Double phase problems with variable growth and convection for the Baouendi–Grushin operator

In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.


Introduction
Let Ω ⊂ R N , N > 1, be a bounded domain with smooth boundary ∂Ω and let n, m be nonnegative integers such that N = n + m. This means that R N = R n × R m and so z ∈ Ω can be written as z = (x, y) with x ∈ R n and y ∈ R m .
We consider the following double phase problem with convection term −Δ G(x,y) u + A(x, y)(|u| G(x,y)−1 + |u| G(x,y)−3 )u = f ((x, y), u, ∇u) inΩ, u = 0 on ∂Ω, (1.1) introduced by Baouendi [7] and Grushin [17]. The Baouendi-Grushin operator can be viewed as the Tricomi operator for transonic flow restricted to subsonic regions. On the other hand, a second-order differential operator T in divergence form on the plane can be written as an operator whose principal part is a Baouendi-Grushin-type operator, provided that the principal part of T is nonnegative and its quadratic form does not vanish at any point, see Franchi and Tesi [15]. In the right-hand side of problem (1.1) we have a nonlinearity f : Ω × R × R N → R which is a Carathéodory function, that is, f (·, s, ξ) is measurable for all (s, ξ) ∈ R×R N and f ((x, y), ·, ·) is continuous for a.a. (x, y) ∈ Ω.
Problem (1.1) is strictly connected with the analysis of nonlinear patterns and stationary waves for transonic flow models. We refer to the pioneering work of Morawetz [20][21][22] on the theory of transonic fluid flow-referring to partial differential equations that possess both elliptic and hyperbolic regionsand 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.
The variable coefficient G(x, y) describes the geometry of a composite realized by using two materials with corresponding behaviour described by |∇ x u| G(x,y) and |∇ y u| G(x,y) . Then in the region {z ∈ Ω : x = 0} the material described by the second integrand is present. In the opposite case, the material described by the first integrand is the only one that creates the composite.
The main goal of our paper is to prove the existence of at least one weak solution of problem (1.1) under very general conditions on the nonlinearity f : Ω×R×R N → R. The novelty of our paper is the fact that we combine a double phase operator driven by the Baouendi-Grushin operator with variable growth and a right-hand side which depends on the gradient of the solution. Such function is called convection term.
It is well known that the Caffarelli-Kohn-Nirenberg inequality is a powerful inequality and it is needed in several ways in the study of partial differential equations. We refer to the works of Adimurthi et al. [2], Baroni et al. [8], Colasuonno and Pucci [12], Colombo and Mingione [13] for relevant applications of the Caffarelli-Kohn-Nirenberg inequality. For recent contributions to the study of double-phase problems we refer to Beck and Mingione [9], Papageorgiou et al. [24,25], and Zhang and Rȃdulescu [31].
The following Caffarelli-Kohn-Nirenberg inequality [10] establishes that for given p ∈ (1, N) and real numbers a, b and q such that there exists a positive constant C a,b such that for all u ∈ C 1 c (Ω) This inequality was extensively studied, see for example Abdellaoui and Peral [1], Adimurthi et al. [2], Bahrouni et al. [4,5], Catrina and Wang [11], and the references therein. In particular, Bahrouni et al. [5] proved a new version of a Caffarelli-Kohn-Nirenberg inequality with variable exponent for the Baouendi-Grushin operator Δ G . More precisely, the following weighted inequality has been proved.
The paper is organized as follows. In Sect. 2 we present the basic properties of variable Lebesgue and Sobolev spaces and state the main tools which will be used later; see Rȃdulescu and Repovš [29] for more details. New properties concerning the Baouendi-Grushin operator will be discussed in Sect. 3, and in the last section we state and prove our main result concerning the existence of a weak solution to problem (1.1).
By W 1,p(·) (Ω) we denote the variable exponent Sobolev space equipped with the norm Then W 1,p(·) (Ω) is a reflexive and separable Banach space. Our main existence result will be based on the following surjectivity result, see Gasinski and Papageorgiou [16]. First, we give the definition of pseudomonotonicity. Definition 2.3. Let X be a reflexive Banach space, X * its dual space and denote by ·, · its duality pairing. Let A : X → X * , then A is called pseudomonotone if u n w → u in X and lim sup n→∞ A(u n ), u n − u ≤ 0 imply Au n w → u and Au n , u n → Au, u .
Theorem 2.4. Let X be a real, reflexive Banach space, and let A : X → X * be a pseudomonotone, bounded, and coercive operator, and b ∈ X * . Then the problem Au = b has at least one solution.

Properties of the double phase operator and the corresponding function space
In this section we recall and prove new results concerning the Baouendi-Grushin operator introduced in Sect. 1. Based on Theorem 1.1, we denote by W the closure of C 1 c (Ω) with respect to the norm Note that the norm · on W is equivalent to From now on we denote the duality pairing between W and its dual space W * by ·, · W . Furthermore, we set G + := sup

G(x, y).
The following compactness property was proved by Bahrouni et al. [5]. Lemma 3.1. Assume that G is a function of class C 1 and that G(x, y) ∈ (2, N) for all (x, y) ∈ Ω. Furthermore, suppose that s ∈ (1, G − ) and 0 < γ < N (G − −s) s . Then W is compactly embedded in L s (Ω). Now, we define ρ : W → R by The following lemma will be helpful in later treatments.  (ii) Let u ∈ W be such that u W < 1, then So, by Proposition 2.1, we get the desired result. (iii) Let u ∈ W be such that u W > 1. By (i), we obtain Then, by the mean value theorem, there exist (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ) ∈ Ω depending on u, G and Ω such that G(x, y) + 1 dx dy Since u W > 1, it follows that This finishes the proof. (i) The functional ρ is of class C 1 , and for all u, v ∈ W we have Proof. (i) This follows directly from the definition of ρ : W → R.
(ii) By Lemma 3.2, for u W > 1, we obtain Proof. By Lemma 3.1 there exists C > 0 such that On the other hand, by Lemma 3.2, for u W > 1 we have Combining the above inequalities we obtain The proof is now complete. Proof. (i) From Lemma 3.3 it is clear that ρ is continuous. Next, we are going to prove that ρ maps bounded sets to bounded sets. By Young's inequality, we obtain G(x, y) + 1 dx dy Hence, from Lemma 3.2, we get which implies that ρ maps bounded sets to bounded sets. The strict monotonicity of ρ is a direct consequence of the well-known Simon inequalities [30, formula (2. 2)]