Systems of elliptic equations with strong interspecific competition arise naturally in several physical and biological models, such as multi-component Gross-Pitaevskii and nonlinear Schrödinger equations. The interaction between different components is typically governed by a real parameter β , whose sign and magnitude describe the nature and intensity of the competition. For instance, in [1], Noris, Tavares, Terracini, and Verzini (2010) established uniform Hölder estimates for the following strongly competitive nonlinear Schrödinger system:

In this setting, the strong competition regime corresponds to the limit β → + ∞ (note that the interaction term carries a negative sign), in which the interaction term strongly penalizes the coexistence of different components. In this limit, solutions are expected to exhibit phase separation phenomena, giving rise to segregated limiting profiles whose supports are essentially disjoint. They showed that uniform L ∞ bounds for u β and v β imply uniform C 0 , α bounds for every α ∈ ( 0 , 1 ) , and their proof is based on a blow-up analysis combined with the Alt-Caffarelli-Friedman monotonicity formula. Up to now, strongly competing models with isotropic diffusion have been extensively studied by many authors [2]-[4]. However, extending these results to the case of anisotropic operators entails significant additional difficulties, mainly due to the lack of rotational invariance and the more complex underlying geometry. Anisotropic diffusion refers to diffusion processes whose rates depend on direction, in [5], Terracini and Soave established an anisotropic monotonicity formula and further investigated anisotropic two-phase problems of the form

where A 1 and A 2 are symmetric positive definite N × N matrices with constant coefficients. Moreover, by means of a suitable change of variables, one may assume without loss of generality that A 1 is diagonal and A 2 is the identity matrix. This is also the system studied in the present paper.

In this paper, we study a genuinely anisotropic two-phase problem

where Ω ⊂ ℝ N is an open set with N ≥ 3 , k > 1 is the competition parameter. We assume that A is a positive definite N × N diagonal matrix with constant coefficients, with the lowest eigenvalue 1: A : = diag ( a 1 , a 2 , ⋯ , a N ) with 1 = a 1 ≤ ⋯ ≤ a N . The exponent γ > 1 describes higher-order competitive interactions. The aim of this paper is to prove the uniform boundedness of positive solutions to the above family of equations in Hölder norms, via the construction of blow-up sequences and the use of monotonicity formulas together with Liouville-type theorems.

To prove the regularity of solutions to the system we study, it is crucial to identify an appropriate montonicity formula. Accordingly, we proceed to consider the positive solutions v 1 , v 2 of the following coupled system, which do not have disjoint supports and belong to H l o c 1 ( ℝ N ) ∩ C ( ℝ N ) , where the system is given by

In order to obtain a Liouville-type result, we need to employ a suitable monotonicity formula. For N ≥ 3 and δ > 0 , we introduce a C 1 auxiliary function

Moreover, we define

where Φ A , δ ( x ) = Φ δ ( A − 1 / 2 x ) , Φ I d , δ ( x ) = Φ δ ( x ) . Then we can observe that m A ( x ) and m ( x ) are both bounded on ℝ N , vanish in ℝ N \ B δ and are nonnegative for almost every x . And since Φ δ coincides with the fundamental solution of the Laplacian away from the origin, Φ A , δ agrees in the set ℰ δ c = { | A − 1 / 2 x | ≥ δ } with the fundamental solution associated with the operator div ( A ∇ ) . We denote this fundamental solution by Γ A , which admits the explicit representation

In particular, Φ A , δ = Γ A in the set ℰ δ c .

The Alt-Caffarelli-Friedman (ACF) monotonicity formula plays a fundamental role in the analysis of two-phase and multi-phase free boundary problems. Its original version was introduced in [6]. In the following, we develop an ACF-type monotonicity formula adapted to the system (2.1) under consideration.

Theorem 2.1 (Monotonicity formula)Let v 1 , v 2 be positive solutions of(2.1)and let ε > 0 be fixed. Then there existsan exponent ν A , N ∈ ( 0 , 2 ) depending on A and N ,and r ¯ = r ¯ ( v 1 , v 2 , ε ) > δ such that the function

is increasing for r ∈ ( r ¯ , + ∞ ) .

ProofOur proof is based on Lemma 2.5 in [1]. Without loss of generality, we consider x 0 = 0 . We introduce

Then, we test the equations for v 1 and v 2 in B r with v 1 Φ A , δ and v 2 Φ I d , δ , respectively. By integration by parts, the definition of m A ( x ) and Young’s inequality, we obtain the estimate

where γ ( Λ i ( 0 , r ) ) = ( N − 2 2 ) 2 + Λ i ( 0 , r ) − N − 2 2 , i = 1 , 2 . Then, we can compute the derivative of J ( r ) for r > δ . Using the above estimate, we obtain that

We proceed by contradiction, assuming that there exists a sequence r n → + ∞ such that

To study the behavior as r n → + ∞ , we perform a suitable normalized rescaling of the solution. By energy boundedness, the rescaled functions are bounded in H 1 ( ∂ B 1 ( 0 ) ) . Then by compactness, we can extract a subsequence that converges weakly to a limiting pair of functions. The crucial step lies in exploiting the competitive term in the system, which allows one to show that the limiting profiles have disjoint supports. This property reduces the problem to the classical Alt-Caffarelli-Friedman framework. Finally, we can obtain that

which contradicts the assumption and thus yields the monotonicity. □

Thus, having obtained a suitable monotonicity formula, we can now prove a Liouville-type result for the system under consideration.

Theorem 2.2 (Liouville-type theorem)Let v 1 , v 2 benon negativesoluctions of(2.1). Suppose that the functions v 1 , v 2 grow at most like | x | α i ,namely | v 1 ( x ) | ≤ C ( 1 + | x | α 1 ) , | v 2 ( x ) | ≤ C ( 1 + | x | α 2 ) for every | x | ∈ ℝ N ,for some C > 0 , with

Then one of the functions is identically zero and the other is a constant.

ProofOwing to the structure of system (2.1), if one of the functions is 0 or a positive constant, then the other must be a constant or 0 respectively. By contradiction we assume that neither v 1 nor v 2 is constant. For simplicity of notation, we write B r = B r ( 0 ) . Then by the maximum principle v 1 and v 2 are positive, and by Theorem 2.1 we know that there exists C > 0 such that

for r sufficiently large. Consider a radial smooth cut-off function η such that 0 ≤ η ≤ 1 , η = 1 in B r ( 0 ) , η = 0 in ℝ N \ B 2 r ( 0 ) , and | ∇ η | ≤ C / r . By texting the equation for v 1 with η 2 Φ A , δ ( x ) v 1 on B 2 r , we have

then integrating by parts further, we obtain

Expanding the right-hand side further, we obtain

Substituting the above expression into (2.3),

By using the Cauchy-Schwarz and Young’s inequalities, we have

Therefore we can obtain that

Recalling that div ( A ∇ Φ A , δ ( x ) ) = − 2 m A ( x ) and testing it with η 2 v 1 2 / 2 in B 2 r , we can obtain

Plugging this into the above gives

Using the definitions of η and Φ A , δ ( x ) , the left-hand side of the preceding inequality can be restricted to B r ,

the final inequality here follows from | v 1 ( x ) | < C ( 1 + | x | ) α 1 . By taking the limits as δ → 0 + , we infer that

Similarly, we obtain

which contradicts (2.2) for r large enough. □

In order to carry out the blow-up analysis, it is crucial to obtain a uniform control on the rescaled solutions. To this aim, we rely on the following technical lemma, which prevents the possible blow-up of subsolutions in the presence of strong absorption terms.

Lemma 2.3Let x 0 ∈ ℝ N , M , δ , ρ , B > 0 and u ∈ H 1 ( B 2 ρ ( x 0 ) ) ∩ C ( B ¯ 2 ρ ( x 0 ) ) be asubsolutionto

for some p ≥ 1 . Then there exists C > 0 , depending only on the dimension N , such that

M u p ( x ) ≤ C B a N ρ 2 + δ for every x ∈ B ρ ( x 0 ) .

ProofLet T : = A 1 2 , and make the linear change of variables x = x 0 + T y , then we get y = T − 1 ( x − x 0 ) . Define a new function v ( y ) : = u ( x 0 + T y ) = u ( x ) , we have that ∇ y v ( y ) = T ∇ x u ( x ) , ∇ x u ( x ) = T − 1 ∇ y v ( y ) , hence

Thus, in the y -coordinates, v satisfies

Notice that B ˜ is an ellipsoid, which satisfies a 1 | y | 2 ≤ | T y | 2 ≤ a N | y | 2 , thus the inclusion holds

In order to directly apply Lemma 2.2 in [2], take ρ y = ρ a N , then B 2 ρ / a N ( 0 ) ⊂ B ˜ . Therefore, we can deduce that

Since v ( y ) = u ( x ) , the estimate holds for all y such that | y | ≤ ρ / a N . For such y , we have

which implies that x ∈ B ρ ( x 0 ) . Therefore, we conclude that

□

The Liouville-type theorem for the limiting system, combined with the monotonicity formula, rules out the existence of nontrivial blow-up limits. This fact will be used in the next section to infer regularity properties of solutions to the original system.

Theorem 3.1Let Ω ⊂ ℝ N be an open set. Under the standing assumptions, assume that N ≥ 3 and γ > 1 . Let u k = ( u 1 , k , u 2 , k ) be a positive solution to the system (1.1) at fixed k > 1 , satisfy that there exists M > 0 with

Then,there exists ν A , N ∈ ( 0 , 2 ) depending only on A and N such that the following holds:for every α ∈ ( 0 , ν A , N 2 ) there exists C > 0 independent of k such that

ProofLet α ∈ ( 0 , ν A , N 2 ) , and suppose for contradiction that { u k } is not bounded in C 0 , α ( Ω ¯ ) , so there exists a sequence k → + ∞ such that

We can assume that L k is attained by u 1 , k at the pair ( x k , y k ) , then

Then we introduce the following blow-up of u k with center x k , with r k → 0 + to be chosen later:

Depending on the asymptotic behavior of the distance d ( x k , ∂ Ω ) , and on r k , we have Ω k → Ω ∞ , where Ω ∞ is either ℝ N or an half-space. The function v k = ( v 1 , k , v 2 , k ) is a positive solution to

where M k = k L k 2 γ r k 2 α γ + 2 . And we denote

Since ‖ u i , k ‖ L ∞ ( Ω k ) ≤ M , L k → + ∞ , r k → 0 + , then we can deduce ‖ f i , k ( ⋅ , v i , k ) ‖ L ∞ ( Ω k ) → 0 as k → + ∞ . Moreover, for every k ,

In this way, by suitable translations and rescalings, we complete the blow-up procedure and normalize the Hölder seminorm of the solutions. At this point, two potential pathological behaviors still need to be excluded: the divergence of the rescaled values at the origin, v k ( 0 ) , and the loss of the competition effects in the limit. Lemma 3.2 and Lemma 3.3 are devoted precisely to ruling out these possibilities, thereby completing the blow-up analysis.

Lemma 3.2Let r k → 0 + as k → + ∞ be such that

1) there exists R ′ > 0 such that | x k − y k | ≤ R ′ r k ;

2) M k ↛ 0 .

Then { v k ( 0 ) } is bounded in k .

ProofAssume by contradiction that { v k ( 0 ) } is unbounded, and Let R ≥ R ′ , then we will show that both v 1 , k and v 2 , k must in fact be bounded.

To begin with, suppose that v 2 , k ( 0 ) is unbounded. Since the v 2 , k is uniformly Hölder continuous and vanish on ∂ Ω k , we can consider k sufficiently large such that B 2 R = B 2 R ( 0 ) ⊂ Ω k , and let R > R ′ .

Let φ ∈ C c ∞ ( B 2 R ) be a nonnegative function such that φ = 1 in B R . By texting the equation for v 1 , k against v 1 , k φ 2 , we obtain that

By further integration by parts and expansion, we obtain

Plugging the above into (3.3) and collecting terms yields

Appling the Cauchy-Schwarz and Young’s inequalities to the above equation,

Substituting the above equation into Equation (3.4), we obtain

Therefore, we have

Hence, by using assumption (ii) and afterwards the boundedness of the oscillation of v 1 , k , we deduce that for every x in B R

where C > 0 depends only on R . Evaluating this inequality at x = 0 , we get

Since | v 2 , k ( 0 ) | → + ∞ and γ > 0 , we can see that v 1 , k ( 0 ) is bounded. In other words, this implies that inf B 2 R | v 2 , k | → + ∞ , sup B 2 R | v 1 , k | is bounded, and from (3.5) we have that actually sup B 2 R v 1 , k → 0 . Let now consider the quantity I k : = M k inf B 2 R v 2 , k γ + 1 → + ∞ , we have

By applying the modified Lemma 2.3,

and hence M k v 1 , k γ v 2 , k γ + 1 → 0 , as k → + ∞ . Moreover, since ‖ r k 2 v 1 , k ( 1 − L k r k α v 1 , k ) ‖ L ∞ ( B R ) → 0 , we have