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A note on nonlinear fourth-order elliptic equations on $$\mathbb R ^N$$

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  • Lin Li
  • Wen-Wu Pan

Abstract

We established the existence of weak solutions of the fourth-order elliptic equation of the form $$\begin{aligned} \Delta ^2 u -\Delta u + a(x)u=\lambda b(x) f(u) + \mu g (x, u), \qquad x \in \mathbb{R }^N, u \in H^2(\mathbb{R }^N), \end{aligned}$$ 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. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Lin Li & Wen-Wu Pan, 2013. "A note on nonlinear fourth-order elliptic equations on $$\mathbb R ^N$$," Journal of Global Optimization, Springer, vol. 57(4), pages 1319-1325, December.
    • Handle: RePEc:spr:jglopt:v:57:y:2013:i:4:p:1319-1325
    DOI: 10.1007/s10898-012-0031-0
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    1. Biagio Ricceri, 2010. "On an elliptic Kirchhoff-type problem depending on two parameters," Journal of Global Optimization, Springer, vol. 46(4), pages 543-549, April.
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