## Abstract and Applied Analysis

### Existence of Positive Solutions of Semilinear Biharmonic Equations

#### Abstract

This paper is concerned with the existence of positive solutions of semilinear biharmonic problem whose associated functionals do not satisfy the Palais-Smale condition.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 624328, 11 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606037

Digital Object Identifier
doi:10.1155/2014/624328

Mathematical Reviews number (MathSciNet)
MR3198221

Zentralblatt MATH identifier
07022756

#### Citation

Zhang, Yajing; Lü, Yinmei; Wang, Ningning. Existence of Positive Solutions of Semilinear Biharmonic Equations. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 624328, 11 pages. doi:10.1155/2014/624328. https://projecteuclid.org/euclid.aaa/1412606037

#### References

• F. Bernis, J. García Azorero, and I. Peral, “Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order,” Advances in Differential Equations, vol. 1, no. 2, pp. 219–240, 1996.
• M. Ramos and P. Rodrigues, “On a fourth order superlinear elliptic problem,” Electronic Journal of Differential Equations, Conference 06, pp. 243–255, 2001.
• F. Ebobisse and M. O. Ahmedou, “On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 52, no. 5, pp. 1535–1552, 2003.
• E. Berchio and F. Gazzola, “Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities,” Electronic Journal of Differential Equations, vol. 2005, 20 pages, 2005.
• M. B. Ayed and M. Hammami, “On a fourth order elliptic equation with critical nonlinearity in dimension six,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 5, pp. 924–957, 2006.
• Y. Liu and Z. Wang, “Biharmonic equations with asymptotically linear nonlinearities,” Acta Mathematica Scientia, vol. 27, no. 3, pp. 549–560, 2007.
• Y. Zhang, “Positive solutions of semilinear biharmonic equations with critical Sobolev exponents,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 1, pp. 55–67, 2012.
• A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” Journal of Functional Analysis, vol. 14, pp. 349–381, 1973.
• M. Ramos, S. Terracini, and C. Troestler, “Superlinear indefinite elliptic problems and Pohožaev type identities,” Journal of Functional Analysis, vol. 159, no. 2, pp. 596–628, 1998.
• 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.
• Z. Liu, S. Li, and Z.-Q. Wang, “Positive solutions of elliptic boundary value problems without the (P.S.) type assumption,” Indiana University Mathematics Journal, vol. 50, no. 3, pp. 1347–1369, 2001.
• A. Bahri and P.-L. Lions, “Solutions of superlinear elliptic equations and their Morse indices,” Communications on Pure and Applied Mathematics, vol. 45, no. 9, pp. 1205–1215, 1992.
• R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, Amsterdam, The Netherlands, 2nd edition, 2009.
• R. C. A. M. van der Vorst, “Best constant for the embedding of the space ${H}^{2}\cap {H}_{0}^{1}(\Omega )$ into ${L}^{2N/(N-4)}(\Omega )$,” Differential and Integral Equations, vol. 6, no. 2, pp. 259–276, 1993.
• K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Basel, Switzerland, 1993.
• H. Hofer, “A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem,” Journal of the London Mathematical Society, vol. 31, no. 3, pp. 566–570, 1985.
• E. Mitidieri, “A Rellich type identity and applications,” Communications in Partial Differential Equations, vol. 18, no. 1-2, pp. 125–151, 1993.
• B. Gidas and J. Spruck, “A priori bounds for positive solutions of nonlinear elliptic equations,” Communications in Partial Differential Equations, vol. 6, no. 8, pp. 883–901, 1981.
• S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,” Communications on Pure and Applied Mathematics, vol. 12, pp. 623–727, 1959. \endinput