Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights

Abstract

It is well-known that it is a quite interesting topic to study the asymptotic expansions of entire large solutions of nonlinear elliptic equations near infinity. But very little is done. In this study, we establish the (m+1)\left(m+1)-expansions of entire kk-convex large solutions near infinity to the kk-Hessian equation Sk(D2u)=b(x)f(u)inRN,{S}_{k}\left({D}^{2}u)=b\left(x)f\left(u)\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where m∈N+m\in {{\mathbb{N}}}_{+}, 1≤kkp\gt k) near infinity or f(u)=up+uqf\left(u)={u}^{p}+{u}^{q} (p>kp\gt k and p>q>−1p\gt q\gt -1) near infinity. In particular, inspired by some ideas in partition theory of integer, we give a recursive formula of the coefficient of (n+1)\left(n+1)-order terms (2≤n≤m)\left(2\le n\le m) of the expansions. And if f(u)=up+uqf\left(u)={u}^{p}+{u}^{q} near infinity, we reveal the influence of the lower term of f(u)f\left(u) on the expansion of any entire large solution.

Keywords

• (m + 1)-expansions of solutions
• entire k-convex large solutions
• k-hessian equations
• 35j60
• 35j15
• 35b08
• 35b40