Nonvariational and singular double phase problems for the Baouendi-Grushin operator

In this paper we introduce a new double phase Baouendi-Grushin type operator with variable coefficients. We give basic properties of the corresponding functions space and prove a compactness result. In the second part, using topological argument, we prove the existence of weak solutions of some nonvariational problems in which this new operator is present. The present paper extends and complements some of our previous contributions related to double phase anisotropic variational integrals.


Introduction
The present paper is motivated by recent fundamental enrichment to the mathematical analysis of nonlinear models with unbalanced growth. We mainly refer to the pioneering contributions of Marcellini [25,26] who studied lower semicontinuity and regularity properties of minimizers of certain quasiconvex integrals. Related problems are inspired by models arising in nonlinear elasticity and they describe the deformation of an elastic body, see Ball [1,2].
More precisely, we are concerned with the following nonlinear equations of double phase Baouendi-Grushin type where N ≥ 3, K ∈ C(R N ), f ∈ C(R), while −∆ G,a stands for a new double phase Baouendi-Grushin type operator with variable exponents (see (1.2)).
The main aim of our work is to introduce a new double phase Baouendi-Grushin type operator with variable exponents and its suitable functions space. Our abstract results related to the new function space are motivated by the existence of solutions for nonvariational problems of type (1.1). The present paper complements our previous contributions related to double phase anisotropic variational integrals, see [3,4,5,6].
First, we recall the notion of Baouendi-Grushin operator with variable growth. Let Ω ⊂ R N , N > 1, be a 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 . In this paper G : Ω → (1, ∞) is supposed to be a continuous function and ∆ G(x,y) stands for the Baouendi-Grushin operator with variable coefficient, which is defined by The differential operator ∆ G(x,y) generalizes the degenerate operator introduced independently by Baouendi [9] and Grushin [21]. 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 & Tesi [19].  [23,36], and Zhang & Rȃdulescu [42]. We refer to Marcellini [27] and Mingione & Rȃdulescu [28] for surveys of recent results on elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators. Now, we are able to introduce the new Baouendi-Grushin type operator with variable coefficients, which is defined by The main goal of our recent paper [6] was to study a singular systems in the whole space R N in which the Baouendi-Grushin operator (−∆ G(x,y) ) is present. So, the main difficulty is the lack of compactness corresponding to the whole Euclidean space. To overcome this difficulty, we proved a related compactness property. However, the interval of compactness is too short. So, we are not able to study a large number of equations driven by −∆ G(x,y) in the whole space R N . For this reason and in order to get a better compactness result, we introduced the new operator −∆ G,a . Our abstract results are motivated by the existence of solutions of the following class of nonlinear equation where Ω ⊂ R N is supposed to be a bounded domain. Another motivation comes from singular problems in the form , (x, y) ∈ R N , where σ(·) ∈ (0, 1) and b is positive function.
The paper is organized as follows. In Section 2 we present the basic properties of variable Lebesgue space and introduce the main tools which will be used later. New properties concerning the new operator (−∆ G,a ) will be discussed in Section 3. In Section 4, combining these abstract results with the topological argument, we study a nonvariational problem in which −∆ G,a is present. In last section, we deal with purely singular double phase equation. We refer to the monograph by Papageorgiou, Rȃdulescu & Repovš [34] as a general reference for the abstract methods used in this paper.

Double phase Baouendi-Grushin operators
In this section we prove new results concerning the new Baouendi-Grushin operator defined in (1.2).
First, we give the hypotheses on continuous functions a, K, G : We need G > 2 in the proof of Lemma 4.5, that is, in the first application. So, it is possible to include the case G = 2 if we do another kind of applications.
is a sequence of Borel sets such that the Lebesgue measure |A n | ≤ R, for all n ∈ N and some r > 0, then In order to treat problem (1.1), let us consider the space: , for all u ∈ X.
This permits us to construct a suitable space endowed with the norm From now on, we shall denote the duality pairing between X and its dual space X * by ·, · X . The following lemma will be helpful in the sequel.
Proof. The proof is similar to that in [5]. (i) The functional ρ is of class C 1 and for all u, v ∈ X we have Proof. The proof is similar to that in Bahrouni, Rȃdulescu & Winkert [5]. Now, we establish the following compactness result. Proof. Let (u n ) be an arbitrary bounded sequence in D 1,G a (R N ). Fix R > 0, s(·) ∈ (1, G * (·)), and set B(0, R) = {x ∈ R N , |x| ≤ R}. We note that u n ⇀ u weakly in L G * (·) (R N ). Thus, for every ϕ ∈ C ∞ 0 (R N ), one has and so (u n ) is bounded in W 1,G 0 (B(0, R)). Therefore, there exist a subsequence (u n k ) and u ∈ W 1,G (B(0, R)), with u = u ↾ B R , such that u n k ⇀ u weakly in W 1,G 0 (B(0, R)). Invoking Theorem 2.4, u n k → u strongly in L s(·) (B(0, R)). Then, taking into account (3.3), we obtain for every ϕ ∈ C ∞ 0 (B(0, R)). This implies that u(x) = u(x) for almost all x ∈ B(0, R), against the fact that u = u ↾ B R . This proves the claim. Hence (u n ) weakly converges to . Applying Theorem 2.4 again, (u n ) strongly converges to u in L s(·) (B(0, R)). This completes the proof of Lemma 3.4. Now, we are ready to prove our compact embedding result in the whole space R N . Let us define, for every s(·) ∈ C + (R N ), the following Lebesgue space K (R N ), for every s(·) ∈ (G(·), G * (·)). Proof. Fix s(·) ∈ (G(·), G * (·)) and ǫ > 0. It is easy to see that Thus, there exist 0 < t 0 < t 1 and a positive constant C > 0 such that Let (u n ) ∈ X be a sequence such that u n ⇀ u in X. It is easy to see that (A(u n )) n is bounded in R. Denoting R n = {x ∈ R N , t 0 < |u n (x)| < t 1 }, we get sup n∈N |A n | < +∞. Hence, by (K), there exists a positive radius r > 0 such that Now, since s(·) ∈ (1, G * (·)) and K ∈ L ∞ (R N ), we deduce, that Here we used Lemma 3.4. Combining (3.4) and (3.5), we conclude for ǫ > 0 small enough, that Consequently, using Proposition 2.1, we infer that K (R N ) for every s(·) ∈ (G(·), G * (·)). This completes the proof of Proposition 3.5.

A nonlinear problem driven by ∆ G,a
As an application of the previous abstract results, the main result of this section concerns the study of both nonvariational and singular aspects of problem (1.1).

Nonvariational case.
In this paragraph, we work under conditions introduced in Proposition 3.5. We are mainly concerned with the following equation The hypotheses on functions f and r are the following: z ∈ R N and for all s ∈ R where • γ(·) ∈ C + (R N ) and γ(·), γ(·) γ(·)−1 ∈ (G(·), G * (·).   The proof of Theorem 4.3 relies on the topological degree theory of (S + )-type mappings. Define the operator L : X → X * by For r > 0, we denote by B r the open ball centered at the origin and of a radius r. Applying the Hölder inequality, we get Again, by Hölder's inequality, we obtain Br |a(z)||u n − u|dz ≤ a G(·) G(·)−1 (Br) u n − u G(·) . Using Lemma 3.4, it follows that   In the same way, we prove that  In what follows, we show that Invoking the Hölder inequality and Proposition 2.1, we obtain

Now, from conditions (H 2 ), (H 3 ) and the Hölder inequality, we deduce that
which, by Proposition 3.5, implies that Consequently, from (4.8) and (4.9), we conclude that Again, using the same argument, we show that This proves Claim 2. Finally, from Claim 1, Claim 2 and (4.2), we infer that Hence, by Lemma 3.3, we get our desired result.
Proof. Let u ∈ X be such that u > 1. Hence, in view of Lemmas 2.3 and 3.2 and Proposition 3.5 and the Hölder inequality, we obtain where C is a positive constant. Choosing u = R large enough, we deduce from the last inequality that L(u), u > 0 for all u ∈ X such that u = R.
This completes the proof of Lemma 4.5.
Proof of Theorem 4.3 completed. It is clear that L is also demicontinuous and bounded. Then, in light of Lemmas 4.4 and 4.5 and using the topological degree theory for (S + ) type mappings, we conclude that where R is defined in Lemma 4.5. Therefore the equation L(u) = 0 has at least one solution u ∈ B(0, R). From assumption (H 1 ), we can conclude that u is a nontrivial weak solution of equation (4.1). This completes the proof of Theorem 4.3.

Singular problem.
In this subsection, we work under conditions introduced in Proposition 3.5. Here, we are interested in weak solutions to nonlinear singular problems. Precisely, we study the following singular double phase equation where σ(·) ∈ C 1 (R N ), 0 < σ(·) < 1. The assumption on function b is the following: Definition 4.6. We say that u ∈ X \ {0} is a weak solution of problem (4.10) Our main result is the following existence theorem. To prove the above theorem, we first consider a perturbation of (4.10) which removes the singularity. So, we consider the following approximation of problem (4.10): (4.11) The main way to deal with this problem is the topological approach. So, given f ∈ L G(·) (R N ), f ≥ 0 and ǫ ∈ (0, 1), we consider the following equation: (4.12) For the above problem we have the following result. Using the Simon inequality (see [40]), B G is bounded, continuous, strictly monotone. Then we consider the map A G : X → X * defined by for all u, v ∈ X. Using the same argument, we can deduce that this operator is bounded continuous, strictly monotone. It follows that the operator V G = A G + B G is bounded continuous, strictly monotone (thus, maximal monotone, too). On the other hand, in light of Lemma 3.3, we have that V is coercive. We know that a maximal monotone coercive operator is surjective. , 0)). Thus, using the fact that (f (·) + ǫ) > 0, we obtain that v ǫ is a nonnegative and v ǫ = 0. Moreover, the strict monotonicity of V (.) implies that this solution is unique. Finally, the anisotropic maximum principle of Zhang [41] implies that v ǫ > 0. This completes the proof of Proposition 4.8.
Using Proposition 4.8, we can define the solution map L ǫ : L G(·) (R N ) → L G(·) (R N ) for problem (4.12) by Proof. In view of Proposition 4.8, we have In (4.14) we choose h = v ǫ = L ǫ (f ) ∈ X and we obtain which implies that there exists a positive constant C such that In what follows, we prove that L ǫ (.) is continuous. To this end, let f n → f in L G(·) (R N ). From (4.15) we have that (L ǫ (f n ) = u n ) n∈N is bounded in X. So, we may assume that u n ⇀ u in X.
Thus, using conditions (B) and (K), we infer that This leads to (4.16) lim Here we used Proposition 3.5. On the other hand, we have hdz, for all h ∈ X and n ∈ N.
So, by Lemma 3.3, If in (4.17) we pass to the limit as n → +∞ and use (4.18), we obtain that and L ǫ (f ) = u.
This proves that L ǫ (.) is continuous. The continuity of L ǫ (.), together with (4.15) and Proposition 3.5, permit the use of the Schauder-Tychonov fixed point theorem (see [30]) and we find u ǫ ∈ X such that L ǫ (u ǫ ) = u ǫ and so, u ǫ is a positive solution of (4.11).
Next we show the uniqueness of this solution. Suppose that v ǫ ∈ X is another positive solution of (4.11). We have Interchanging the roles of u ǫ and v ǫ in the above argument, we also have that v ǫ ≤ u ǫ , therefore u ǫ = v ǫ . This completes the proof of Proposition 4.9. Now, we prove the following monotonicity property of the map ǫ → u ǫ . Proof. Let 0 < ǫ ′ < ǫ ≤ 1 and let u ǫ , u ǫ ′ ∈ X be the corresponding unique positive solutions of problem (4.11).
We define the following function: We set F ǫ (z, x) = x 0 f ǫ (z, s)ds and we introduce the functional I ǫ : X → R defined by Evidently I ǫ is of class C 1 . If u ∈ X is large enough, we have G(·)−1 ǫ γ + . Therefore, I ǫ is coercive. On the other hand, by condition (B), we can prove that I ǫ is weakly lower semicontinuous. Then, invoking the Weierstrass-Tonelli theorem, we can find v ǫ ∈ X such that I ǫ (v ǫ ) = inf u∈X I ǫ (u).
In (4.20) we choose h = u n and use Proposition 4.10. Hence which implies that (u n ) is bounded in X. Therefore, we can find u ∈ X such that u n ⇀ u in X and u n → u a.e in R N .
Then, by (4.21) and (4.22) and passing to the limit as n → +∞ in (4.20), we conclude that This proves that u is a weak solution of problem (4.10). Since u 1 ≤ u n for all n ∈ N, we have u > 0. Finally, we show the uniqueness of this positive solution. So, suppose that v ∈ X is another positive solution of equation (4.10). As in the proof of Proposition 4.10, we can prove that u = v. The proof of Theorem 4.7 is now complete.