Semiclassical solutions of perturbed biharmonic equations with critical nonlinearity

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.