Large solutions for a system of elliptic equations arising from fluid dynamics

This paper is concerned with the elliptic system (0.1) Delta upsilon=phi, Delta phi=vertical bar del upsilon vertical bar(2) posed in a bounded domain Omega subset of R-N, N is an element of N. Specifically, we are interested in the existence and uniqueness or multiplicity of "large solutions," that is, classical solutions of (0.1) that approach infinity at the boundary of Omega. Assuming that Omega is a ball, we prove that the system (0.1) has a unique radially symmetric and nonnegative large solution with v(0) = 0 (obviously, v is determined only up to an additive constant). Moreover, if the space dimension N is sufficiently small, there exists exactly one additional radially symmetric large solution with v(0) = 0 (which, of course, fails to be nonnegative). We also study the asymptotic behavior of these solutions near the boundary of Omega and determine the exact blow-up rates; those are the same for all radial large solutions and independent of the space dimension. Our investigation is motivated by a problem in fluid dynamics. Under certain assumptions, the unidirectional flow of a viscous, heat-conducting fluid is governed by a pair of parabolic equations of the form (0.2) upsilon(t) -Delta upsilon=theta, theta t-Delta theta=vertical bar del upsilon vertical bar(2), where v and theta represent the fluid velocity and temperature, respectively. The system (0.1), with phi = -theta, is the stationary version of (0.2)

Large radial solutions of a polyharmonic equation with superlinear growth

This paper concerns the equation ∆mu = |u| p, where m ∈ N, p ∈ (1, ∞), and ∆ denotes the Laplace operator in RN, for some N ∈ N. Specifically, we are interested in the structure of the set L of all large radial solutions on the open unit ball B in RN . In the well-understood second-order case, the set L consists of exactly two solutions if the equation is subcritical, of exactly one solution if it is critical or supercritical. In the fourth-order case, we show that L is homeomorphic to the unit circle S 1 if the equation is subcritical, to S 1 minus a single point if it is critical or supercritical. For arbitrary m ∈ N, the set L is a full (m − 1)-sphere whenever the equation is subcritical. We conjecture, but have not been able to prove in general, that L is a punctured (m − 1)-sphere whenever the equation is critical or supercritical. These results and the conjecture are closely related to the existence and uniqueness (up to scaling) of entire radial solutions. Understanding the geometric and topological structure of the set L allows precise statements about the existence and multiplicity of large radial solutions with prescribed center values u(0), ∆u(0), . . . , ∆m−1u(0)

Periodic Solutions and Invariant Manifolds for an Even-Order Differential Equation with Power Nonlinearity

The PDE ∆^m u = u|u|^(p-1), with m ∈ N and p ∈ (1,∞), serves as a paradigm for a large class of higher-order elliptic equations with power-like nonlinearities. In studying radially symmetric solutions of this PDE and their asymptotic behavior, it is of critical importance to understand the dynamics of the associated ODE, u^(2m)= u|u|^(p-1), which is the subject of the present paper. Most solutions of the ODE blow up in finite time, diverging to either ∞ or −∞. In the phase space R^(2m), the orbits of these two types of solutions are separated by a (2m−1)-dimensional manifold M, which is unordered and homeomorphic to each of the coordinate hyperplanes under orthogonal projection. Solutions with orbits on M do not blow up or,else,are unbounded from above and from below (oscillatory blow-up). In the second-order case, M coincides with the stable manifold of the trivial equilibrium. In the fourth-order case, which is our main focus, M contains a two-dimensional manifold M_0 of periodic orbits. There exists a unique periodic solution ũ with ũ (0)=1 and ũ’(0)=0, which shares most of the symmetry properties of the common cosine; all nontrivial periodic solutions are obtained from ũ via scaling and phase-shifting. Every solution with orbit on M \M_0 converges to either the trivial equilibrium or one of the nontrivial periodic orbits; oscillatory blow-up does not occur. In higher-order cases, the structure of the manifold M remains largely open

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation ∆^2 u = u|u|^(p-1) with p ∈ (1,∞) has positive solutions on R^n if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on R^n in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball

Convergence versus periodicity in a single-loop positive-feedback system, 1. Convergence to equilibrium

We study a parameter-dependent single-loop positive-feedback system in the nonnegative orthant of R-n, with n is an element of N, that arises in the analysis of the blow-up behavior of large radial solutions of polyharmonic PDEs with power nonlinearities. We describe the global dynamics of the system for arbitrary n and prove that, in every dimension n <= 4, all forward-bounded solutions converge to one of two equilibria (one stable, the other unstable). In Part 2 of the paper, we will establish the existence of nontrivial periodic orbits in every dimension n >= 12

Multiplicity results for a quasilinear elliptic system via Morse theory

In this work we prove some multiplicity results for solutions of a system of elliptic quasilinear equations, involving the p-Laplace operator (p > 2). The proofs are based on variational and topological arguments and make use of new perturbation results in Morse theory for the Banach space W^{1,p}_0