Connected sets of positive solutions of elliptic systems in exterior domains

Aleksandra Orpel

doi:10.1007/s00605-019-01343-0

Abstract

The existence of infinitely many connected sets of positive solutions for a certain elliptic system is investigated in this paper. We consider semilinear equations with perturbed Laplace operators described in an exterior domain. We show that each of these solutions mathbf {u}=( u_{1},u_{2}) has the minimal asymptotic decay, namely u_{i}(x)=O(||x||^{2-n}) as ||x||rightarrow infty ,i=1,2, and finite energy in a neighborhood of infinity. Our main tool is the sub and super-solutions theorem which is based on the Sattinger’s iteration procedure. We do not need any growth assumptions concerning nonlinearities.

Highlights

The purpose of this paper is to formulate conditions which guarantee the existence of continua of positive solutions of the following system involving perturbed Laplace operators div(a1(||x||)∇u1(x)) + f1(x, u1(x), u2(x)) + g1(x)x · ∇u1(x) = 0 (1)
Many problems modeled by similar systems arise in various areas of applied mathematics, in biological, chemical or physical phenomena, for example in pseudoplastic fluids [9], reaction–diffusion processes, chemical heterogeneous catalysts [4] or heat conduction in electrically conducting materials [18]
The existence and multiplicity of solutions for such elliptic systems considered in unbounded domains has been widely discussed in the literature

Summary

Introduction

In that paper the variational approach allowed the authors to show the existence of at least nine solutions in the case when is a bounded regular domain in Rn, the right-hand side is a Carathé odory function and satisfies, among others, some growth conditions (see Th.1.1). There exists a vector function u = (u1, u2) ∈ Cl2o+cα(G R) 2 satisfying (1)–(2) and such that for i = 1, 2, ui ≤ ui ≤ ui in G R and ui (x) = ui (x) on ∂ G R This approach allows us to prove two main results: the first one is associated with the existence of sequences of uncountable sets of solutions and the other one gives additional information concerning the asymptotics of solutions and their gradients.

Existence of continua of super-solutions

Proof of the main results

Final remarks and examples

2: Now we fix k0