Opuscula Math. 35, no. 1 (2015), 5-19

Opuscula Mathematica

Positive solutions with specific asymptotic behavior for a polyharmonic problem on Rn

Abdelwaheb Dhifli

Abstract. This paper is concerned with positive solutions of the semilinear polyharmonic equation \((-\Delta)^{m} u = a(x){u}^{\alpha}\) on \(\mathbb{R}^{n}\), where \(m\) and \(n\) are positive integers with \(n\gt 2m\), \(\alpha\in (-1,1)\). The coefficient \(a\) is assumed to satisfy \[a(x)\approx{(1+|x|)}^{-\lambda}L(1+|x|)\quad \text{for}\quad x\in \mathbb{R}^{n},\] where \(\lambda\in [2m,\infty)\) and \(L\in C^{1}([1,\infty))\) is positive with \(\frac{tL'(t)}{L(t)}\longrightarrow 0\) as \(t\longrightarrow \infty\); if \(\lambda=2m\), one also assumes that \(\int_{1}^{\infty}t^{-1}L(t)dt\lt \infty\). We prove the existence of a positive solution \(u\) such that \[u(x)\approx{(1+|x|)}^{-\widetilde{\lambda}}\widetilde{L}(1+|x|) \quad\text{for}\quad x\in \mathbb{R}^{n},\] with \(\widetilde{\lambda}:=\min(n-2m,\frac{\lambda-2m}{1-\alpha})\) and a function \(\widetilde{L}\), given explicitly in terms of \(L\) and satisfying the same condition at infinity. (Given positive functions \(f\) and \(g\) on \(\mathbb{R}^{n}\), \(f\approx g\) means that \(c^{-1}g\leq f\leq cg\) for some constant \(c\gt 1\).)

Keywords: asymptotic behavior, Dirichlet problem, Schauder fixed point theorem, positive bounded solutions.

Mathematics Subject Classification: 34B18, 35B40, 35J40.

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  • Abdelwaheb Dhifli
  • Campus Universitaire, Faculté des Sciences de Tunis, Département de Mathématiques, 2092 Tunis, Tunisia
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2014-03-03.
  • Revised: 2014-04-04.
  • Accepted: 2014-04-24.
  • Published online: 2014-11-12.
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Cite this article as:
Abdelwaheb Dhifli, Positive solutions with specific asymptotic behavior for a polyharmonic problem on Rn, Opuscula Math. 35, no. 1 (2015), 5-19, http://dx.doi.org/10.7494/OpMath.2015.35.1.5

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