Bifurcation along curves for the p-Laplacian with radial symmetry

We study the global structure of the set of radial solutions of a nonlinear Dirichlet problem involving the p-Laplacian with p>2, in the unit ball of $R^N$, N \ges 1. We show that all non-trivial radial solutions lie on smooth curves of respectively positive and negative solutions and bifurcating from the line of trivial solutions. This involves a local bifurcation result of Crandall-Rabinowitz type, and global continuation arguments relying on monotonicity properties of the equation. An important part of the analysis is dedicated to the delicate issue of differentiability of the inverse p-Laplacian. We thus obtain a complete description of the global continua of positive/negative solutions bifurcating from the first eigenvalue of a weighted, radial, p-Laplacian problem, by using purely analytical arguments, whereas previous related results were proved by topological arguments or a mixture of analytical and topological arguments. Our approach requires stronger hypotheses but yields much stronger results, bifurcation occuring along smooth curves of solutions, and not only connected sets.Comment: Minor changes to the statement and proof of Theorem 1

Landesman-Lazer conditions at half-eigenvalues of the p-Laplacian

We study the existence of solutions of the Dirichlet problem {gather} -\phi_p(u')' -a_+ \phi_p(u^+) + a_- \phi_p(u^-) -\lambda \phi_p(u) = f(x,u), \quad x \in (0,1), \label{pb.eq} \tag{1} u(0)=u(1)=0,\label{pb_bc.eq} \tag{2} {gather} where $p>1$, \phi_p(s):=|s|^{p-1}\sgn s for $s \in \mathbb{R}$, the coefficients $a_\pm \in C^0[0,1]$, $\lambda \in \mathbb{R}$, and $u^\pm := \max\{\pm u,0\}$. We suppose that $f\in C^1([0,1]\times\mathbb{R})$ and that there exists $f_\pm \in C^0[0,1]$ such that $\lim_{\xi\to\pm\infty} f(x,\xi) = f_\pm(x)$, for all $x \in [0,1]$. With these conditions the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to have a `jumping nonlinearity'. We also suppose that the problem {gather} -\phi_p(u')' = a_+ \phi_p(u^+) - a_- \phi_p(u^-) + \lambda \phi_p(u) \quad\text{on} \ (0,1), \tag{3} \label{heval_pb.eq} {gather} together with \eqref{pb_bc.eq}, has a non-trivial solution $u$. That is, $\lambda$ is a `half-eigenvalue' of \eqref{pb_bc.eq}-\eqref{heval_pb.eq}, and the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to be `resonant'. Combining a shooting method with so called `Landesman-Lazer' conditions, we show that the problem \eqref{pb.eq}-\eqref{pb_bc.eq} has a solution. Most previous existence results for jumping nonlinearity problems at resonance have considered the case where the coefficients $a_\pm$ are constants, and the resonance has been at a point in the `Fucik spectrum'. Even in this constant coefficient case our result extends previous results. In particular, previous variational approaches have required strong conditions on the location of the resonant point, whereas our result applies to any point in the Fucik spectrum.Comment: 14 page

Instability of an integrable nonlocal NLS

In this note we discuss the global dynamics of an integrable nonlocal NLS on $\mathbb{R}$, which has been the object of recent investigation by integrable systems methods. We prove two results which are in striking contrast with the case of the local cubic focusing NLS on $\mathbb{R}$. First, finite time blow-up solutions exist with arbitrarily small initial data in $H^s(\mathbb{R})$, for any $s\geqslant0$. On the other hand, the solitons of the local NLS, which are also solutions of the nonlocal equation, are unstable by blow-up for the latter.Comment: title and some nomenclature changed; journal versio