We prove a new result on bifurcating branches of bound states for the nonlinear radially symmetric Schrödinger equation $$ -\Delta u =w(|x|)|u|^{\sigma}u -\lambda^2u \ \ \text{ on ${\mathbb{R}}^N$.} $$ We show that, under suitable assumptions on $w$ and $\sigma$, there exist infinitely many continua of nontrivial bound states $u_\lambda$ which emanate from the trivial solution branch at $\lambda=0$. These continua reach arbitrarily large values of $\lambda$, and they are distinguished by the number of nodal domains of the corresponding solutions $u_\lambda$.

In this paper, we prove the existence of infinitely many radial solutions having a singular behaviour at the origin for a superlinear problem of the form $-\Delta_pu=|u|^{\delta-1}u$ in $B(0,1)\setminus\{0\}\subset\mathbb R^N$,\, $u=0$ for $|x|=1$, where $N>p>1$ and $\delta>p-1$. Solutions are characterized by their nodal properties. The case $\delta+1 <\frac{Np}{N-p}$ is treated. The study of the singularity is based on some energy considerations and takes into account the classification of the behaviour of the possible solutions available in the literature. By following a shooting approach, we are able to deduce the main multiplicity result from some estimates on the rotation numbers associated to the solutions.

We study a class of elliptic equations with different degeneracies in different directions such as $$ \left( |u_x|^{m-2}u_x\right)_x + \left( |u_y|^{q-2}u_y\right)_y =0 $$ with unequal parameters $m$ and $q$, both greater than one. We show that any bounded solution of such an equation must have a bounded gradient. The ideas also apply to a much more general class of degenerate equations, which we describe in some detail.

We consider the problem: $ (P_{\varepsilon}):\, -\Delta u_\varepsilon = u_\varepsilon^{5},\, u_\varepsilon >0 $ in $ A_\varepsilon; \, u_\varepsilon= 0\,\, \mbox{ on } \partial A_\varepsilon $, where $\{A_{\varepsilon } \subset {\mathbb{R}}^3 : {\varepsilon } >0\}$ is a family of bounded annulus-shaped domains such that $A_{\varepsilon }$ becomes ``thin'' as ${\varepsilon }\to 0$. We show that, for any given constant $C>0,$ there exists $\varepsilon_0>0$ such that for any $\varepsilon < \varepsilon_0$, the problem $(P_{\varepsilon })$ has no solution $u_\varepsilon,$ whose energy, $\int_{A_\varepsilon}|\nabla u_\varepsilon |^2,$ is less than C. Such a result extends to dimension three a result previously known in higher dimensions. Although the strategy to prove this result is the same as in higher dimensions, we need a more careful and delicate blow up analysis of asymptotic profiles of solutions $u_{\varepsilon }$ when ${\varepsilon }\to 0$.