Radially symmetric solutions of elliptic PDEs with uniformly negative weight

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Abstract

We consider the perturbed Hammerstein integral equation
$$\begin{aligned} y(t)=\gamma (t)H(\varphi (y))+\lambda \int _0^1G(t,s)f(s,y(s))\, \mathrm{d}s \end{aligned}$$
in the case where it may hold that \(f(t,y)<0\), for each \((t,y)\in [0,1]\times [0,+\infty )\), and \(\lim _{y\rightarrow \infty }f(t,y)=-\infty \); in other words, f may be a strictly negative function on its entire domain and may uniformly blow up to \(-\infty \) as \(y\rightarrow +\infty \). We apply our results, in part, to radially symmetric solutions of PDEs of the form
$$\begin{aligned} -\Delta u(\varvec{x})=\lambda a(|\varvec{x}|)g(u(\varvec{x})) \end{aligned}$$
subject to nonlocal boundary conditions and show that this problem can possess a positive solution even if \(\lim _{u\rightarrow \infty }g(u)=-\infty \). By using a nonstandard cone and attendant open set, these results are able to be guaranteed by imposing relatively straightforward conditions. In addition, our results apply to forcing terms f and g with polynomial growth at \(+\infty \) of any degree. We demonstrate that, in principle, our results can be applied to ecological modeling with density-dependent growth and nonlocal boundary conditions.

Keywords

Hammerstein integral equation Negative weight Coercivity Radially symmetric solution Density-dependent growth 

Mathematics Subject Classification

Primary 35B09 35J25 45G10 45M20 47H30 Secondary 34B10 47H07 92D40 

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCreighton Preparatory SchoolOmahaUSA

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