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A Landesman–Lazer Local Condition for Semilinear Elliptic Problems

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Abstract

The objective of this paper is to study the existence, multiplicity and non existence of solutions for semilinear elliptic problems under a local Landesman–Lazer condition. There is no growth restriction at infinity on the nonlinear term and it may change sign. In order to establish the existence of solution we combine the Lyapunov–Schmidt reduction method with truncation and approximation arguments via bootstrap methods. In our applications we also consider the existence of a bifurcation point which may have multiple positive solutions for a fixed value of the parameters.

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References

  • Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Commun. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  Google Scholar 

  • Ahmad, S., Lazer, A.C., Paul, J.L.: Elementary critical point theory and perturbations of elliptic boundary value problems at resonance. Indiana Univ. Math. J. 25, 933–944 (1976)

    Article  MathSciNet  Google Scholar 

  • Alama, S., Del Pino, M.: Solutions of elliptic equations with indefinite nonlinearities via morse theory and linkings. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 95–115 (1996)

  • Alama, S., Tarantello, G.: On semilinear elliptic equations with indefinite nonlinearities. Calc. Var. Partial Differ. Equ. 1, 439–475 (1993)

    Article  MathSciNet  Google Scholar 

  • Alama, S., Tarantello, G.: Elliptic problems with nonlinearities indefinite in sign. J. Funct. Anal. 141, 159–215 (1996)

    Article  MathSciNet  Google Scholar 

  • Amann, H.: Saddle points and multiple solutions of differential equations. Math. Z. 196, 127–166 (1979)

    Article  MathSciNet  Google Scholar 

  • Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519–543 (1994)

    Article  MathSciNet  Google Scholar 

  • Arcoya, D., Gámez, J.L.: Bifurcation theory and related problems: anti-maximum principle and resonance. Commun. Partial Differ. Equ. 26(9–10), 1879–19011 (2001)

    Article  MathSciNet  Google Scholar 

  • Bartolo, P., Benci, V.: Fortunato, Abstract criticl point theorems and application to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983)

    Article  MathSciNet  Google Scholar 

  • Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal. 4, 59–78 (1994)

    Article  MathSciNet  Google Scholar 

  • Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical exponents. Commun. Pure Appl. Math. 34, 437–477 (1983)

    Article  MathSciNet  Google Scholar 

  • Castro, A.: Reduction Methods via Minimax. First Latin American School of Differential Equations (São Paulo, Brazil, 1981). Lecture Notes in Mathematics, vol. 957, pp. 1–20. Springer, Berlin (1982)

  • Castro, A., Lazer, A.C.: Critical point theory and the number of solutions of a nonlinear Dirichlet problem. Ann. Mat. Pura Appl. 120, 113–137 (1979)

    Article  MathSciNet  Google Scholar 

  • Chang, K.C., Jiang, M.Y.: Dirichlet problem with indefinite nonlinearities. Calc. Var. Partial Differ. Equ. 20, 257–282 (2004)

    Article  MathSciNet  Google Scholar 

  • Cossio, J.: Contribution to the study of elliptic partial differential equations. \((\text{Spanish})\) Mathematics and statistics \((\text{ Spanish })\). Rev. Acad. Colomb. Cienc. Exact. Fis. Natur. 28, 135–145 (2004)

  • Costa, D.G., Tehrani, H.: Existence of positive solutions for a class of indefinite elliptic problems in \(\mathbb{R}^N\). Calc. Var. Partial Differ. Equ. 13, 159–189 (2001)

    Article  Google Scholar 

  • De Figueiredo, D.G., Gossez, J.P., Ubilla, P.: Local superlinearity and sublinearity for indefinite semilinear elliptic problems. J. Funct. Anal. 199, 452–467 (2003)

    Article  MathSciNet  Google Scholar 

  • De Figueiredo, D.G., Gossez, J.P., Ubilla, P.: Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity. J. Eur. Math. Soc. 8, 269–286 (2006)

    Article  MathSciNet  Google Scholar 

  • Landesman, E. M., Lazer, A. C.: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19, 609–623 (1969/1970)

  • Landesman, E.M., Lazer, A.C., Meyers, D.R.: On Saddle Point Problems in the calculus of Variations, the Ritz Algorithm, and Monotone Convergence. J. Math. Anal. Appl. 52, 594–614 (1975)

    Article  MathSciNet  Google Scholar 

  • Medeiros, E.S., Severo, U.B., Silva, E.A.B.: On a class of elliptic problems with indefinite nonlinearities. Calc. Var. Partial Differ. Equ. 50, 751–777 (2014)

    Article  MathSciNet  Google Scholar 

  • Ouyang, T.: On the positive solutions of semilinear equations \(\Delta u+\lambda u+hu^p=0\) on compact manifolds II. Indiana Univ. Math. J. 40, 1083–1141 (1991)

    Article  MathSciNet  Google Scholar 

  • Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)

    Article  MathSciNet  Google Scholar 

  • Rabinowitz, P.H.: Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations, Nonlinear Analysis (A Collection of Papers in Honor of Erich Röthe), pp. 161–177. Academic Press, New York (1978)

  • Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC by the American Mathematical Society, Providence, RI, (1986)

  • Ruiz, D.: A priori estimates and existence of positive solutions for strongly nonlinear problems. J. Differ. Equ. 199, 96–114 (2004)

    Article  MathSciNet  Google Scholar 

  • Shaw, H.: Nonlinear elliptic boundary value problems at resonance. J. Differ. Equ. 26, 335–346 (1977)

    Article  MathSciNet  Google Scholar 

  • Silva, E.A.B.: Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal. 16, 455–477 (1991)

    Article  MathSciNet  Google Scholar 

  • Silva, E.A.B., Silva, M.L.: Continuous dependence of solutions for indefinite semilinear elliptic problems. Electron. J. Differ. Equ. 239, 1–17 (2013)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, Brasília, DF, CEP 70910-900, Brazil

    M. C. M. Rezende & E. A. B. Silva

  2. Faculdade de Matemática, Universidade Federal de Mato Grosso, Cuiabá, MT, CEP 78060-900, Brazil

    P. M. Sánchez-Aguilar

Authors
  1. M. C. M. Rezende
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  2. P. M. Sánchez-Aguilar
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  3. E. A. B. Silva
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Correspondence to E. A. B. Silva.

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Rezende, M.C.M., Sánchez-Aguilar, P.M. & Silva, E.A.B. A Landesman–Lazer Local Condition for Semilinear Elliptic Problems. Bull Braz Math Soc, New Series 50, 889–911 (2019). https://doi.org/10.1007/s00574-019-00132-5

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