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

Symmetries of solutions for nonlinear Schrödinger equations: Numerical and theoretical approaches

On a bounded domain of $IR^N$, we are interested in the nonlinear Schrödinger problem $-Delta u + V(x)u = vert uvert^{p-2}u$ submitted to the Dirichlet boundary conditions or Neumann boundary conditions. This equation has many interests in astrophysics and quantum mechanics. Depending on the domain and the potential $V$, we are studying numerically (by making and computing algorithms) and theoretically the structure of ground state (resp. least energy nodal) solution, i.e. one-signed (resp. sign-changing) solutions with minimal energy. We prove some symmetry and symmetry breaking results and make a lot of conjectures. We also pay attention to the $p$-Laplacian case and we change the nonlinearity $vert uvert^{p-2}u$.Doctorat en science

Lane–Emden problems: Asymptotic behavior of low energy nodal solutions

We study the nodal solutions of the Lane Emden Dirichlet problem $-\Delta u = |u|^{p-1}u with DBC on a smooth bounded domain$\Omega$in$\IR^2$and where$p>1$. We consider solutions$u_p$satisfying$p \int_{\Omega}\abs{\nabla u_p}^2\to 16\pi e\quad\hbox{as}p\rightarrow+\infty\qquad (*)$and we are interested in the shape and the asymptotic behavior as$p\rightarrow+\infty$. First we prove that (*) holds for least energy nodal solutions. Then we obtain some estimates and the asymptotic profile of this kind of solutions. Finally, in some cases, we prove that$pu_p$can be characterized as the difference of two Green's functions and the nodal line intersects the boundary of$\Omega$, for large$p$.Comment: Annales de l'Institut Henri Poincar\'e, 201

Lane Emden problems with large exponents and singular Liouville equations

We consider the Lane-Emden Dirichlet problem [GRAPHICS] where p > 1 and B denotes the unit ball in R-2. We study the asymptotic behavior of the least energy nodal radial solution u(p), as p -> +infinity. Assuming w.l.o.g. that u(p)(0) < 0, we prove that a suitable rescaling of the negative part u(p)(-) converges to the unique regular solution of the Liouville equation in R-2, while a suitable rescaling of the positive part u(p)(+) converges to a (singular) solution of a singular Liouville equation in R-2. We also get exact asymptotic values for the L-infinity-norms of u(p)(-) and u(p)(+), as well as an asymptotic estimate of the energy. Finally, we have that the nodal line N-p := {x is an element of B: vertical bar x vertical bar = r(p)} shrinks to a point and we compute the rate of convergence of r(p). (C) 2014 Published by Elsevier Masson SAS