We are interested in questions related with existence, multiplicity, positivity and behaviour of solutions of elliptic boundary value problems of second and higher order. In general problems $(-\Delta)^m u=f$ in $\Omega\subset\mathbb{R}^2$, $\partial^j / (\partial\nu)^j u=0$ on $\partial\Omega$, where $m>1$, $0\le j\le m-1$ do not satisfy a maximum principle or the positivity preserving property. We will show that for domains near to a circle positivity preserving property is satisfied. Then we will give some results of existence and multiplicity of solutions of the Steklov problem of second and fourth order. Finally we will characterize singular radial solutions of $\Delta^2u=\lambda e^u$ in the unit disk, with boundary conditions $u=\partial u/ \partial\nu=0$. We will show that its radial singular solutions are weakly singular, it means $\lim_{r\rightarrow 0} ru'(r)\in\mathbb{R}$ exists.
