Abstract
We established the existence of weak solutions of the fourth-order elliptic equation of the form
where \(\lambda \) is a positive parameter, \(a(x)\) and \(b(x)\) are positive functions, while \(f : \mathbb{R }\rightarrow \mathbb{R }\) is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that the problem has at least three solutions.
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Acknowledgments
The authors are very grateful to the anonymous referees for their knowledgeable reports, which helped us to improve our manuscript. This work supported by National Natural Science Foundation of China (No. 11201323), the Fundamental Research Funds for the Central Universities (No. XDJK2013D007), Scientific Research Fund of SUSE (No. 2011KY03) and Scientific Reserch Fund of SiChuan Provincial Education Department (No. 12ZB081).
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Li, L., Pan, WW. A note on nonlinear fourth-order elliptic equations on \(\mathbb R ^N\) . J Glob Optim 57, 1319–1325 (2013). https://doi.org/10.1007/s10898-012-0031-0
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DOI: https://doi.org/10.1007/s10898-012-0031-0
Keywords
- Multiple solutions
- Three critical points theorem
- Nonlinear fourth-order elliptic equations
- Variational method