We consider the problem of transforming a symmetric differential expression of even or odd order with both real and complex coefficients with a Kummer-Liouville transformation. An existence proof is given which yields an algorithm for computing exactly the coefficients of the transformed equation. By using the concept of a modified Kummer-Liouville transformation we derive explicit expressions for the coefficients of the transformed equation which are correct modulo Levinson terms only. However, the application of Levinson's theorem to asymptotic integration shows that Levinson terms have no effect on the asymptotics of the eigenfunctions.

We study the theory of existence and uniqueness of nonlinear diffusion equations of the form \[ u_{t}=\Delta \varphi(u)\quad\text{in}\quad\mathbb{R}^{N} \times(0,\infty), \] where $\varphi:{{\mathbb{R}^N}}_+\to{{\mathbb{R}^N}}_+$ is a continuous and increasing function. We focus on the fast diffusion type by imposing the growth condition : $$ \qquad {\displaystyle} m_1\le \dfrac{s\varphi'(s)}{\varphi(s)} \le m_2, $$ for some constants $0 < m_1\le m_2 < 1$ and all $s>0$. Moreover, $m_1>(N-2)/N$. Existence is obtained for an optimal class of initial data, namely, for any nonnegative Borel measure (not necessarily a locally finite measure). Uniqueness is proven for concave $\varphi$'s. The results extend to the general equation the optimal theory with measure-valued data now available for the equation with power functions $\varphi(s)=s^{m}$. The asymptotic behaviour is also studied.

We consider the supercritical Dirichlet problem $$\left(P_{\varepsilon}\right)\qquad -\Delta u=u^{{N+2\over N-2}+{\varepsilon}}\ \hbox{in $\Omega,$}\ u>0\ \hbox{in $\Omega,$}\ u=0\ \hbox{on $\partial\Omega$} $$ where $N\ge3,$ ${\varepsilon}>0$ and $\Omega\subset{\mathbb R}^N$ is a smooth bounded domain with a small hole of radius $d.$ When $\Omega$ has some symmetries, we show that $\left(P_{\varepsilon}\right)$ has an arbitrary number of solutions for ${\varepsilon}$ and $d$ small enough. When $\Omega$ has no symmetries, we prove the existence, for $d$ small enough, of solutions blowing up at two or three points close to the hole as ${\varepsilon}$ goes to zero.

This work deals with the study of the optimal constants of Sobolev and Hardy-Sobolev inequalities with weights and their relations with the behavior of some mixed Dirichlet-Neumann boundary conditions. More precisely, we analyze the attainability of the Sobolev constant \begin{equation}\label{eq:sobb00} S^2_{\gamma}({\Omega},\Sigma_1)=\inf_{u\in {E_{\Sigma_1}^{2,\gamma}(\Omega)};u\not\equiv 0}\frac{{\displaystyle\int}_\Omega {{|x|^{-2\gamma}}} |{\nabla} u|^2dx}{ \Big ({\displaystyle\int}_\Omega |x|^{-2^*\gamma}|u|^{2^*}dx \Big )^{\frac{2}{2^*}}}, \end{equation} and the Hardy-Sobolev constant \begin{equation}\label{HS00} {\Lambda}_{N,\gamma}(\Omega,\Sigma_1)=\inf_{u\in{E_{\Sigma_1}^{2,\gamma}({\Omega})} , u\not\equiv 0} \frac{{\displaystyle\int}_\Omega {{|x|^{-2\gamma}}} |{\nabla} u|^2dx}{ {\displaystyle\int}_\Omega \frac{|u|^2}{|x|^{2(\gamma+1)}} dx} \end{equation} where $\Omega\subset{{\rm I\! R^{N}}}$, $N\ge 3$, is a smooth bounded domain such that $0\in\Omega$, $-\infty <\gamma <\frac{N-2}{2}$, $2^*=\frac{2N}{N-2}$, and ${E_{\Sigma_1}^{2,\gamma}({\Omega})}$ is a Sobolev space that we will define later. The deep relation between the geometry of the domain, the boundary conditions and the attainability of the critical constants is studied. As a direct application we will consider elliptic problems of the form \begin{equation}\label{P0} \left\{\begin{array}{rcl} -\div({{|x|^{-2\gamma}}} {{\nabla}} u) & = & {\lambda} \frac {u^q}{|x|^{2(\gamma +1)}}+ \frac {u^r}{|x|^{(r+1)\gamma}}, \quad u > 0{\quad\mbox{in }}\Omega,\\ u & = & 0{\quad\mbox{on }}\Sigma_1,\\ {{|x|^{-2\gamma}}} \frac {{\partial} u}{{\partial} \nu} & = & 0{\quad\mbox{on }} \Sigma_2, \end{array}\right. \end{equation} where $q$ and $r$ are given real parameters under convenient hypotheses and $\overline \Sigma_1$, $\overline \Sigma_2$, is a smooth partition of $\partial\Omega$.