Electron. J. Differential Equations, Vol. 2017 (2017), No. 19, pp. 1-15.

Semiclassical solutions of perturbed biharmonic equations with critical nonlinearity

Yubo He, Xianhua Tang, Wen Zhang

Abstract:
We consider the perturbed biharmonic equations
$$ \varepsilon^4 \Delta^2 u+V(x)u=f(x,u),\quad x\in\mathbb{R}^N $$
and
$$ \varepsilon^4 \Delta^2 u+V(x)u=Q(x)|u|^{2^{\ast\ast}-2}u+f(x,u), \quad x\in\mathbb{R}^N $$
where $\Delta^2$ is the biharmonic operator, $N\geq 5$, $2^{\ast\ast}=\frac{2N}{N-4}$ is the Sobolev critical exponent, Q(x) is a bounded positive function. Under some mild conditions on V and f, we show that the above equations have at least one nontrivial solution provided that $\varepsilon \leq \varepsilon_0$, where the bound $\varepsilon_0$ is formulated in terms of N, V, Q and f.

Submitted November 15, 2016. Published January 16, 2017.
Math Subject Classifications: 35J35, 35J60, 58E05, 58E50.
Key Words: Perturbed biharmonic equation; semiclassical solution; critical nonlinearity.

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Yubo He
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: hybmath@163.com
Xianhua Tang
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: tangxh@mail.csu.edu.cn
Wen Zhang
School of Mathematics and Statistics
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: zwmath2011@163.com

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