On the large solutions to a class of k-Hessian problems

1 Introduction

This presentation is to investigate the following k-Hessian problem

where Ω ⊂ R N ( N ≥ 2 ) is a C ^∞-smooth bounded strictly (k − 1)-convex domain, k ∈ {1, …, N}, λ ⃗ = ( λ 1 , ⋅ ⋅ ⋅ , λ N ) and λ ₁, ⋅⋅⋅, λ _N are the eigenvalues of the Hessian of u ∈ C ²(Ω), and

denotes kth elementary symmetric function given in [1] and [2]; furthermore, S 0 ( λ ⃗ ) = 1 given in [3]. The last condition u = +∞ on ∂Ω means that u(x) → +∞ as d(x) ≔ dist(x, ∂Ω) → 0 and the solution is called “large solution”, “blow-up solution” or “explosive solution”. The weight b and nonlinearity f satisfy

• (b ₁ ) b ∈ C ^∞(Ω) is positive in Ω;

• (f ₁ ) f ∈ C[0, ∞) ∩ C ¹(0, ∞) is positive and increasing on (0, ∞) with f(0) = 0 (or ( f 01 ) f ∈ C 1 ( R ) is positive and increasing on R );

• (f ₂ ) the Keller–Osserman type condition ∫ t ∞ ( F ( s ) ) − 1 / ( k + 1 ) d s < ∞ , ∀ t > 0 , F ( t ) = ∫ 0 t f ( s ) d s .

We obtain by Definition 1.1 of [3] that u ∈ C ²(Ω) is (strictly) k-convex if S i ( D 2 u ) = S i ( λ ⃗ ) ( > ) ≥ 0  in  Ω  for  i = 1 , ⋅ ⋅ ⋅ , k . Moreover, it follows by Definition 1.2 of [3] that Ω is (strictly) l-convex if

where κ i ( x ̄ ) ( i = 1 , ⋅ ⋅ ⋅ , N − 1 ) are the principal curvatures of ∂Ω at x ̄ .

If k = 1, then problem (1.1) reduces to the semilinear elliptic problem

The study to this problem on the existence, uniqueness and boundary behavior has a long history. In 1916, problem (1.3) first appeared in the work of Bieberbach [4] for N = 2 in connection with geometric problem of Riemannian surfaces of constant negative curvatures. Then, Rademacher [5], using the idea of Bieberbach, showed the results still hold for N = 3. Later, Lazer and McKenna [6] extended the above results in bounded domain Ω ⊂ R N ( N ≥ 1 ) with a uniform outer sphere condition. Let b ≡ 1, f satisfy (f ₁ ), Keller [7] and Osserman [8] showed problem (1.3) has solutions if and only if (f ₂ ) holds with k = 1. For further insights on problem (1.3) and some related problems, we refer the readers to [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] and the references therein.

Now, let us return to problem (1.1). If k = N, then the problem is Monge–Ampère problem

Problem (1.4) arised from a few geometric problems and was first studied by Cheng and Yau [20], [21] for f(u) = exp(Ku). Many related works have been considered after Cheng and Yau’s results to problem (1.4). Lazer and McKenna [22] studied the existence, uniqueness and global estimate of strictly convex solutions to problem (1.4), where 0 < b ∈ C ∞ ( Ω ̄ ) , f(u) = u ^γ or f(u) = exp(u). Especially, when f(u) = u ^γ with γ ∈ (0, N], they investigated the nonexistence of solutions. Matero [23], for a class of more general Monge–Ampère problem (i.e., b(x)f(u) in (1.4) is replaced by f ̃ ( x , u ) ), studied the existence, uniqueness and boundary behavior of strictly convex solutions. Guan and Jian [24] generalized the results of Cheng and Yau [20], [21], in which various existence and nonexistence results were shown for rather general Monge–Ampère equations with gradient terms in bounded (strictly) convex domains. In particular, they also studied the global estimate of strictly convex solutions to the problem in bounded strictly convex domains. Then, the results were extended by Jian [25] to the k-Hessian problem, where the existence and global estimate of viscosity solutions and the nonexistence of classical solutions were established. In [26], Mohammed studied the existence of strictly convex solutions to problem (1.4) when the following Dirichlet problem

has a strictly convex solution. In Theorem 1.1 of [27], Caffarelli et al. showed that problem (1.5) admits a convex solution if b ∈ C ∞ ( Ω ̄ ) . Cheng and Yau [28] proved that problem (1.5) possesses a strictly convex solution if 0 < b(x) < C(d(x)) ^δ−N−1 in Ω for some constants δ > 0 and C > 0. Mohammed [29] showed that if b(x) > C(d(x))^−N−1 in Ω with C > 0, then problem (1.5) has no strictly convex solution. Let b ∈ C ∞ ( Ω ̄ ) be positive in Ω and nonnegative on ∂Ω. Under suitable conditions on f, Cîrstea and Trombetti [30] investigated the existence of strictly convex solutions to problem (1.4) in smooth, bounded, strictly convex domains. And they further gave the boundary behavior and uniqueness of strictly convex solutions by using Karamata regular variation theory. Then, the results were further generalized by Huang [31] to problem (1.1). When f(u) = exp(u) or f(u) = u ^γ , γ > N, and b satisfies (b ₁ ) and grows like a negative power of d(x) near boundary, Yang and Chang [32] showed the existence, uniqueness, nonexistence and global estimate of strictly convex solutions to problem (1.4). And when Ω is a ball, they obtained the exact boundary behavior of large solutions. Recently, under the hypotheses (f ₁ ), (f ₂ ) and some additional structural conditions, the existence and boundary behavior of solutions were further studied by Zhang and Du [33] and Zhang [34], [35], [36] to problem (1.4) and by Ma and Li [37], Wan and Shi [38], Zhang and Feng [39] and Zhang [40] to problem (1.1). Especially, the authors in [37] investigated the existence and boundary behavior of viscosity solutions, and the author in [35] studied the asymptotic behavior of solutions when some indexes (which is relevant to b) tend to the corresponding critical values. For other related works, we refer the readers to [3], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57] and the references therein.

For convenience, we introduce the class of Karamata regularly varying functions as below.

In this paper, we first establish the first expansion (equality) of k-convex solutions near the boundary when f is borderline regularly varying at infinity (i.e., f ∈ N RV _k ) and has a lower term. We reveal that the most accurate influences of some indexes of f and principal curvatures of ∂Ω on the first expansion of solutions. It should be pointed out that the boundary behavior of solutions to problem (1.1) is a hard issue when f is borderline regularly varying at infinity. We analyze briefly the cause of the difficulty by Remark 2.3 (see page 5). Then, we establish the first expansion of solutions to problem (1.1) when f is a Γ-varying function at infinity (defined on page 5) and b may be vanishing or singular (including borderline singular) on ∂Ω. We find the principal curvatures of ∂Ω have no influence on the expansions. As we know, Γ-varying functions is a class of very special rapidly varying functions defined by Appendix A.1 (see Appendix A). By analyzing the first boundary expansions of solutions, we find an interesting fact: Roughly speaking, if f ∈ N RV _k  (f ∈ Γ), then the solutions near the boundary can be expressed by a rapidly (slowly) varying function. Our results and methods are quite different from the existing ones. On the other hand, if Ω is a ball, b is radially symmetric in Ω and has high singularity on ∂Ω, f satisfies the following Keller–Osserman type condition

we prove problem (1.1) has infinitely many positive k-convex radial solutions, which give a positive answer to the open problem (where Ω is a general (k − 1)-convex domain, b satisfies (b ₁ ) and has high singularity on ∂Ω, f satisfies (1.7)) in the case of radial symmetry in the ball. Our result and method are quite different from the ones of Zhang in [35] and Feng and Zhang in [43]. Furthermore, we study the global estimates of all positive radial solutions to this radially symmetric problem.

2 Main results

First, we study the influences of some indexes of f and principal curvatures of ∂Ω on the first expansion of k-convex large solutions when f is borderline regularly varying at infinity. To our aim, we show some further hypotheses on b and f as follows:

• (b ₂ ) there exist positive constants b 0 , δ ̃ , δ 0 ( δ ̃ ≤ δ 0 ) and non-increasing function θ ∈ Λ such that

  b ( x ) = b 0 θ k + 1 ( d ( x ) ) , x ∈ Ω δ ̃ ,

  where Ω δ ̃ ≔ { x ∈ Ω : d ( x ) < δ ̃ } , Λ (following [11], [12], [30] and [19]) denotes the set of all positive monotonic functions θ ∈ C ¹(0, δ ₀] ∩ L ¹(0, δ ₀] which satisfy

  lim t → 0 + d d t Θ ( t ) θ ( t ) ≔ D θ ∈ [ 0 , ∞ ) , Θ ( t ) = ∫ 0 t θ ( s ) d s .

• (f ₃ ) there exist some constant t ₀ > 0 and functions f ₁, f ₂ such that f(t) ≔ f ₁(t) + f ₂(t), t ≥ t ₀, where f ₁ ∈ C ²[t ₀, ∞), g ( t ) ≔ t f 1 ′ ( t ) f 1 ( t ) − k , t ≥ t 0 . And g, f ₂ satisfy the following conditions:

• (S ₁ ) g ( t ) ≥ 0 , t ≥ t 0 , lim t → ∞ g ( t ) = 0 , lim t → ∞ t g ′ ( t ) g ( t ) = 0 ,

  lim t → ∞ t g ′ ( t ) g 2 ( t ) = K g ≠ 0 , 1 k + 1 + K g ≠ 0  and  lim t → ∞ t k f 1 ( t ) g 3 ( k + 1 ) ( t ) = 0 ;

• (S ₂ ) for any ξ > 0, lim t → ∞ f 2 ( ξ t ) f 2 ( t ) = ξ k , and there exists C 1 ≠ 0 such that

  lim t → ∞ f 2 ( t ) g ( t ) f 1 ( t ) = C 1 ,

  or

• (S ₃ ) lim t → ∞ f 2 ( t ) g ( t ) f 1 ( t ) = 0 and there exists μ ≤ k such that for any ξ > 0,

  lim t → ∞ f 2 ( ξ t ) f 2 ( t ) = ξ μ .

Next, we show the first expansions of k-convex solutions to problem (1.1) when f is Γ-varying at infinity. For convenience, we introduce Γ-varying functions as below:

Some typical Γ-varying functions are

1. f(t) = exp(t ^p ), p > 0, t > 0, where the auxiliary function χ(t) = p ⁻¹ t ^1−p ;

2. f(t) = exp(tlnt), t > 0, where the auxiliary function is given by

   χ ( t ) = 1 ,    if  t ∈ ( 0,1 ] , ( ln ⁡ t ) − 1 ,    if  t ∈ ( 1 , ∞ ) ;

3. f(t) = exp(exp(t)), t > 0, where the auxiliary function χ(t) = exp(−t).

If f is non-decreasing on (0, ∞), then by Theorem 1.28 in [58] we see that the following statements are equivalent:

1. f is Γ-varying at infinity;

2. there exist some constant B _∞ > 0 and positive function T ∈ C ¹[B _∞, ∞) with lim _t→∞ T′(t) = 0 such that

   (2.6) f ( t ) ∼ f ̂ ( t ) ≔ exp ∫ B ∞ t d s T ( s )  as  t → ∞ ,

   where “ f ( t ) ∼ f ̂ ( t ) as t → ∞” means that f ( t ) / f ̂ ( t ) → 1 as t → ∞;

3. lim t → ∞ f ( t ) ∫ 0 t F ( s ) d s ( F ( t ) ) 2 = 1 ,  where  F ( t ) = ∫ 0 t f ( s ) d s .

When Ω is a spherical region, we show the existence of positive k-convex radial solutions to problem (1.1) as follows. Without loss of generality, we suppose Ω is a unit ball.

To investigate the global estimates of radial solutions, we extend the class of functions L 2 by the following way. For each L ∈ L 2 , let L ̃ ∈ C 1 ( 0 , ∞ ) be a positive differential extension of L. Define

For two functions h ₁ and h ₂ defined on S, h ₁(x) ≍ h ₂(x), x ∈ S means that there exist positive constants c ₁, c ₂ such that c ₁ h ₁(x) ≤ h ₂(x) ≤ c ₂ h ₁(x), x ∈ S. Without loss of generality, let Ω be still a unit ball. We next show the global estimates of positive k-convex radial solutions to problem (1.1).

Obviously, in Theorems 2.11, 2.13 and Corollary 2.14, b ̃ ( r ) is controlled by a regularly varying function. Now, we consider the global estimate of positive k-convex radial solutions to problem (1.1) when b ̃ ( r ) is controlled by a function which is rapidly varying to infinity at zero. For convenience, we now introduce a new class of rapidly varying functions as follows.

Let Λ_∞ denote the set of all positive decreasing functions θ _∞ ∈ C ¹(0, 1] which satisfy

Let Q (see [33], [39]) denote the set of all the decreasing functions q ∈ C(0, ∞) which satisfy

For more general class Q , we have the following results.

3 Some preliminary results

In this section, we collect some well-known results which are important to our results on the boundary behaviors of solutions.

By Lemma 3.3, we see that

Our proofs of Theorems are given in Sections 4–9. Some auxiliary lemmas and preliminaries of Karamata regular (rapid) variation theory are given in Appendix A and B.

4 Proof of Theorem 2.1

In this section, we prove Theorem 2.1.