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Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products

  • Márcio Batista , Giovanni Molica Bisci EMAIL logo , Henrique F. de Lima and Wallace F. Gomes
Published/Copyright: October 15, 2024
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

Our purpose is to establish nonexistence results concerning complete noncompact mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products, under mild constraints on the warping and soliton functions. Applications to self-shrinkers in the Euclidean space, as well as to mean curvature flow solitons in other important warped product models such as the Schwarzschild and Reissner-Nordström spaces, and Robertson-Walker spacetimes such as the Einstein-de Sitter spacetime, are also given. Furthermore, we study the nonexistence of entire solutions to the mean curvature flow equation. Our approach is based on a suitable conformal change of metric jointly with a maximum principle for complete noncompact Riemannian manifolds with polynomial volume growth due to Alías et al.

Keywords: semi-Riemannian warped products; complete mean curvature flow solitons; polynomial volume growth; mean curvature flow equation; Schwarzschild and Reissner-Nordström spaces; Robertson-Walker spacetimes
MSC 2010: 53C42; 53E10

1 Introduction

Let ψ : Σ n → R ν n + 1 be an n -dimensional ( n ≥ 2 ) hypersurface in the ( n + 1 ) -dimensional semi-Euclidean R ν n + 1 of index ν ∈ { 0 , 1 } , where ν = 0 implies that R ν n + 1 is just the Euclidean space R n + 1 and ν = 1 means that R ν n + 1 corresponds to the Lorentz-Minkowski space L n + 1 and (in both cases) endowed with its standard metric

g ¯ = ( − 1 ) ν d x 1 2 + ∑ i = 2 n + 1 d x i 2 .

If the position vector ψ evolves in the direction of the mean curvature vector H → , then it gives rise to a solution to mean curvature flow

Ψ : [ 0 , T ) × Σ n → R ν n + 1

satisfying Ψ ( 0 , p ) = ψ ( p ) , for all p ∈ Σ n , and

(1.1) ∂ Ψ ∂ t ( t , p ) = H → ( t , p ) ,

where H → ( t , p ) stands for the (non-normalized) mean curvature vector of the hypersurface Σ t n = Ψ ( t , Σ n ) at a point Ψ ( t , p ) . This equation is called the mean curvature flow equation.

The study of the mean curvature flow from the perspective of partial differential equations was started with Huisken [33] on the flow of convex hypersurfaces, and it has motivated several works dealing with the singularities of equation (1.1), the so-called mean curvature flow solitons (see, for instance, [1,17–21,24,25,27,28]). In the Lorentzian setting, this interest is also justified by the fact that the existence (or nonexistence) of spacelike mean curvature flow solitons may give insight into the structure of certain spacetimes at null infinity and have possible applications in general relativity (for more details, see [26]).

More recently, Alías et al. [10] extended these investigations by introducing a general definition of self-similar mean curvature flow in a Riemannian manifold M ¯ n + 1 endowed with a conformal vector field K and establishing the corresponding notion of mean curvature soliton. In particular, when M ¯ n + 1 is a warped product of the type I × ρ M n and K = ρ ( t ) ∂ t , they applied certain maximum principles to guarantee that a complete n -dimensional mean curvature flow soliton is a slice of M ¯ n + 1 . Colombo et al. [22] also studied some properties of mean curvature flow solitons in general Riemannian manifolds and warped products, focusing on Riemannian space forms and Schwarzschild and Reissner-Nordström spaces. Then, Alías et al. [11] established a natural framework for the stability of mean curvature flow solitons in warped product spaces. Regarding these solitons as stationary immersions for a weighted volume functional, they were able to find geometric conditions for finiteness of the index and some characterizations of stable solitons. They also proved some nonexistence results for solitons as applications of a comparison principle, which suits well the structure of the diffusion elliptic operator associated with the weighted measures they were considering. Next, the third author jointly with de Lima et al. [23] applied suitable maximum principles to guarantee that a mean curvature flow soliton is a slice of a warped product space, as well as to obtain nonexistence results concerning these geometric objects. In particular, they studied entire graphs constructed over the fiber of the ambient space and which are mean curvature flow solitons, and also they inferred the stability of such a compact (without boundary) soliton with respect to an appropriate stability operator.

Meanwhile, Guilfoyle and Klingenberg [29] proved the longtime existence for mean curvature flow of a smooth n -dimensional spacelike submanifold of an ( n + m )-dimensional manifold whose metric satisfies the so-called timelike curvature condition. Then, Lambert and Lotay [35] proved long-time existence and convergence results for spacelike solutions to mean curvature flow in the n -dimensional semi-Euclidean space R ν n of index ν ≥ 1 , which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. Gürbüz and Yoon [30] studied Hasimoto surfaces for the second and third classes of curve evolution corresponding to a Frenet frame in Minkowski 3-space. Then, they derived some formulas for the differentials of the second and third Hasimoto-like transformations associated with the repulsive-type nonlinear Schrödinger equation (for a throughout discussion related to interesting connections between moving curves and soliton equations, we also refer [34,38,41]). When the ambient space is a Lorentzian product space, the first and third authors [15] established nonexistence results for complete spacelike translating solitons under suitable curvature constraints on the curvatures of the Riemannian base of the ambient space.

Here, our purpose is to establish nonexistence results concerning complete noncompact mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products, under mild constraints on the warping and soliton functions (Theorems 3.1 and 3.7). Applications to self-shrinkers in the Euclidean space, as well as to mean curvature flow solitons in other important warped product models such as the Schwarzschild and Reissner-Nordström spaces, and Robertson-Walker (RW) spacetimes such as the Einstein-de Sitter spacetime, are also given (see corollaries in Sections 3.1 and 3.2). Furthermore, we study the nonexistence of entire solutions of the mean curvature flow equation (Section 4). Our approach is based on a suitable change of metric (already used in the context of generalized Robertson-Walker (GRW) spacetimes and Riemannian warped products; see [13,14]) jointly with a maximum principle for complete noncompact Riemannian manifolds with polynomial volume growth due to Alías et al. [6, Lemma 2.2].

2 Preliminaries

Let M ¯ n + 1 be a connected semi-Riemannian manifold endowed with a metric g ¯ of index ν ∈ { 0 , 1 } and with semi-Riemannian connection ∇ ¯ . Let us denote by D ( M ¯ ) the ring of smooth functions f : M ¯ n + 1 → R and X ( M ¯ ) the algebra of smooth vector fields on M ¯ n + 1 . For a smooth vector field X ∈ X ( M ¯ ) , we define ε X ≔ g ¯ ( X , X ) . So, we say that such a smooth vector field X is unit when ε X = ± 1 and that it is timelike when ε X = − 1 .

In all that follows, we consider Riemannian immersions ψ : Σ n → M ¯ n + 1 , namely, immersions from a connected n -dimensional ( n ≥ 2 ) orientable differentiable manifold Σ n into M ¯ , such that the induced metric g = ψ * ( g ¯ ) turns Σ n into a Riemannian manifold (in the Lorentz case ν = 1 , we refer to ( Σ n , g ) as being a spacelike hypersurface of M ¯ n + 1 ), with Levi-Civita connection ∇ . We orient Σ n by the choice of a unit normal vector field N on it.

2.1 Riemannian immersions in semi-Riemannian warped products

Let ( M n , g M ) be a connected, n -dimensional ( n ≥ 2 ), oriented Riemannian manifold, I ⊂ R an open interval and ρ : I → R a positive smooth function. Also, in the product manifold M ¯ n + 1 = I × M n , let π I and π M denote the canonical projections onto the factors I and M n , respectively. The class of semi-Riemannian manifolds that will be of our concern here is the one obtained by furnishing M ¯ n + 1 with the metric g ¯ given by

(2.1) g ¯ ( u , v ) p = ε g I ( ( π I ) ∗ v , ( π I ) ∗ v ) π I ( p ) + ( ρ ∘ ( π I ) ) 2 ( p ) g M ( ( π M ) ∗ v , ( π M ) ∗ v ) π M ( p ) ,

for all p ∈ M ¯ n + 1 and u , v ∈ T p M ¯ , where ε = ε ∂ t , ∂ t = ∂ ∂ t stands for the coordinate vector field tangent to I and g I stands for the standard metric of I ⊂ R . Along this work, ( M ¯ n + 1 , g ¯ ) will be called a warped product with warping function ρ and fiber M n , and we will simply write

(2.2) M ¯ n + 1 = ε I × ρ M n .

In the Lorentzian context ε = − 1 , according to the nomenclature established in [12], we say that M ¯ n + 1 is a GRW spacetime with warping function f and Riemannian fiber M n . When M n has constant sectional curvature, − I × ρ M n has been known in the mathematical literature as a RW spacetime, an allusion to the fact that, for n = 3 , it is an exact solution of Einstein’s field equations (see, for instance, [39, Chapter 12]).

We will also consider the conformal closed vector field K = ρ ( π I ) ∂ t globally defined on M ¯ n + 1 (cf. [36,37]). From the relationship between the Levi-Civita connections of M ¯ n + 1 and those ones of I and M n (see [39, Proposition 7.35]), it follows that

(2.3) ∇ ¯ V K = ρ ′ ( π I ) V ,

for all V ∈ X ( M ¯ ) , where ∇ ¯ is the Levi-Civita connection of g ¯ .

Given a connected Riemannian immersion ψ : Σ n → ε I × ρ M n oriented by the unit vector field N , we have that ε = ε ∂ t = ε N . So, we denote by A ≔ − ∇ ¯ N and H ≔ ε tr ( A ) the Weingarten operator and the (non-normalized) mean curvature function with respect to N .

Remark 2.1

We note that there exists a remarkable family of Riemannian immersions in the semi-Riemannian warped product M ¯ n + 1 = ε I × ρ M n (where ε = ± 1 ): its slices M t ∗ = { t ∗ } × M , with t ∗ ∈ I . The Weingarten operator and the mean curvature of M t ∗ with respect to N = ∂ t are, respectively, A t ∗ = − ε ρ ′ ( t ∗ ) ρ ( t ∗ ) I , where I denotes the identity operator, and H t ∗ = − ε n ρ ′ ( t ∗ ) ρ ( t ∗ ) .

Now, we consider two particular functions naturally attached to a Riemannian immersion ψ : Σ n → ε I × ρ M n , namely, the height function h = ( π I ) ∣ Σ and the angle function Θ = g ¯ ( N , ∂ t ) . A simple computation shows that

(2.4) ∇ ¯ π I = ε g ¯ ( ∇ ¯ π I , ∂ t ) ∂ t = ε ∂ t .

So, from (2.4), we have

(2.5) ∇ h = ( ∇ ¯ π I ) ⊤ = ε ∂ t ⊤ = ε ∂ t − Θ N .

Thus, (2.5) gives the following relation:

(2.6) ∣ ∇ h ∣ 2 = ε ( 1 − Θ 2 ) ,

where ε = ± 1 and ∣ ⋅ ∣ stands for the norm of a tangent vector field on Σ n in the metric g .

On the other hand, from (2.3), we have that

(2.7) ∇ ¯ V ∂ t = ρ ′ ( π I ) ρ ( π I ) { V + g ¯ ( V , ∂ t ) ∂ t } .

Consequently, from (2.5) and (2.7), we deduce that, for any X ∈ X ( Σ ) , the Hessian of h in the metric g is given by

(2.8) ∇ 2 h ( X , X ) = g ( ∇ X ∇ h , X ) = ρ ′ ( h ) ρ ( h ) { ε ∣ X ∣ 2 − g ( X , ∇ h ) 2 } + g ( A X , X ) Θ .

Hence, from (2.8), we obtain that the Laplacian of h in the metric g is

(2.9) Δ h = ρ ′ ( h ) ρ ( h ) { ε n − ∣ ∇ h ∣ 2 } + ε H Θ .

2.2 Maximum principle for complete noncompact Riemannian manifold

For our purpose, we will also need to quote a suitable maximum principle that will be used to prove our nonexistence results. For this, let ( Σ n , g ) be a connected, oriented, complete noncompact Riemannian manifold. We denote by B ( p , t ) the geodesic ball centered at p and with radius t . Given a polynomial function σ : ( 0 , + ∞ ) → ( 0 , + ∞ ) , we say that Σ n has polynomial volume growth like σ ( t ) if there exists p ∈ Σ n such that

vol ( B ( p , t ) ) = O ( σ ( t ) ) ,

as t → + ∞ , where vol denotes the standard Riemannian volume related to the metric g . As it was already observed in the beginning of Section 2 in [6], if p , q ∈ Σ n are at distance d from each other, we can verify that

vol ( B ( p , t ) ) σ ( t ) ≥ vol ( B ( q , t − d ) ) σ ( t − d ) ⋅ σ ( t − d ) σ ( t ) .

So, the choice of p in the notion of volume growth is immaterial. For this reason, we will just say that Σ n has polynomial volume growth.

Keeping in mind this previous digression, we close this section quoting the following key lemma, which corresponds to a particular case of a new maximum principle due to Alías et al. (see [6, Theorem 2.1]).

Lemma 2.2

Let ( Σ n , g ) be a connected, oriented, complete noncompact Riemannian manifold, and let u ∈ C ∞ ( Σ ) be a nonnegative smooth function such that Δ u ≥ a u on Σ n , for some positive constant a ∈ R . If Σ n has polynomial volume growth and ∣ ∇ u ∣ is bounded on Σ n , then u vanishes identically on Σ n .

3 Mean curvature flow solitons in semi-Riemannian warped products

We recall that the mean curvature flow Ψ : [ 0 , T ) × Σ n → M ¯ n + 1 of a Riemannian immersion ψ : Σ n → M ¯ n + 1 in a ( n + 1 ) -dimensional semi-Riemannian manifold M ¯ n + 1 , satisfying Ψ ( 0 , ⋅ ) = ψ ( ⋅ ) , looks for solutions of the equation

∂ Ψ ∂ t = H → ,

where H → ( t , ⋅ ) is the (non-normalized) mean curvature vector of Σ t n = Ψ ( t , Σ n ) . In our context, according to [10, Definition 1.1] and [22, Definition 1.1], a Riemannian immersion ψ : Σ n → M ¯ n + 1 in a semi-Riemannian warped product M ¯ n + 1 = ε I × ρ M n is called a mean curvature flow soliton with respect to K = ρ ( t ) ∂ t and with soliton constant c ∈ R if its (non-normalized) mean curvature function satisfies

(3.1) H = c ρ ( h ) Θ .

In the Lorentzian case ε = − 1 , we will consider the future mean curvature function, i.e., the mean curvature related to the orientation N such that Θ ≤ − 1 , and we will use the nomenclature spacelike mean curvature flow soliton.

Adopting the terminology introduced in [10] and [22], we will also deal with the soliton function

(3.2) ζ c ( t ) = n ρ ′ ( t ) + c ρ 2 ( t ) .

Taking into account Remark 2.1, it is not difficult to verify that each slice M t ∗ = { t ∗ } × M n is a mean curvature flow soliton with respect to K = ρ ( t ) ∂ t and with soliton constant c given by

(3.3) c = − n ρ ′ ( t ∗ ) ρ ( t ∗ ) 2 .

Moreover, t ∗ is implicitly given by the condition ζ c ( t ∗ ) = 0 .

Along our main results, we will consider the modified soliton function as the function

(3.4) ζ ¯ c ( t ) ≔ ρ ′ ( t ) ζ c ( t ) ,

where ζ c is the soliton function defined in (3.2).

We will also suppose that Σ n lies in a slab of ε I × ρ M n , which means that Σ n is contained in a bounded region of the type

[ t 1 , t 2 ] × M n = { ( t , p ) ∈ ε I × ρ M n : t 1 ≤ t ≤ t 2 and p ∈ M n } .

Furthermore, we will assume that the warping function ρ of the ambient space M ¯ n + 1 = ε I × ρ M n satisfies the following inequality:

(3.5) ( log ρ ) ″ ≤ γ [ ( log ρ ) ′ ] 2 ,

for some nonnegative constant γ . As it was observed in [5], inequality (3.5) is a mild hypothesis due to the fact that, for instance, when a GRW spacetime M ¯ n + 1 = − I × ρ M n obeys the null convergence condition (i.e., when its Ricci tensor Ric ¯ is such that Ric ¯ ( Z , Z ) ≥ 0 , for all null vector field Z ∈ X ( M ¯ ) ) and its Riemannian fiber M n is Ricci-flat, we have that (3.5) is automatically satisfied (for more details about the null convergence condition, see also [37]).

3.1 Nonexistence results for mean curvature flow solitons in I × ρ M n

Now, we present our first nonexistence result concerning mean curvature flow solitons.

Theorem 3.1

Let M ¯ n + 1 = I × ρ M n be a warped product whose warping function ρ satisfies inequality (3.5). There does not exist complete noncompact mean curvature flow soliton ψ : Σ n → M ¯ n + 1 with respect to K = ρ ( t ) ∂ t with soliton constant c ≠ 0 and mean curvature bounded away from zero, having polynomial volume growth and lying in a slab [ t 1 , t 2 ] × M n with ζ ¯ c ( t ) < 0 for all t ∈ [ t 1 , t 2 ] .

Proof

Let us suppose, by contradiction, the existence of such a mean curvature flow soliton ψ : Σ n → M ¯ n + 1 , and let us consider on Σ n the metric g ˆ = 1 ρ ( h ) 2 g , which is conformal to its induced metric g . If we denote by Δ ˆ the Laplacian with respect to the metric g ˆ , from (2.6) and (2.9), we obtain

(3.6) Δ ˆ h = ρ ( h ) 2 Δ h − ( n − 2 ) ρ ( h ) ρ ′ ( h ) ∣ ∇ h ∣ 2 = n ρ ( h ) ρ ′ ( h ) Θ 2 + ρ ( h ) ρ ′ ( h ) ∣ ∇ h ∣ 2 + H ρ ( h ) 2 Θ .

With a straightforward computation, from (3.20), we obtain

(3.7) Δ ˆ ρ ( h ) = ρ ″ ( h ) g ˆ ( ∇ ^ h , ∇ ^ h ) + ρ ′ ( h ) Δ ˆ h = ρ ″ ( h ) f ( h ) 2 ∣ ∇ h ∣ 2 + ρ ′ ( h ) ( n ρ ( h ) ρ ′ ( h ) Θ 2 + ρ ( h ) ρ ′ ( h ) ∣ ∇ h ∣ 2 + H ρ ( h ) Θ ) = n ρ ( h ) ρ ′ ( h ) 2 + H ρ ′ ( h ) ρ ( h ) 2 Θ + ρ ( h ) 3 ( log ρ ) ″ ( h ) − ( n − 2 ) ρ ′ ( h ) 2 ρ ( h ) 2 ∣ ∇ h ∣ 2 .

Given a positive real number α , we have that

(3.8) Δ ˆ ρ ( h ) − α = α ( α + 1 ) ρ ( h ) − α − 2 g ˆ ( ∇ ^ ρ ( h ) , ∇ ^ ρ ( h ) ) − α ρ ( h ) − α − 1 Δ ˆ ρ ( h ) .

Substituting (3.7) into (3.8), we obtain

(3.9) Δ ˆ ρ ( h ) − α = − α n ρ ( h ) − α ρ ′ ( h ) 2 − α H ρ ′ ( h ) ρ ( h ) − α + 1 Θ + α ( α + 1 ) ρ ( h ) − α ρ ′ ( h ) 2 ∣ ∇ h ∣ 2 − α ρ ( h ) − α + 2 ( log ρ ) ″ ( h ) − ( n − 2 ) ρ ′ ( h ) 2 ρ ( h ) 2 ∣ ∇ h ∣ 2 .

But, from (2.6), we have

(3.10) − α n ρ ( h ) − α ρ ′ ( h ) 2 = − α n ρ ( h ) − α ρ ′ ( h ) 2 ∣ ∇ h ∣ 2 − α n ρ ( h ) − α ρ ′ ( h ) 2 Θ 2 .

Thus, from (3.9), (3.10), (3.1), and (3.4), we obtain

(3.11) Δ ˆ ρ ( h ) − α = − α ρ ( h ) − α ζ ¯ c ( h ) Θ 2 − α ρ ( h ) − α + 2 { ( log ρ ) ″ ( h ) − ( α − 1 ) [ ( log ρ ) ′ ( h ) ] 2 } ∣ ∇ h ∣ 2 .

Now, taking into account hypothesis (3.5) and choosing α = 1 + γ > 0 , from (3.11), we obtain

(3.12) Δ ˆ ρ ( h ) − α ≥ − ζ ¯ c ( h ) α ρ ( h ) − α Θ 2 .

At this point, we observe that, since c ≠ 0 , Σ n ⊂ [ t 1 , t 2 ] × M n , and H is bounded away from zero, from (3.1), we see that Θ 2 is also bounded away from zero. So, since we are also assuming that ζ ¯ c ( t ) < 0 for all t ∈ [ t 1 , t 2 ] , from (3.12), we reach at the following inequality:

Δ ˆ ρ ( h ) − α ≥ a ρ ( h ) − α ,

where a = α inf Σ ∣ ζ ¯ c ( h ) ∣ .

Moreover, it is not difficult to verify that

(3.13) ∣ ∇ ˆ ρ ( h ) − α ∣ g ˆ = α ρ ( h ) − α ∣ ρ ′ ( h ) ∣ ∣ ∇ h ∣ ≤ α ρ ( h ) − α ∣ ρ ′ ( h ) ∣ .

So, since Σ n ⊂ [ t 1 , t 2 ] × M n , from (3.13), we conclude that ∣ ∇ ˆ ρ ( h ) − α ∣ g ˆ is bounded on Σ n .

On the other hand, considering the coefficients of conformal metric g ˆ i j = 1 ρ ( h ) 2 g i j , where g i j stands for the coefficients of the induced metric g , we have that

(3.14) G ˆ = det ( g ˆ i j ) = ρ ( h ) − 2 n det ( g i j ) = ρ ( h ) − n G .

In particular, using once more that Σ n ⊂ [ t 1 , t 2 ] × M n , from (3.14) jointly with the hypothesis that Σ n has polynomial volume growth with respect to g , we guarantee that the same holds with respect to the conformal metric g ˆ .

Therefore, we are in position to apply Lemma 2.2 to infer that ρ ( h ) − α vanishes identically on Σ n , which contradicts the fact that ρ is a positive function.□

Let o = ( 0 , … , 0 ) be the origin of the ( n + 1 ) -dimensional Euclidean space R n + 1 . We have that R n + 1 \ { o } is isometric to R + × t S n (see [36, Section 4, Example 1]), whose slides { t } × S n are isometric to n -dimensional Euclidean spheres S n ( t ) of radius t ∈ R + . In this setting, the mean curvature flow solitons with respect to K = t ∂ t with soliton constant c = − 1 are just the self-shrinkers. So, from (3.3), we conclude that S n ( n ) ≡ { n } × S n is the only slice, which is a self-shrinker.

It is not difficult to verify that we obtain from Theorem 3.1 the following result concerning the nonexistence of complete self-shrinkers:

Corollary 3.2

There does not exist complete noncompact n-dimensional self-shrinker immersed in R n + 1 with mean curvature bounded away from zero, having polynomial volume growth and lying in the closure of an n-dimensional annulus with with inner radius r i r > n .

Remark 3.3

We note that, for each 1 ≤ m ≤ n − 1 , the cylinder S m ( m ) × R n − m is a self-shrinker immersed in R n + 1 with mean curvature ∣ H ∣ = m n and having polynomial volume growth; however, it does not belong to any n -dimensional annuli. It is also worth to point out that Cao and Li [17] proved that an n -dimensional complete self-shrinker immersed in R n + p , with polynomial volume growth and whose second fundamental form satisfies ∣ A ∣ 2 ≤ 1 , must be isometric to one of the followings: a round sphere S n ( n ) , a cylinder S m ( m ) × R n − m , with 1 ≤ m ≤ n − 1 , or a hyperplane R n , all of them immersed in R n + 1 .

According to [40], warped products of the type I × e t M n are called pseudo-hyperbolic spaces. This terminology is due to the fact that the ( n + 1 ) -dimensional hyperbolic space H n + 1 is isometric to the warped product R × e t R n , where the slices constitute a family of horospheres sharing a same fixed point in the asymptotic boundary ∂ ∞ H n + 1 and giving a complete foliation of H n + 1 (for more details about pseudo-hyperbolic spaces, see, for instance, [9,36,40]). From (3.3), we conclude that the slice { log ( − n c ) } × M n is the only one that is a mean curvature flow soliton with respect to K = e t ∂ t with soliton constant c < 0 .

From Theorem 3.1, we also obtain the following consequence:

Corollary 3.4

Let M ¯ n + 1 = I × e t M n be a pseudo-hyperbolic space. There does not exist complete noncompact mean curvature flow soliton ψ : Σ n → M ¯ n + 1 with respect to K = e t ∂ t with soliton constant c < 0 and mean curvature bounded away from zero, having polynomial volume growth and lying in a slab [ t 1 , t 2 ] × M n with t 1 > log − n c .

Given a mass parameter m > 0 , the Schwarzschild space is defined to be the product

M ¯ n + 1 = ( r 0 ( m ) , + ∞ ) × S n

furnished with the metric g ¯ = V m ( r ) − 1 d r 2 + r 2 g S n , where g S n is the standard metric of S n , V m ( r ) = 1 − 2 m r 1 − n stands for its potential function, and r 0 ( m ) = ( 2 m ) 1 ⁄ ( n − 1 ) is the unique positive root of V m ( r ) = 0 . Its importance lies in the fact that the manifold R × M ¯ n + 1 equipped with the Lorentzian static metric − V m ( r ) d t 2 + g ¯ is a solution of the Einstein field equation in vacuum with zero cosmological constant (see, for instance, [39, Chapter 13] for more details concerning Schwarzschild geometry).

As it was observed in [22, Example 1.3], M ¯ n + 1 can be reduced in the form I × ρ S n with metric (2.1) via the following change of variables:

(3.15) t = ∫ r 0 ( m ) r d σ V m ( σ ) , ρ ( t ) = r ( t ) , I = R + .

As it was noted in [22, Example 4.1], since V m ( r ) is strictly increasing on ( r 0 ( m ) , + ∞ ) , it follows from (3.15) that the warping function ρ satisfies

(3.16) ρ ′ ( t ) = d r d t = V m ( r ( t ) ) > 0 and ρ ″ ( t ) = 1 2 d V m d r ( r ( t ) ) > 0 .

Thus, from (3.3) and (3.16), we can verify that a slice { t ∗ } × S n is a mean curvature flow soliton with respect to ρ ( t ) ∂ t = r V m ( r ) ∂ r with soliton constant c < 0 when t ∗ = t ( r ∗ ) with r ∗ > r 0 ( m ) solving the following equation:

(3.17) V m ( r ) = c 2 n 2 r 4 .

We note that such a solution exists if and only if the function φ m ( t ) = c 2 n 2 t 4 + 2 m t n − 1 − 1 has a zero on ( r 0 ( m ) , + ∞ ) . Note that φ m is a convex function that goes to infinity if t goes to 0 or + ∞ and so φ m has a unique minimal point in ( 0 , ∞ ) . Such value r ˆ is given implicitly by φ ′ ( r ˆ ) = 0 , i.e.,

4 c 2 n 2 r ˆ 3 − 2 m ( n − 1 ) r ˆ n = 0 .

Therefore, equation (3.17) has a solution if and only if r ˆ > r 0 ( m ) and φ m ( r ˆ ) ≤ 0 . The last condition can be rewritten in the following way:

(3.18) r ˆ = m ( n − 1 ) n 2 2 c 2 1 ⁄ ( n + 3 ) ≥ m ( n + 3 ) 2 1 ⁄ ( n − 1 ) .

In particular, there are two solutions r 0 ( m ) < r ∗ , − < r ˆ < r ∗ , + if the strict inequality holds in (3.18), and a unique solution r ∗ = r ˆ if equality holds.

Taking into account the previous setting, from Theorem 3.1, we obtain

Corollary 3.5

Let M ¯ n + 1 = I × ρ S n be the Schwarzschild space, where the warping function ρ is obtained from (3.15). There does not exist complete noncompact mean curvature flow soliton ψ : Σ n → M ¯ n + 1 with respect to K = ρ ( t ) ∂ t with soliton constant c < 0 and mean curvature bounded away from zero, having polynomial volume growth and lying in a slab [ t 1 , t 2 ] × S n with ρ ( t 1 ) ≥ − n c .

Proof

First, using (3.16) and taking the positive constant γ = n − 1 2 V m ( r ( t 1 ) ) , we can verify that inequality (3.5) is satisfied. Moreover, since V m ( r ( t ) ) < 1 for all t ∈ I , r ( t 1 ) = ρ ( t 1 ) ≥ − n c implies

V m ( r ( t ) ) < 1 ≤ c 2 n 2 r ( t ) 4 ,

and, consequently, we have that

ζ ¯ c ( t ) = n V m ( r ( t ) ) + c r ( t ) 2 V m ( r ( t ) ) < 0 ,

for all t ≥ t 1 . Therefore, we can apply Theorem 3.1 to conclude our result.□

Given a mass parameter m > 0 and an electric charge q ∈ R , with ∣ q ∣ ≤ m , the Reissner-Nordström space is defined to be the product

M ¯ n + 1 = ( r 0 ( m , q ) , + ∞ ) × S n

endowed with the metric g ¯ = V m,q ( r ) − 1 d r 2 + r 2 g S n , where g S n is the standard metric of S n , V m,q ( r ) = 1 − 2 m r 1 − n + q 2 r 2 − 2 n stands for its potential function and r 0 ( m,q ) = q 2 m − m 2 − q 2 1 ⁄ ( n − 1 ) is the largest positive zero of V m,q ( r ) . The importance of this model lies in the fact that the manifold R × M ¯ n + 1 equipped with the Lorentzian static metric − V m,q ( r ) d t 2 + g ¯ is a charged black-hole solution of the Einstein field equation in vacuum with zero cosmological constant.

As in the case of the Schwarzschild space, M ¯ n + 1 can be reduced in the form I × ρ S n with metric (2.1) via the same change of variables as in (3.15). Furthermore, following the same previous steps, the warping function ρ has positive first and second derivatives. Moreover, we can verify that a slice { t ∗ } × S n is a mean curvature flow soliton with respect to ρ ( t ) ∂ t = r V m,q ( r ) ∂ r with soliton constant c < 0 when t ∗ = t ( r ∗ ) with r ∗ > r 0 ( m,q ) solving the following equation:

(3.19) V m,q ( r ) = c 2 n 2 r 4 .

We observe that such a case is more complicated to explicit all the values, but qualitatively, we can say that such a solution of (3.19) exists if and only if the function

φ m,q ( x ) = c 2 n 2 x 4 + 2 m x n − 1 − q 2 x 2 n − 2 − 1

has a zero on ( r 0 ( m ) , + ∞ ) . Note that φ m,q goes to positive infinity if x goes to positive infinity and φ m,q goes to negative infinity if x goes to zero. So, φ m,q has at least one root in ( 0 , + ∞ ) , and if such roots are greater than r 0 ( m , q ) , we obtain the desired solutions r ∗ .

We can reason as in the proof of Corollary 3.5 to obtain the following nonexistence result:

Corollary 3.6

Let M ¯ n + 1 = I × ρ S n be the Reissner-Nordström space, where the warping function ρ is obtained from (3.15). There does not exist complete noncompact mean curvature flow soliton ψ : Σ n → M ¯ n + 1 with respect to K = ρ ( t ) ∂ t with soliton constant c < 0 and mean curvature bounded away from zero, having polynomial volume growth and lying in a slab [ t 1 , t 2 ] × S n with V m,q ( r ( t ) ) < c 2 n 2 r ( t ) 4 for all t ∈ [ t 1 , t 2 ] .

3.2 Nonexistence results for mean curvature flow solitons in − I × ρ M n

In the Lorentzian setting, we obtain the following nonexistence result:

Theorem 3.7

Let M ¯ n + 1 = − I × ρ M n be a GRW spacetime whose warping function ρ satisfies inequality (3.5). There does not exist complete noncompact spacelike mean curvature flow soliton ψ : Σ n → M ¯ n + 1 with respect to K = ρ ( t ) ∂ t with soliton constant c ≠ 0 , bounded mean curvature and polynomial volume growth, lying in a slab [ t 1 , t 2 ] × M n , with ζ ¯ c ( t ) > 0 for all t ∈ [ t 1 , t 2 ] .

Proof

As in the proof of Theorem 3.1, let us consider on Σ n the conformal metric g ˆ = 1 ρ 2 ( h ) g . Denoting by Δ ˆ the Laplacian with respect to the metric g ˆ , from (2.6) and (2.9), we have

(3.20) Δ ˆ h = − ρ ( h ) ρ ′ ( h ) { n + ( n − 1 ) ∣ ∇ h ∣ 2 } − H ρ 2 ( h ) Θ .

Thus, from (3.20), we obtain

(3.21) Δ ˆ ρ ( h ) = − n ρ ( h ) ( ρ ′ ( h ) ) 2 − H ρ ′ ( h ) ρ 2 ( h ) Θ + ρ 3 ( h ) { ( log ρ ) ″ ( h ) − ( n − 2 ) ( ( log ρ ) ′ ( h ) ) 2 } ∣ ∇ h ∣ 2 .

For any positive real number α , with a straightforward computation from (3.21), we obtain

(3.22) Δ ˆ ρ − α ( h ) = − α ρ − α − 1 ( h ) { − n ρ ( h ) ( ρ ′ ( h ) ) 2 − H ρ ′ ( h ) ρ 2 ( h ) Θ + ρ 3 ( h ) ( ( log ρ ) ″ ( h ) − ( n + α − 3 ) ( ( log ρ ) ′ ( h ) ) 2 ) ∣ ∇ h ∣ 2 } = − α ρ − α − 1 ( h ) { − n ρ ( h ) ( ρ ′ ( h ) ) 2 Θ 2 − H ρ ′ ( h ) ρ 2 ( h ) Θ + ρ 3 ( h ) ( ( log ρ ) ″ ( h ) − ( α − 3 ) ( ( log ρ ) ′ ( h ) ) 2 ) ∣ ∇ h ∣ 2 } .

Hence, from (3.1), (3.4), and (3.22), we reach at

(3.23) Δ ˆ ρ − α ( h ) = α ρ − α ( h ) Θ 2 ζ ¯ c ( h ) − α ρ 2 − α ( h ) { ( log ρ ) ″ ( h ) − ( α − 3 ) ( ( log ρ ) ′ ( h ) ) 2 } ∣ ∇ h ∣ 2 .

So, observing that Θ 2 ≥ 1 and choosing α = 3 + γ , we can use (3.5) and the assumption that ζ ¯ c ( h ) > 0 on Σ n to obtain from (3.23) the following estimate:

(3.24) Δ ˆ ρ ( h ) − α ≥ α ζ ¯ c ( h ) ρ ( h ) − α .

Consequently, since we are assuming that Σ n ⊂ [ t 1 , t 2 ] × M n , from (3.24), we obtain

(3.25) Δ ˆ ρ ( h ) − α ≥ a ρ ( h ) − α ,

where a = α inf Σ ζ ¯ c ( h ) > 0 .

Moreover, we have that

(3.26) ∣ ∇ ˆ ρ ( h ) − α ∣ g ˆ = α ρ ( h ) − α ∣ ρ ′ ( h ) ∣ ∣ ∇ h ∣ ≤ α ρ ( h ) − α ∣ ρ ′ ( h ) ∣ ∣ Θ ∣ = ∣ c − 1 ∣ α ρ ( h ) − α − 1 ∣ ρ ′ ( h ) ∣ ∣ H ∣ .

So, since Σ n ⊂ [ t 1 , t 2 ] × M n and H is bounded on Σ n , from (3.26), we conclude that ∣ ∇ ˆ ρ ( h ) − α ∣ g ˆ is also bounded on Σ n .

But, as in the proof of Theorem 3.1, our hypotheses also guarantee that Σ n has polynomial volume growth with respect to the conformal metric g ˆ . Therefore, we can apply once more Lemma 2.2 to infer that ρ ( h ) − α vanishes identically on Σ n , contradicting the fact that ρ is a positive function.□

The four-dimensional Einstein-de Sitter spacetime − R + × t 2 3 R 3 , where R 3 stands for the three-dimensional Euclidean space endowed with its canonical metric, is a classical exact solution to the Einstein field equation without cosmological constant. It is an open Friedmann-RW model, which incorporates homogeneity and isotropy (the cosmological principle) and permitted expansion (for more details, see [39, Chapter 12]). Here, we consider the ( n + 1 ) -dimensional Einstein-de Sitter spacetime − R + × t 2 3 R n . From (3.3), we conclude that the slice { ( − 2 n 3 c ) 3 5 } × R n is the only one that is a spacelike mean curvature flow soliton with respect to K = t 2 3 ∂ t and with soliton constant c < 0 .

From Theorem 3.7, we obtain the following consequence.

Corollary 3.8

Let M ¯ n + 1 = − R + × t 2 3 R n be the ( n + 1 ) -dimensional Einstein-de Sitter spacetime. There does not exist a complete noncompact spacelike mean curvature flow soliton ψ : Σ n → M ¯ n + 1 with respect to K = t 2 3 ∂ t with soliton constant c ≥ 0 , whose mean curvature is bounded bounded, having polynomial volume growth and lying in a slab of M ¯ n + 1 .

According to the terminology introduced by Albujer and Alías [2], a GRW spacetime − R × e t M n is called a steady-state-type spacetime. This terminology is due to the fact that the steady-state model of the universe ℋ 4 , proposed by Bondi and Gold [16] and Hoyle [32] when looking for a model of the universe, which looks the same not only at all points and in all directions (i.e., spatially isotropic and homogeneous) but also at all times, is isometric to the RW spacetime − R × e t R 3 (for more details, see [31]). From (3.3), we conclude that the slice { log ( − n c ) } × M n is the only one that is a spacelike mean curvature flow soliton with respect to K = e t ∂ t and with soliton constant c < 0 . In this context, Theorem 3.7 gives the following:

Corollary 3.9

Let M ¯ n + 1 = − R × e t M n be a steady-state-type spacetime. There does not exist a complete noncompact spacelike mean curvature flow soliton ψ : Σ n → M ¯ n + 1 with respect to K = e t ∂ t with soliton constant c ≥ 0 , whose mean curvature is bounded, having polynomial volume growth and lying in a slab of M ¯ n + 1 .

From [37, Example 4.2], the ( n + 1 ) -dimensional de Sitter space S 1 n + 1 is isometric to the RW spacetime − R × cosh t S n , where S n denotes the n -dimensional unit Euclidean sphere endowed with its standard metric. Taking into account the terminology introduced in [3], the open half-space R + × S n ⊂ S 1 n + 1 (respect. R − × S n ⊂ S 1 n + 1 ) is called the chronological future (respect. past) of S 1 n + 1 with respect to the totally geodesic equator { 0 } × S n . From (3.3), we see that the equator is a spacelike mean curvature flow soliton with respect to K = cosh t ∂ t and constant soliton c = 0 and the slices { sinh − 1 ( − n ± n 2 − 4 c 2 2 c ) } × S n are spacelike mean curvature flow soliton with respect to K = cosh t ∂ t and with soliton constant 0 < ∣ c ∣ ≤ n 2 .

We close this section with the following consequence of Theorem 3.7:

Corollary 3.10

There does not exist a complete noncompact spacelike mean curvature flow soliton ψ : Σ n → S 1 n + 1 with respect to K = cosh t ∂ t having soliton constant c ≥ 0 (respect. c ≤ 0 ), whose mean curvature is bounded, having polynomial volume growth and lying in a slab contained in the chronological future (respect. past) of S 1 n + 1 with respect to the equator { 0 } × S n .

4 Nonexistence of entire solutions of the mean curvature flow equation

Let Ω ⊆ M n be a connected domain, and let u ∈ C ∞ ( Ω ) be a smooth function such that u ( Ω ) ⊆ I , then Σ ( u ) will denote the (vertical) graph over Ω determined by u , i.e.,

Σ ( u ) = { ( u ( x ) , x ) : x ∈ Ω } ⊂ M ¯ n + 1 = ε I × ρ M n .

The graph is said to be entire if Ω = M n . Observe that h ( u ( x ) , x ) = u ( x ) , for all x ∈ Ω . Hence, h and u can be identified in a natural way. The metric induced on Ω from the semi-Riemannian metric of the ambient via Σ ( u ) is

(4.1) g u = ε d u 2 + ρ 2 ( u ) g M .

It is not difficult to verify that the following unit vector field gives an orientation for Σ ( u ) such that Θ < 0 ,

(4.2) N ( x ) = − ε ρ ( u ( x ) ) ρ ( u ( x ) ) 2 + ε ∣ D u ( x ) ∣ M 2 ∂ t ∣ ( u ( x ) , x ) − ε D u ( x ) ρ ( u ( x ) ) 2 , x ∈ Ω .

Moreover, from (4.2), we obtain the corresponding Weingarten operator

(4.3) A X = − 1 ρ ( u ) ρ ( u ) 2 + ε ∣ D u ∣ M 2 D X D u + ε ρ ′ ( u ) ρ ( u ) 2 + ε ∣ D u ∣ M 2 X − ε − g M ( D X D u , D u ) ρ ( u ) ( ρ ( u ) 2 + ε ∣ D u ∣ M 2 ) 3 ⁄ 2 − ε ρ ′ ( u ) g M ( D u , X ) ( ρ ( u ) 2 + ε ∣ D u ∣ M 2 ) 3 ⁄ 2 D u ,

for any vector field X tangent to Ω , where D is the Levi-Civita connection of ( M n , g M ) .

Consequently, being Σ ( u ) a graph over a domain Ω ⊆ M n and denoting by div M the divergence operator computed in the metric g M , it is not difficult to verify from (4.3) that the mean curvature function H ( u ) of Σ ( u ) is given by

(4.4) H ( u ) = − ε div M D u ρ ( u ) ρ ( u ) 2 + ε ∣ D u ∣ M 2 + ρ ′ ( u ) ρ ( u ) 2 + ε ∣ D u ∣ M 2 n − ε ∣ D u ∣ M 2 ρ ( u ) 2 .

Hence, from (3.1) and (4.4), we have that Σ ( u ) is a mean curvature flow soliton with respect to K = ρ ( t ) ∂ t with soliton constant c if, and only if, u is a solution of the following nonlinear differential equation, which is called the mean curvature flow equation:

(4.5) div M D u ρ ( u ) ρ ( u ) 2 + ∣ D u ∣ M 2 = ε 1 ρ ( u ) 2 + ∣ D u ∣ M 2 c ρ ( u ) 2 + ρ ′ ( u ) n − ε ∣ D u ∣ M 2 ρ ( u ) 2 .

Our next result corresponds to a nonparametric version of Theorem 3.1.

Theorem 4.1

Let M ¯ n + 1 = I × ρ M n be a warped product whose fiber M n is complete noncompact, having polynomial volume growth and with its warping function ρ satisfying inequality (3.5). Suppose, in addition, that c ≠ 0 is a constant such that the modified soliton function ζ ¯ c ( t ) < 0 for all t ∈ I . There does not exist a bounded smooth function u : M n → I , which is solution of the mean curvature flow soliton equation (4.5) and such that ∣ D u ∣ M is bounded on M n .

Proof

Let u ∈ C ∞ ( M ) be such a solution of equation (4.5). We start observing that, since M n is complete and inf M ρ ( u ) > 0 (due to the boundedness of u ), from (4.1), we conclude that the entire graph Σ ( u ) must be complete.

From (4.2), we obtain

(4.6) Θ ( u ) = − ρ ( u ) ρ ( u ) 2 + ∣ D u ∣ M 2 .

Thus, since we are assuming that u and ∣ D u ∣ M are bounded, from (3.1), and (4.6), we obtain that H ( u ) is bounded away from zero.

On the other hand, reasoning as in the proof of [8, Theorem 1], we deduce from the induced metric (4.1) that d Σ = ∣ G ∣ d M , where d M and d Σ stand for the Riemannian volume elements of ( M n , g M ) and ( Σ ( u ) , g u ) , respectively, and (as in the proof of Theorem 3.1) G = det ( g i j ) with

(4.7) g i j = g u ( E i , E j ) = ρ 2 ( u ) δ i j + E i ( u ) E j ( u ) .

Here, { E 1 , … , E n } denotes a local orthonormal frame with respect to the metric g M . So, it is not difficult to verify that

(4.8) ∣ G ∣ = ρ 2 ( n − 1 ) ( u ) ( ρ 2 ( u ) + ∣ D u ∣ M 2 ) .

Then, from (4.7) and (4.8), we obtain

(4.9) d Σ = ρ n − 1 ( u ) ρ 2 ( u ) + ∣ D u ∣ M 2 d M .

Hence, since we are supposing that ( M n , g M ) has polynomial volume growth, we can use once more the hypotheses that u and ∣ D u ∣ M are bounded jointly with relation (4.9) to obtain that ( Σ ( u ) , g u ) also has polynomial volume growth. Therefore, we are in position to apply Theorem 3.1 and conclude that Σ ( u ) cannot exist.□

Taking into account [4, Lemma 17] jointly with equation ( 5.9 ) in the proof of [7, Corollary 5.1], it is not difficult to see that we can reason in a similar way of the proof of Theorem 4.1 to obtain the following nonparametric version of Theorem 3.7.

Theorem 4.2

Let M ¯ n + 1 = − I × ρ M n be a GRW spacetime whose Riemannian fiber M n is complete noncompact, having polynomial volume growth and with its warping function ρ satisfying inequality (3.5). Suppose, in addition, that c ≠ 0 is a constant such that the modified soliton function ζ ¯ c ( t ) > 0 for all t ∈ I . There does not exist a bounded smooth function u : M n → I , which is solution of the mean curvature flow soliton equation (4.5) and such that ∣ D u ∣ M ≤ β ρ ( u ) , for some constant 0 < β < 1 .

Acknowledgement

The authors would like to thank the referees for their valuable comments and suggestions, which helped them to improve and clarify the present manuscript.

  1. Funding information: Márcio Batista was partially supported by FAPEAL, Brazil. Márcio Batista and Henrique F. de Lima were partially supported by CNPq, Brazil, Grants 308440/2021-8, 405468/2021-0 to M.B. and 305608/2023-1 to H.dL. Giovanni Molica Bisci was partially supported by INdAM-GNAMPA Research Project 2020 titled Equazioni alle derivate parziali: problemi e modelli. Wallace F. Gomes was partially supported by FACEPE, Brazil, Grant BFP-0182-1.01/23. This work has been funded by the European Union – NextGenerationEU within the framework of PNRR Mission 4 – Component 2 – Investment 1.1 under the Italian Ministry of University and Research (MUR) programme “PRIN 2022” – Grant Number 2022BCFHN2 – Advanced theoretical aspects in PDEs and their applications – CUP:H53D23001960006.

  2. Author contributions: All authors developed the theoretical formalism, performed the analytic calculations, discussed the results, and contributed to write the manuscript.

  3. Conflict of interest: The authors have no relevant financial or non-financial interests to disclose.

  4. Data availability statement: No data were used for the research described in the article.

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Received: 2024-04-19
Revised: 2024-07-10
Accepted: 2024-07-22
Published Online: 2024-10-15

© 2024 the author(s), published by De Gruyter

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  17. Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
  18. A minimization problem with free boundary for p-Laplacian weakly coupled system
  19. Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
  20. k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
  21. Infinitely many solutions for Hamiltonian system with critical growth
  22. On optimal control in a nonlinear interface problem described by hemivariational inequalities
  23. Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
  24. Potential and monotone homeomorphisms in Banach spaces
  25. Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
  26. Supercritical Hénon-type equation with a forcing term
  27. Concentration of blow-up solutions for the Gross-Pitaveskii equation
  28. Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
  29. Regularity for critical fractional Choquard equation with singular potential and its applications
  30. Double phase anisotropic variational problems involving critical growth
  31. Normalized solutions of NLS equations with mixed nonlocal nonlinearities
  32. Nontrivial solutions for resonance quasilinear elliptic systems
  33. Standing waves for Choquard equation with noncritical rotation
  34. Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
  35. Modified quasilinear equations with strongly singular and critical exponential nonlinearity
  36. Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
  37. Nests of limit cycles in quadratic systems
  38. Regularity of minimizers for double phase functionals of borderline case with variable exponents
  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
https://doi.org/10.1515/anona-2024-0034
Keywords for this article
semi-Riemannian warped products; complete mean curvature flow solitons; polynomial volume growth; mean curvature flow equation; Schwarzschild and Reissner-Nordström spaces; Robertson-Walker spacetimes
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  4. Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
  5. Existence and uniqueness of solution for a singular elliptic differential equation
  6. The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
  7. Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
  8. Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
  9. Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
  10. Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
  11. Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
  12. On a nonlinear Robin problem with an absorption term on the boundary and L1 data
  13. Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
  14. Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
  15. Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
  16. Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
  17. Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
  18. A minimization problem with free boundary for p-Laplacian weakly coupled system
  19. Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
  20. k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
  21. Infinitely many solutions for Hamiltonian system with critical growth
  22. On optimal control in a nonlinear interface problem described by hemivariational inequalities
  23. Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
  24. Potential and monotone homeomorphisms in Banach spaces
  25. Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
  26. Supercritical Hénon-type equation with a forcing term
  27. Concentration of blow-up solutions for the Gross-Pitaveskii equation
  28. Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
  29. Regularity for critical fractional Choquard equation with singular potential and its applications
  30. Double phase anisotropic variational problems involving critical growth
  31. Normalized solutions of NLS equations with mixed nonlocal nonlinearities
  32. Nontrivial solutions for resonance quasilinear elliptic systems
  33. Standing waves for Choquard equation with noncritical rotation
  34. Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
  35. Modified quasilinear equations with strongly singular and critical exponential nonlinearity
  36. Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
  37. Nests of limit cycles in quadratic systems
  38. Regularity of minimizers for double phase functionals of borderline case with variable exponents
  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
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