Opuscula Math. 35, no. 1 (), 5-19
http://dx.doi.org/10.7494/OpMath.2015.35.1.5
Opuscula Mathematica

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

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.