TY  - JOUR
ID  - Dhifli2015
LB  - Dhifli2015
AU  - Dhifli, Abdelwaheb
TI  - Positive solutions with specific asymptotic behavior for a polyharmonic problem on R<sup>n</sup>
ST  - Positive solutions with specific asymptotic behavior for a polyharmonic problem on R<sup>n</sup>
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2015
VL  - 35
IS  - 1
SP  - 5
EP  - 19
DO  - http://dx.doi.org/10.7494/OpMath.2015.35.1.5
UR  - http://dx.doi.org/10.7494/OpMath.2015.35.1.5
AB  - 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 \(c\gt 1\).)
KW  - asymptotic behavior
KW  - Dirichlet problem
KW  - Schauder fixed point theorem
KW  - positive bounded solutions
SN  - 1232-9274
ER  -