We study a semilinear fourth order elliptic problem with exponential nonlinearity. Motivated by a question raised in [Li], we partially extend known results for the corresponding second order problem. Several new difficulties arise and many problems still remain to be solved. We list the ones we feel particularly interesting in the final section.
We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, their stability, as well as bifurcation diagrams. As a case study, these methods are applied to the Kuramoto-Sivashinski equation. This equation has been investigted extensively, and its bifurcation diagram is well known from… (More)
The existence of nontrivial solutions of quasilinear elliptic equations at critical growth is proved. The solutions are obtained by variational methods: as the corresponding functional is nonsmooth, the analysis of Palais-Smale sequences requires suitable generalizations of the techniques involved in the study of the corresponding semilinear problem with… (More)
Existence results available for the semilinear Brezis-Nirenberg eigenvalue problem suggest that the compactness problems for the corresponding action functionals are more serious in small dimensions. In space dimension n = 3, one can even prove nonexistence of positive solutions in a certain range of the eigenvalue parameter. In the present paper we study a… (More)
We develop a computer-assisted technique for constructing and analyzing orbits of dissipative evolution equations. As a case study, the methods are applied to the Kuramoto–Sivashinski equation, for which we prove the existence of a hyperbolic periodic orbit. 1. Introduction. In this paper we consider the problem of investigating the flow of dis-sipative… (More)
We prove the existence of the critical fixed point (F, G) for MacKay's renormaliza-tion operator for pairs of maps of the plane. The maps F and G commute, are area-preserving, reversible, real analytic, and they satisfy a twist condition.
We consider the equation −∆u = wu 3 on a square domain in R 2 , with Dirichlet boundary conditions, where w is a given positive function that is invariant under all (Euclidean) symmetries of the square. This equation is shown to have a solution u, with Morse index 2, that is neither symmetric nor antisymmetric with respect to any nontrivial symmetry of the… (More)
All attempts of aeroelastic explanations for the torsional instability of suspension bridges have been somehow criticised and none of them is unanimously accepted by the scientific community. We suggest a new nonlinear model for a suspension bridge and we perform numerical experiments with the parameters corresponding to the collapsed Tacoma Narrows Bridge.… (More)
We consider the equation −∆u = wf ′ (u) on a symmetric bounded domain in R n with Dirichlet boundary conditions. Here w is a positive function or measure that is invariant under the (Euclidean) symmetries of the domain. We focus on solutions u that are positive and/or have a low Morse index. Our results are concerned with the existence of non-symmetric… (More)
We consider the nonlinear wave equation u tt − u xx = ±u 3 and the beam equation u tt + u xxxx = ±u 3 on an interval. Numerical observations indicate that time-periodic solutions for these equations are organized into structures that resemble branches and seem to undergo bifurcations. Besides describing our observations, we prove the existence of… (More)