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

1 Introduction

In this work, we study the ( m + 1 ) -expansions of entire k -convex large solutions to the k -Hessian equation

where m ∈ N + , 1 ≤ k < N ⁄ 2 , N ≥ 3 . Moreover, D 2 u denotes the Hessian of u ∈ C 2 ( R N ) and

where λ = ( λ 1 , … , λ N ) are the eigenvalues of D 2 u in R N and σ k ( λ ) denotes the k th elementary symmetric function for λ ∈ R N . We call u ∈ C 2 ( R N ) an entire large solution to equation (1.1) if u is a k -convex solution of equation (1.1) in R N and u ( x ) → ∞ as ∣ x ∣ → ∞ . The function u is called k -convex if λ ( D 2 u ) ∈ Γ k , where Γ k = { λ ∈ R N : σ l ( λ ) > 0 , 1 ≤ l ≤ k } .

When k = 1 , equation (1.1) reduces to the semilinear elliptic equation

The study of entire large solutions to equation (1.2) has a long history. In 1957, a famous theorem was established by Keller [27] and Osserman [32]: if f ∈ C [ 0 , ∞ ) is positive and increasing on ( 0 , ∞ ) and f ( 0 ) = 0 , then equation (1.2) with b ≡ 1 has an entire positive large solution if and only if f satisfies the Keller-Osserman condition ∫ 1 ∞ ( F ( t ) ) − 1 ⁄ 2 d t = ∞ , where F ( t ) = ∫ 0 t f ( s ) d s . We know that the existence of positive solutions to equation (1.2) has a wide range of applications in conformal geometry. Let g = ( δ i j ) be the usual Euclidean metric on R N . Kazdan and Warner [26] (or Ni [31]) provided a brief description of how the entire positive solution to equation (1.2) with f ( u ) = u ( N + 2 ) ⁄ ( N − 2 ) gives a new metric g ˜ such that − b is the scalar curvature of g ˜ and g ˜ is conformal to g . If f ( u ) = u p with p > 1 and there exists some positive constant C 0 such that ( 1 ⁄ C 0 ) ∣ x ∣ − γ < b ( x ) < C 0 ∣ x ∣ − γ near infinity for some constant γ > 2 , Cheng and Ni [11] studied the existence, convergence and the complete classification of entire positive solutions. Especially, for the maximal solution U , they showed there exists a positive constant C such that

For further study on the existence of entire large solutions to (1.2), we refer the readers to the works of Lair [28], Cîrstea and Rădulescu [13], Tao and Zhang [35], Ye and Zhou [40], Yang [39], and Dupaigne et al. [19]. On the other hand, we know that the first, second, or higher expansions of solutions to semilinear (or quasilinear) elliptic boundary blow-up problems in bounded domains have been studied by many authors (refer for instance, Anedda and Porru [1,2], Alarcón et al. [3], Bandle and Marcus [4], Bandle [5], Cîrstea and Rădulescu [14–16], del Pino and Letelier [17], Du and Guo [18], Huang et al. [20], Mohammed [30], Takimoto and Zhang [34], Zhang et al. [41], and references therein), but very little is considered when the domain is R N . Recently, Wan [36] and Wan et al. [37] investigated the first expansions of entire large solutions near infinity to equation (1.2) by utilizing a fine truncation technique. In this study, our results will include the arbitrary expansions of entire large solutions near infinity to (1.2).

Now, let us return to equation (1.1). If k > 1 , then equation (1.1) is a fully nonlinear equation. When f ≡ 1 , equation (1.1) becomes

When k = N and b ≡ 1 , a famous result of Jörgens [25] for N = 2 , Calabi [10] for N ≤ 5 , and Pogorelov [33] for N ≥ 2 asserts that each classical convex solution of equation (1.3) is sure to be a quadratic polynomial. Then, Caffarelli [9] extended the Bernstein-type result for classical solutions to viscosity solutions. Bao et al. [6] investigated a similar result for a Hessian quotient equation. If b is a measurable function and there exists some positive constant C 0 such that C 0 − 1 ≤ b ≤ C 0 in R N , then Chou and Wang [12] showed that equation (1.3) with k = N has infinitely many convex solutions. Then, Jian and Wang [24] extended the existence of solutions in [12] to the case of nonnegative functions b . If f satisfies the doubling condition, then they showed that the solution is of polynomial growth. When f ≢ 1 , Jin et al. [22] proved that if b ≡ 1 in R N and f ( u ) = u p with p > k , then equation (1.1) has no entire k -convex positive subsolution. If f ( u ) = u p is replaced by f and f ∈ C [ 0 , ∞ ) is a positive non-decreasing function on ( 0 , ∞ ) and f ( 0 ) = 0 , then Ji and Bao [21] showed that this equation has an entire k -convex positive large subsolution if and only if f satisfies ∫ 1 ∞ ( F ( t ) ) − 1 ⁄ k d t = ∞ , where F ( t ) = ∫ 0 t f ( s ) d s . Recently, Bao and Feng [7] proved the necessary and sufficient conditions for the existence of subsolutions to a class of p - k -Hessian equations. For the existence of radial entire large solutions and bounded solutions of equation (1.1) (or systems) with radially symmetric weights, we refer the readers to the works of Zhang and Zhou [42]. Recently, when b ∈ C ∞ ( R N ) is positive in R N and

Wan et al. [38] proved the existence of classical entire k -convex large solutions to equation (1.1). Furthermore, we established the first expansions of entire large solutions near infinity to equation (1.1). As Bhattacharya and Mohammed mentioned in Appendix B of [8] that the condition (1.4) implies 1 ≤ k < N ⁄ 2 . Most recently, under some appropriate assumptions on b , Zhang and Xia [43] gave a necessary and sufficient condition on f for the existence of radial entire large solutions to (1.1). Li and Bao [29] studied the existence and nonexistence results for non-radial entire large solutions to equation (1.1) with 0 < p < k . In particular, they proved that if b ( x ) = ∣ x ∣ − γ + O ( ∣ x ∣ − m ) near infinity for γ ≤ k − 1 and m > γ + ( 2 k − γ ) k ⁄ ( k − p ) , then equation (1.1) admits an entire k -convex positive solution satisfying ( 1 ⁄ C ) ∣ x ∣ 2 k − γ k − p ≤ u ( x ) ≤ C ∣ x ∣ 2 k − γ k − p as ∣ x ∣ → ∞ , where C is a positive constant.

Inspired by the above works, in this study, for an arbitrary m ∈ N + , we established the ( m + 1 ) -expansions of entire k -convex large solutions as ∣ x ∣ → ∞ to equation (1.1). Our results are new even for the case of equation (1.2).

2 Main results

For 1 ≤ k < N ⁄ 2 , Wan et al. [38] proved the existence of classical entire k -convex large solutions to equation (1.1). Moreover, inspired by the class of Karamata functions introduced by Cîrstea and Rădulescu [14–16] for non-decreasing functions and by Mohammed [30] for non-increasing functions, Wan et al. [38] defined a class of functions Λ and further obtained the first expansions of entire large solutions to equation (1.1) with the help of Λ . The set Λ is defined as follows: Λ denotes the set of all positive non-increasing functions θ ∈ C 1 [ R , ∞ ) ∩ L 1 [ R , ∞ ) , which satisfy

Choose a positive constant λ such that λ > k + 1 . Define θ ∞ ( t ) ≔ k k + 1 k λ − k − 1 − 1 k + 1 t − λ k + 1 , t ≥ R and Θ ∞ ( t ) ≔ ∫ t ∞ θ ∞ ( s ) d s = k λ − k − 1 k k + 1 t k + 1 − λ k + 1 , t ≥ R . By a direct calculation, we see that θ ∞ ∈ Λ with D θ = k + 1 λ − k − 1 . We suppose that b satisfies

1. b ∈ C ( R N ) is positive in R N ;

2. there exists some constant λ > k + 1 such that b ( x ) = ∣ x ∣ − λ − k + 1 ( 1 + o ( 1 ) ) as ∣ x ∣ → ∞ ,

1. f ∈ C 1 [ 0 , ∞ )   ( or f ∈ C 1 ( R ) ) is positive and non-decreasing on [ 0 , ∞ ) (or on R ),

1. there exists some sufficiently large positive constant t * such that f ( t ) = t p (with p > k ) for all t ∈ [ t * , ∞ ) ;

1. there exists some sufficiently large positive constant t * such that f ( t ) = t p + t q (with p > k and p > q > − 1 ) for all t ∈ [ t * , ∞ ) .

Next, we establish the ( m + 1 ) -expansions of entire large solutions to equation (1.1) with 1 ≤ k < N ⁄ 2 . To our aims, we further assume that b satisfies

1. let m ≥ 1 be a positive integer and there exist some constant λ > k + 1 , T n ∈ R and σ ∈ ( m , m + 1 ) such that

   b ( x ) = ∣ x ∣ − λ − k + 1 1 + ∑ n = 1 m T n ∣ x ∣ − n + o ( ∣ x ∣ − σ ) a s ∣ x ∣ → ∞ .

3 Proof of Theorem 2.3

In this section, we prove Theorem 2.3. Lemma 2.1 is a useful auxiliary result in our proof.