On explosive solutions for a class of quasi-linear elliptic equations

We study existence, uniqueness, multiplicity and symmetry of large solutions for a class of quasi-linear elliptic equations. Furthermore, we characterize the boundary blow-up rate of solutions, including the case where the contribution of boundary curvature appears.Comment: 34 page

Asymptotic profile and Morse index of nodal radial solutions to the H\'enon problem

We compute the Morse index of nodal radial solutions to the H\'enon problem $$\left\{\begin{array}{ll} -\Delta u = |x|^{\alpha}|u|^{p-1} u \qquad & \text{ in } B, \newline u= 0 & \text{ on } \partial B, \end{array} \right.$$ where $B$ stands for the unit ball in ${\mathbb R}^N$ in dimension $N\ge 3$, $\alpha>0$ and $p$ is near at the threshold exponent for existence of solutions $p_{\alpha}=\frac{N+2+2\alpha}{N-2}$, obtaining that \begin{align*} m(u_p) & = m \sum\limits_{j=0}^{1+\left[{\alpha}/{2}\right]} N_j \quad & \mbox{ if $\alpha$ is not an even integer, or} \newline m(u_p)& = m\sum\limits_{j=0}^{ \alpha /2} N_j + (m-1) N_{1+\alpha/ 2} & \mbox{ if $\alpha$ is an even number.} \end{align*} Here $N_j$ denotes the multiplicity of the spherical harmonics of order $j$. The computation builds on a characterization of the Morse index by means of a one dimensional singular eigenvalue problem, and is carried out by a detailed picture of the asymptotic behavior of both the solution and the singular eigenvalues and eigenfunctions. In particular it is shown that nodal radial solutions have multiple blow-up at the origin, where each node converges (up to a suitable rescaling) to the bubble shaped solution of a limit problem. As side outcome we see that solutions are nondegenerate for $p$ near at $p_{\alpha}$, and we give an existence result in perturbed balls.Comment: 47 page

Bifurcation and symmetry breaking for the H\'enon equation

In this paper we consider the H\'enon problem in a ball. We prove the existence of (at least) one branch of nonradial solutions that bifurcate from the radial ones and that this branch is unbounded

Nonradial sign changing solutions to Lane Emden equation

In this paper we prove the existence of continua of nonradial solutions for the Lane-Emden equation. In a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones. Next, using the fixed point index in cone, we produce nonradial solutions with a new type of symmetry. This result also applies to solutions with fixed signed, showing that the set of solutions to the Lane Emden problem has a very rich and complex structure.Comment: 13 p