., 18(1988), 287–303 \ref\key 16
  • D. G. de Figueiredo, Lectures on Boundary Value Problems of Ambrosetti–Prodi Type, Atas do 12$^\circ$ Seminario Brasileiro de Análise, São Paulo (1980) \ref\key 17
  • S. Fučik, Solvability of Nonlinear Equations and Boundary Value Problems, D. Reidel Publ. Co., Dordrecht (1980) \ref\key 18
  • J. L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations , Comm. Pure Appl. Math., XXVIII (1975), 567–597 \ref\key 19
  • T. Küpper and C. A. Stuart, Bifurcation into gaps in the essencial spectrum , J. Reine Angew. Math., 409 (1990), 1–34 \ref\key 20
  • A. C. Lazer and P. J. McKenna, Nonlinear perturbations of linear elliptic boundary value problems at resonance , J. Math. Mech., 19 (1973), 63–72 \ref\key 21
  • A. Marino, La biforcazione nel caso variazionale , 132, Confer. Sem. Mat. Univ. Bari (1977) \ref\key 22
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag (1989) \ref\key 23
  • D. Mitrović and D. Zubrinić, Fundamentals of Applied Functional Analysis, Pitman Mongraphs and Surveys in Pure and Applied Mathematics No 91, Addison Wesley Longman(1998) \ref \key 24
  • P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations , 65 AMS Conf. Ser. Math. (1989) \ref\key 25
  • B. Ruf and P. N. Srikanth, Multiplicity results for superlinear elliptic problems with partial interference with spectrum , J. Math. Anal. Appl., 118 (1986), 15–23 \ref\key 26
  • G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent , Ann. Inst\.H. Poincaré Anal. Non Linéaire, 9 (1992), 281–304 \ref\key 27
  • M. Willem, Minimax Theorems, Progr. in Nonlinear Differential Equations Appl. (1996)