Bifurcation analysis for degenerate problems with mixed regime and absorption

We are concerned with the study of a bifurcation problem driven by a degenerate operator of Baouendi–Grushin type. Due to its degenerate structure, this differential operator has a mixed regime. Studying the combined effects generated by the absorption and the reaction terms, we establish the bifurcation behavior in two cases. First, if the absorption nonlinearity is dominating, then the problem admits solutions only for high perturbations of the reaction. In the case when the reaction dominates the absorption term, we prove that the problem admits nontrivial solutions for all the values of the parameter. The analysis developed in this paper is associated with patterns describing transonic flow restricted to subsonic regions.


Introduction
This paper deals with the bifurcation analysis of solutions for a class quasilinear degenerate of mixed-type. Problems with mixed regime are strictly connected with the analysis of nonlinear patterns and stationary waves for transonic flow models. We refer to the pioneering work of  on the theory of transonic fluid flow -referring to partial differential equations that possess both elliptic and hyperbolic region -and this remains the most fundamental mathematical work on this subject. The flow is supersonic in the elliptic region, while a shock wave is created at the boundary between the elliptic and hyperbolic regions. In the 1950s, Morawetz used functional-analytic methods to study boundary value problems for such transonic problems.
To the best of our knowledge, the study of elliptic-problems has been initiated by Tricomi [31] who studied problems driven by the operator We remark that the Tricomi operator is degenerate in the region Ω 0 := {(x, y) ∈ R 2 : x = 0}. The Tricomi operator has important applications in the theory of planar transonic flow, see Manwell [16].
There are many generalizations of the Tricomi operator T , which can be obtained by substituting x with a function g(x). For instance, Baouendi [3] and Grushin [13] studied the degenerate operator This operator has been studied in a pioneering paper by Mitidieri and Pohozaev [19]. A related differential operator was studied by Bahrouni et al. [2] in the framework of double-phase transonic flow problems with variable growth.
We give in what follows some details about the differential operator that describes our problem. Assume that the Euclidean space R N (N ≥ 2) can be written as R N = R m × R n and for all z ∈ Ω ⊂ R N (Ω is an arbitrary open set) we denote z = (x, y), where x ∈ R n and y ∈ R m . Let γ be a nonnegative real number. Then the differential operator that describes the problem studied in this paper involves the sub-elliptic gradient, which is the N -dimensional vector field given by where |x| (respectively, |y|) is the Euclidean norm in R m (respectively, R n ) and with the corresponding Baouendi-Grushin vector fields Then the Baouendi-Grushin operator on Ω ⊂ R N = R m+n is defined by where Δ x and Δ y are the Laplace operators in the variables x and y, respectively. We point out that the Baouendi-Grushin operator for an even positive integer γ is a sum of squares of C ∞ vector fields satisfying the Hörmander condition rank Lie[X 1 , . . . , X m , Y 1 , . . . , Y n ] = N.
We can define on R N = R m+n the anisotropic dilation δ λ attached to Δ γ as An important element in the mathematical analysis of PDEs driven by the Baouendi-Grushin operator is the homogeneous dimension with respect to the dilation δ λ , which is defined by We notice that Q = N + γn > N ≥ 2. By the change of variables formula for the Lebesgue measure we obtain Assuming that γ = 1/2, then the corresponding Baouendi-Grushin operator is intimately connected to the sub-Laplacians in groups of Heisenberg type, see Garofalo and Lanconelli [11].
Finally, we point out that if f = f (r) ∈ C 2 (0, ∞) and define u = f (r) then we have the following useful formula: In this paper, we use the Baouendi-Grushin differential operator in a quasilinear nonradial setting.
Some of the abstract methods used in this paper are developed in the recent monograph by Papageorgiou et al. [26].

Statement of the Problem
Our purpose in this paper is to study a class of degenerate bifurcation problems described by a nonlinear differential operator of mixed-type.
Let Ω ⊂ R N = R m+n (N ≥ 2) be a bounded domain with smooth boundary. We suppose that Ω intersects the degeneracy set [x = 0], that is We are concerned with the study of the following degenerate bifurcation problem: A related problem was studied for exterior domains and for the p-Laplace operator by Yu [32] and by Filippucci et al. [8].
Throughout this paper, we denote by · s the norm in the Lebesgue space L s (Ω) for 1 ≤ s ≤ ∞.
We shall distinguish two cases in the analysis developed in this paper. We first consider the case where the absorption term |u| q−2 u in problem (3) is dominating with respect to the reaction |u| r−2 u. Mathematically, this corresponds to q > r. In this case, due to the dominance of the absorption, we prove that solutions exist only for high perturbations of the reaction, that is, for sufficiently big values of the parameter λ. Next, if the reaction is the dominating nonlinearity (that is, if r > q)  Bull. Math. Sci. 2021.11. Downloaded from www.worldscientific.com by 86.120.251.98 on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.

Degenerate problems with mixed regime and absorption
we prove that the influence of the reaction is sufficient to guarantee solutions for all values of the positive parameter λ.
Let X be the closure of C ∞ c (Ω) with respect to the norm Let Q * denote the Sobolev critical exponent associated with the homogeneous dimension Q defined in (2), that is, Q * = 2Q/(Q−2). We observe that 2 < Q * < 2 * .
We assume throughout this paper that γ > 0 and q, r ∈ (2, Q * ). We say that u ∈ X \{0} is a solution of problem (3) if for all v ∈ X , we have In this case, we say that λ is an eigenvalue of problem (3) and the corresponding u ∈ X \{0} is an eigenfunction of (3). These terms are in accordance with the related notions introduced by Fučik et al. [10, p. 117] in the context of nonlinear operators. Indeed, if we denote Taking v = u in relation (4), we deduce that any solution u of problem (3) verifies This implies that problem (3) does not have any solution if λ ≤ 0. So, we shall be concerned only with positive values of the bifurcation parameter λ.
We first consider the case 2 < r < q < Q * .
We state in what follows the first result of this paper.  (3) has a variational structure and the associated energy functional should be well defined. For this purpose, we need Sobolev-type embeddings. More details will be given in Sec. 3.
The proof of Theorem 2.1 is developed in Sec. 3, while Sec. 4 will include the details of the proof of Theorem 2.2. We conclude this paper with some open problems and perspectives.

Proof of Theorem 2.1
The energy functional associated to problem (3) is J : X → R defined by We prove that J is well defined. For this purpose, we need suitable Sobolev-type embeddings. For Sobolev-Poincaré inequalities for weighted gradients we refer to Franchi et al. [9], Garofalo and Nhieu [12], Laptev et al. [14], Lu [15], and Song and Li [30]. Partial results are also given by Bahrouni et al. [2,Theorem 3.2] in the anisotropic setting.
In this paper, we refer to Monti [20] who obtained the following Sobolev-Poincaré inequality. We recall that in (2) we defined the homogeneous dimension Q as Q = m + (1 + γ)n = N + γn > N ≥ 2.
Then there exists a positive constant C = C(m, n, γ) such that for all u ∈ D 1 (R m+n ), Theorem 3.1 implies the following compact embedding theorem for function spaces with weighted gradients. R N (N ≥ 2). Then the space X is continuously embedded into L 2Q/(Q−2) (Ω) and is compactly embedded into L s (Ω) for all 1 < s < 2Q/(Q − 2).

Corollary 3.1. Let Ω be a bounded domain with smooth boundary in
Using Corollary 3.1, we deduce that the energy functional J is well defined.

Degenerate problems with mixed regime and absorption
We first observe that J is coercive. Indeed, since r < q, then the Hölder inequality yields for all u ∈ X Ω |u| r dxdy ≤ Fix positive numbers a and b. Since 0 < r < q then there exists C(a, b) such that We conclude that for all u ∈ X hence J is coercive. This means that any minimizing sequence (u n ) ⊂ X of J is bounded. Thus, by the reflexivity of X and Corollary 3.1, we can assume that (u n ) converges weakly in X and strongly in L q (Ω) and L r (Ω) to u. Since J (|u|) ≤ J (u), we can assume that u ≥ 0. Finally, by lower semicontinuity, we deduce that u is a global minimizer of the energy functional J . Next, we prove that solutions of problem (3) could exist only in the case of high perturbations of the reaction, that is, for large values of the positive parameter λ.
By Young's inequality and using the basic hypothesis 2 < r < q, we deduce that for all u ∈ X \{0} Assume that u is a solution of problem (3). It follows that Combining (6) and (7), we obtain that any solution u of (3) satisfies By Corollary 3.1, relation (8) yields 2050017-7 Bull. Math. Sci. 2021.11. Downloaded from www.worldscientific.com by 86.120.251.98 on 05/01/21. Re-use and distribution is strictly not permitted, except for Open Access articles.
By the Sobolev embedding theorem, this relation yields Combining relations (8) and (10), we obtain Recall that 2 < r < q. Relation (11) implies that there exists such that if problem (3) has solutions, then λ ≥ λ 0 . We now prove that the global minimizer u ≥ 0 of J is nontrivial. For this purpose, it is sufficient to show that the corresponding energy level is negative. The idea is to construct a natural constrained minimization problem and to show that the corresponding value is negative, provided that λ is large enough.
If (v n ) ⊂ X is a corresponding minimizing sequence, then (v n ) is bounded. Thus, up to a subsequence, We conclude that v 0 r r = 1 and
We prove in what follows that problem (3) admits solutions for all λ > λ * . The basic abstract tool is the comparison principle for degenerate operators, see Pucci and Serrin [28, Theorem 3.6.1].
We claim that problem (3) for λ =λ has a nonnegative upper solution U such that U ≥ u ν in Ω. For this purpose, we define the following minimization problem: With the same arguments as for the previous constrained minimization problem, we obtain that this problem has a solution U ≥ u ν , which proves our claim. We conclude that problem (3) admits at least one nonnegative solution for all λ > λ * . Now, assuming that there exists 0 < μ < λ * such that (3) for λ = μ has a solution, then the previous argument shows that problem (3) has solutions for all λ ≥ μ, which contradicts the definition of λ * . It follows that (3) does not have any solution for all λ ∈ (0, λ * ).
We now establish that problem (3) has a solution for λ = λ * . The idea is to consider a sequence of real numbers (λ n ) such that λ n λ * as n → ∞. Let u n ≥ 0 be a solution of (3) for λ = λ n . It follows that for all v ∈ X We have already remarked that the sequence (u n ) ⊂ X is bounded. Thus, up to a subsequence, So, taking n → ∞ in relation (13), we deduce that u * ≥ 0 is a solution of problem (3) for λ = λ * . To conclude the proof, it remains to argue that u * is nontrivial. Returning to the sequence of solutions (u n ) corresponding to λ n λ * , we have λ n u n r r = u n 2 + u n q q for all n ≥ 1.

A. Alsaedi, V. D. Rȃdulescu & B. Ahmad
If C is the best constant of the continuous embedding of X into L r (Ω), it follows that λ n C u n r ≥ u n 2 for all n ≥ 1, hence lim inf n→∞ u n > 0. We obtain lim inf n→∞ u n r > 0.
We have proved that problem (3) has a nontrivial solution u if and only if λ ≥ λ * . This solution is nonnegative, since the above arguments show that we can replace u by |u|. By symmetry reasons, problem (3) admits also nonpositive solutions for all λ ≥ λ * .

Proof of Theorem 2.2
The proof relies on the mountain pass theorem, which we shall use in the following form (see Brezis and Nirenberg [6, p. 943]).
Consider the family P of all continuous paths p joining 0 and v 0 and set Then there exists a sequence (u n ) in X such that If, in addition, we assume (P S) c with c given in (15), then c is a critical value of F .
In our case, we observe that for all u ∈ X where C denotes the best Sobolev constant of the embedding X ⊂ L r (Ω).
Since r > 2, there are positive numbers ρ and c 0 such that J (u) ≥ c 0 for all u ∈ X with u = ρ.
Next, let v ∈ X \{0}. Then for all t > 0 and Let (u n ) ⊂ X be a (P S) c sequence of J , that is, Using the expression of the energy functional J , these conditions yield Combining these relations, we obtain Since 2 < q < r, this estimate shows that the arbitrary (P S) c sequence (u n ) ⊂ X is convergent. We now claim that the (P S) c sequence (u n ) ⊂ X is relatively compact. Here, we use a method developed by Brezis and Nirenberg [6], which is based on the qualitative properties of the differential operator that describes problem (3).
We first observe that for all v ∈ X where satisfies By (17), our claim follows if we prove that the sequence (φ(u n )) is relatively compact in X * . By Theorem 3.1, this is done if we prove that (φ(u n )) converges, up to a subsequence, in (L 2Q/(Q−2) (Ω)) * = L 2Q/(Q+2) (Ω).
To conclude the proof, it is enough to show that can be made arbitrarily small.
We first observe that So, choosing η > 0 small enough, the right-hand side of this inequality can be made as small as we wish. Next, we observe that for all ε > 0 small enough, there exists C ε > 0 such that for all n ≥ 1 By Theorem 3.1 and since (u n ) ⊂ X is bounded, this relation implies that provided that |ω| is sufficiently small. The proof of the claim is complete. So, by Theorem 4.1, problem (3) admits a solution for all λ > 0. Finally, we argue that this solution is positive. This follows by a standard truncation argument, which consists in replacing the reaction |u| r−2 u of problem (3) by Then, with the same arguments as above, one can prove that the problem  (20) and integrating, we find It follows that U − = 0, that is, U is a positive solution of (20). It follows that g(U ) = U r−1 , hence U is also a solution of problem (3).
In the final part of this section, we emphasize two key facts concerning the proof of Theorem 2.2.
We first observe that the proof of the Palais-Smale condition also holds if the reaction of problem (3) has an "almost" critical growth. Indeed, in the estimate (19) we only need that φ(u) = o(|u| (Q+2)/(Q−2) ) as |u| → +∞.

Case of small perturbations
In a seminal paper, Baras and Pierre [4] established a striking result concerning the existence of solutions for perturbed nonlinear elliptic equations. They studied the problem ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −Δu = u γ + μg in Ω, where g ≡ 0, μ ≥ 0 and γ > 1. Baras and Pierre [4] proved that problem (21) does not have any solution if μ is bigger than a certain critical value. More precisely, they established that problem (21) has a solution for all μ ≤ 1 if and only if for all ϕ ∈ W 1,∞ 0 (Ω) ∩ W 2,∞ (Ω) nonnegative and superharmonic function with compact support. In terms of duality, this condition expresses that a certain norm