Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem $$\begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases}$$ that branch out from the constant solution $u=1$ as $p$ grows from $2$ to $+\infty$. The non-zero constant positive solution is the unique positive solution for $p$ close to $2$. We show that there exist arbitrarily many positive solutions as $p\to\infty$ (in particular, for supercritical exponents) or as $R \to \infty$ for any fixed value of $p>2$, answering partially a conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for $p$ and $R$ so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.Comment: 37 pages, 24 figure

Isomorphism Theorem for BSS Recursively Enumerable Sets over Real Closed Fields

The main result of this paper lies in the framework of BSS computability : it shows roughly that any recursively enumerable set S in R N 6 1, where R is a real closed field, is isomorphic to R dimS by a bijection ' which is decidable over S. Moreover the map S 7! ' is computable

Entire radial and nonradial solutions for systems with critical growth

In this paper we establish existence of radial and nonradial solutions to the system \begin{equation*} \begin{cases} \displaystyle -\Delta u_1 = F_1(u_1,u_2) &\text{in }\R^N,\\ -\Delta u_2 = F_2(u_1,u_2) &\text{in }\R^N,\\ u_1\geq 0,\ u_2\geq 0 &\text{in }\R^N,\\[1\jot] u_1,u_2\in D^{1,2}(\R^N), \end{cases} \end{equation*} where $F_1,F_2$ are nonlinearities with critical behavior

Asymptotic symmetries for fractional operators

We investigate existence of ground states and nodal ground states for a class of nonlocal equations. We also study the problem numerically