Multiple positive solutions for a class of Neumann problems

We study the existence of multiple positive solutions of the Neumann problem \begin{equation*} \begin{split} -u''(x)&=\lambda f(u(x)), \qquad x\in(0,1),\\ u'(0)&=0=u'(1), \end{split} \end{equation*} where $\lambda$ is a positive parameter, $f\in C([0,\infty),\mathbb{R})$ and for some $\beta>0$ such that $f(0)=0$, $f(s)>0$ for $s\in(\beta,\infty)$, $f(s)\beta)$ is the unique positive zero of $F(s)=\int_0^sf(t)\,dt$. In particular, we prove that there exist at least $2n+1$ positive solutions for $\lambda\in \big(\frac{n^2\pi^2}{f'(\beta)},\infty\big)$, where $n\in \mathbb{N}$. The proof of our main result is based upon the bifurcation and continuation methods

Periodic solutions and torsional instability in a nonlinear nonlocal plate equation

A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of periodic solutions are proved. The natural phase space is a particular second order Sobolev space that can be orthogonally split into two subspaces containing, respectively, the longitudinal and the torsional movements of the plate. Sufficient conditions for the stability of periodic solutions and of solutions having only a longitudinal component are given. A stability analysis of the so-called prevailing mode is also performed. Some numerical experiments show that instabilities may occur. This plate can be seen as a simplified and qualitative model for the deck of a suspension bridge, which does not take into account the complex interactions between all the components of a real bridge.Comment: 34 pages, 4 figures. The result of Theorem 6 is correct, but the proof was not correct. We slightly changed the proof in this updated versio

Global Behavior of the Components for the Second Order m-Point Boundary Value Problems

We consider the nonlinear eigenvalue problems u″+rf(u)=0, 00 for i=1,…,m−2, with ∑i=1m−2αi<1; r∈â„Â; f∈C1(â„Â,â„Â). There exist two constants s2<0<s1 such that f(s1)=f(s2)=f(0)=0 and f0:=limu→0(f(u)/u)∈(0,∞), f∞:=lim|u|→∞(f(u)/u)∈(0,∞). Using the global bifurcation techniques, we study the global behavior of the components of nodal solutions of the above problems

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

Adaptive control in rollforward recovery for extreme scale multigrid

With the increasing number of compute components, failures in future exa-scale computer systems are expected to become more frequent. This motivates the study of novel resilience techniques. Here, we extend a recently proposed algorithm-based recovery method for multigrid iterations by introducing an adaptive control. After a fault, the healthy part of the system continues the iterative solution process, while the solution in the faulty domain is re-constructed by an asynchronous on-line recovery. The computations in both the faulty and healthy subdomains must be coordinated in a sensitive way, in particular, both under and over-solving must be avoided. Both of these waste computational resources and will therefore increase the overall time-to-solution. To control the local recovery and guarantee an optimal re-coupling, we introduce a stopping criterion based on a mathematical error estimator. It involves hierarchical weighted sums of residuals within the context of uniformly refined meshes and is well-suited in the context of parallel high-performance computing. The re-coupling process is steered by local contributions of the error estimator. We propose and compare two criteria which differ in their weights. Failure scenarios when solving up to $6.9\cdot10^{11}$ unknowns on more than 245\,766 parallel processes will be reported on a state-of-the-art peta-scale supercomputer demonstrating the robustness of the method