Abstract and Applied Analysis

On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers

Claudianor O. Alves and Marco A. S. Souto

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Abstract

We prove that the semilinear elliptic equation Δ u = f ( u ) , in Ω , u = 0 , on Ω has a positive solution when the nonlinearity f belongs to a class which satisfies μ t q f ( t ) C t p at infinity and behaves like t q near the origin, where 1 < q < ( N + 2 ) / ( N 2 ) if N 3 and 1 < q < + if N = 1 , 2 . In our approach, we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity does not satisfy any hypotheses such those required by the blowup method. Furthermore, we do not impose any restriction on the growth of p .

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 578417, 6 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969173

Digital Object Identifier
doi:10.1155/2008/578417

Mathematical Reviews number (MathSciNet)
MR2411040

Zentralblatt MATH identifier
1187.35064

Citation

Alves, Claudianor O.; Souto, Marco A. S. On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers. Abstr. Appl. Anal. 2008 (2008), Article ID 578417, 6 pages. doi:10.1155/2008/578417. https://projecteuclid.org/euclid.aaa/1220969173


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References

  • A. Ambrosetti and P. H. Rabinowitz, ``Dual variational methods in critical point theory and applications,'' Journal of Functional Analysis, vol. 14, no. 4, pp. 349--381, 1973.
  • B. Gidas and J. Spruck, ``Global and local behavior of positive solutions of nonlinear elliptic equations,'' Communications on Pure and Applied Mathematics, vol. 34, no. 4, pp. 525--598, 1981.
  • D. G. de Figueiredo and J. Yang, ``On a semilinear elliptic problem without ($PS$) condition,'' Journal of Differential Equations, vol. 187, no. 2, pp. 412--428, 2003.
  • C. Azizieh and P. Clément, ``A priori estimates and continuation methods for positive solutions of $p$-Laplace equations,'' Journal of Differential Equations, vol. 179, no. 1, pp. 213--245, 2002.
  • S. Li and Z. Liu, ``Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents,'' Topological Methods in Nonlinear Analysis, vol. 28, no. 2, pp. 235--261, 2006.
  • J. Chabrowski and J. Yang, ``Existence theorems for elliptic equations involving supercritical Sobolev exponent,'' Advances in Differential Equations, vol. 2, no. 2, pp. 231--256, 1997.