Existence of positive solutions to the biharmonic equations in \(\mathbb {R}^{N}\)

This article considers the biharmonic equation \(\begin{aligned} \Delta ^{2}u=K(x)f(u)\quad \text {in }~\mathbb { R}^{N}. \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}^{2}u=K(x)f(u)\text{in}{\mathbb{R}}^N.\end{array}$$ Under suitable assumptions, the existence of positive solutions is obtained. The methods used here contain the integral operator and the Schauder fixed point theory. Since the form of fundamental solution of \(\Delta ^{2}u=0\) ${\mathrm{\Delta }}^{2}u=0$ in \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ depends on N, we divide our discussions into three cases as (a) \(N=2\) $N=2$ ; (b) \(N=4\) $N=4$ ; (c) \(N>2\) $N>2$ but \(N\ne 4\) $N\ne 4$ . The fundamental solution of \(\Delta ^{2}\) ${\mathrm{\Delta }}^2$ plays an essential role in our results.