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

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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 k k -convex large solutions near infinity to the k k -Hessian equation S k ( D 2 u ) = b ( x ) f ( u ) in R N , {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 ≤ k < N ⁄ 2 1\le k\lt N/2 , N ≥ 3 N\ge 3 , f ( u ) = u p f\left(u)={u}^{p} ( p > k p\gt k ) near infinity or f ( u ) = u p + u q f\left(u)={u}^{p}+{u}^{q} ( p > k p\gt k and p > q > − 1 p\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 ) = u p + u q f\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.