A Semilinear Fourth Order Elliptic Problem with Exponential Nonlinearity

Abstract

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.

Radial entire solutions for supercritical biharmonic equations

“…Let us mention that the subcritical case p ≤ n+4 n−4 is by now well-established, see [5,Theorems 1.3,1.4]. There are several motivations for the study of (1). Let us try to explain them in some detail.…”

“…Most of the methods employed for the proof of Proposition 1 are special for second order equations and do not apply to (1). For instance, qualitative properties of solutions require a detailed analysis of a dynamical system in the corresponding phase space which is two dimensional for (2), whereas it is four dimensional for (1).…”

“…For instance, qualitative properties of solutions require a detailed analysis of a dynamical system in the corresponding phase space which is two dimensional for (2), whereas it is four dimensional for (1). And in four dimensional spaces powerful tools such as the Poincaré-Bendixson theory are no longer available.…”

“…This means that existence of a solution immediately implies the existence of infinitely many solutions, each one of them being characterized by its value at the origin. To ensure smoothness of the solution, one needs to require that u ′ (0) = u ′′′ (0) = 0 for (1) and u ′ (0) = 0 for (2). But contrary to (2), solutions of (1) may be determined only by fixing a priori also the value of u ′′ (0).…”

“…It states that the shooting concavity u ′′ (0) enables to find both positive and negative finite time blow up solutions for (1). Since no free parameter is available, no such solutions exist for (2).…”