Skip to main content
SpringerLink
Log in

On the existence of positive solutions to a certain class of semilinear elliptic equations

  • Original Paper
  • Published:
Partial Differential Equations and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study the following semilinear elliptic equation

$$\begin{aligned} \Delta u = \varphi \left( V(x) u - f(x,u(x)) \right) \quad \text {in} \,\, \mathbf {R}^N, \quad u \in H^1( \mathbf {R}^N ) \end{aligned}$$

where \(N \ge 1\) and \(\varphi (s)\), V(x), f(x, s) are given functions. Under some conditions on \(\varphi (s), V(x), f(x,s)\), we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berestycki, H., Gallouët, T., Kavian, O.: Équations de champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris Sér. I Math. 297(5), 307–310 (1983)

    MathSciNet  MATH  Google Scholar 

  2. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983)

    Article  MathSciNet  Google Scholar 

  3. Birindelli, I., Galise, G., Leoni, F., Pacella, F.: Concentration and energy invariance for a class of fully nonlinear elliptic equations. Calc. Var. Partial Differ. Equ. 57(6), 22 (2018). Art. 158

  4. Bony, J.M.: Principe du maximum dans les espaces de Sobolev. C. R. Acad. Sci. Paris Sér. A-B 265, A333–A336 (1967)

    MathSciNet  MATH  Google Scholar 

  5. Busca, J., Esteban, M.J., Quaas, A.: Nonlinear eigenvalues and bifurcation problems for Pucci’s operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(2), 187–206 (2005)

    Article  MathSciNet  Google Scholar 

  6. Cerami, G.: Un critero di esistenza per i punti critici su varietá illimitate. Ren. Acad. Sci. Lett. Ist. Lomb. 112, 332–336 (1978)

    MATH  Google Scholar 

  7. Ekeland, I.: Convexity Methods in Hamiltonian Mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 19. Springer, Berlin (1990)

    Book  Google Scholar 

  8. Evans, L.C.: Personal communication. In: Shokaku Mathematical Lecture Series at Tohoku University, (2016)

  9. Felmer, P., Ikoma, N.: Existence and nonexistence of positive solutions to some fully nonlinear equation in one dimension. J. Funct. Anal. 275(8), 2162–2196 (2018)

    Article  MathSciNet  Google Scholar 

  10. Felmer, P.L., Quaas, A.: On critical exponents for the Pucci’s extremal operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(5), 843–865 (2003)

    Article  MathSciNet  Google Scholar 

  11. Felmer, P.L., Quaas, A.: Positive radial solutions to a ‘semilinear’ equation involving the Pucci’s operator. J. Differ. Equ. 199(2), 376–393 (2004)

    Article  MathSciNet  Google Scholar 

  12. Felmer, P.L., Quaas, A., Tang, M.: On uniqueness for nonlinear elliptic equation involving the Pucci’s extremal operator. J. Differ. Equ. 226(1), 80–98 (2006)

    Article  MathSciNet  Google Scholar 

  13. Figueiredo, G.M., Ikoma, N., Santos Júnior, J.R.: Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Ration. Mech. Anal. 213(3), 931–979 (2014)

    Article  MathSciNet  Google Scholar 

  14. Galise, G., Leoni, F., Pacella, F.: Existence results for fully nonlinear equations in radial domains. Commun. Partial Differ. Equ. 42(5), 757–779 (2017)

    Article  MathSciNet  Google Scholar 

  15. Hirata, J., Ikoma, N., Tanaka, K.: Nonlinear scalar field equations in \(\mathbb{R}^N\): mountain pass and symmetric mountain pass approaches. Topol. Methods Nonlinear Anal. 35(2), 253–276 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Ikoma, N.: Existence of solutions of scalar field equations with fractional operator. J. Fixed Point Theory Appl. 19(1), 649–690 (2017)

    Article  MathSciNet  Google Scholar 

  17. Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on \(\mathbb{R}^N\). Proc. R. Soc. Edinb. Sect. A 129(4), 787–809 (1999)

    Article  MathSciNet  Google Scholar 

  18. Jeanjean, L., Lu, S.S.: Nonlinear scalar field equations with general nonlinearity. Nonlinear Anal. 190, 111604 (2020)

    Article  MathSciNet  Google Scholar 

  19. Jeanjean, L., Tanaka, K.: A remark on least energy solutions in \(\mathbf{R}^N\). Proc. Am. Math. Soc. 131(8), 2399–2408 (2003)

    Article  Google Scholar 

  20. Jeanjean, L., Tanaka, K.: A note on a mountain pass characterization of least energy solutions. Adv. Nonlinear Stud. 3(4), 445–455 (2003)

    Article  MathSciNet  Google Scholar 

  21. Li, W., Cheng, J.: Nonnegative radial solutions to a class of elliptic equations involving the Pucci’s extremal operator. J. Math. Anal. Appl. 341(1), 386–397 (2008)

    Article  MathSciNet  Google Scholar 

  22. Lions, P.-L.: A remark on Bony maximum principle. Proc. Am. Math. Soc. 88(3), 503–508 (1983)

    Article  MathSciNet  Google Scholar 

  23. Quaas, A.: Existence of a positive solution to “semilinear’’ equation involving Pucci’s operator in a convex domain. Differ. Integral Equ. 17(5–6), 481–494 (2004)

    MathSciNet  MATH  Google Scholar 

  24. Quaas, A., Sirakov, B.: Existence results for nonproper elliptic equations involving the Pucci operator. Commun. Partial Differ. Equ. 31(7–9), 987–1003 (2006)

    Article  MathSciNet  Google Scholar 

  25. Rabier, P.J.: On the Ekeland–Ghoussoub–Preiss and Stuart criteria for locating Cerami sequences. Ric. Mat. 61(1), 19–29 (2012)

    Article  MathSciNet  Google Scholar 

  26. Willem, M.: Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser Boston, Inc., Boston (1996)

Download references

Acknowledgements

The author would like to thank Lawrence Craig Evans for pointing out the variational structure of (1.3) to him. He also would like to thank Hitoshi Ishii for helpful comments on the consideration of (1.3). The author was supported by JSPS KAKENHI Grant Numbers JP 17H02851, 19H01797 and 19K03590.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Technology, Keio University, Kanagawa, Japan

    Norihisa Ikoma

Authors
  1. Norihisa Ikoma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Norihisa Ikoma.

Additional information

This article is part of the section "Viscosity solutions - Dedicated to Hitoshi Ishii on the award of the 1st Kodaira Kunihiko Prize" edited by Kazuhiro Ishige, Shigeaki Koike, Tohru Ozawa, Senjo Shimizu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ikoma, N. On the existence of positive solutions to a certain class of semilinear elliptic equations. Partial Differ. Equ. Appl. 2, 28 (2021). https://doi.org/10.1007/s42985-021-00079-7

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s42985-021-00079-7

Keywords

Mathematics Subject Classification

Search

Navigation