The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

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                </mml:math></jats:alternatives></jats:disp-formula> on a bounded domain Ω in <jats:inline-formula><jats:alternatives><jats:tex-math>${\mathbb{R}}^{N}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                </mml:math></jats:alternatives></jats:inline-formula> boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.</jats:p>


Introduction
Partial differential equations involving the p(x)-Laplacian arise, for instance, in nonlinear elasticity, fluid mechanics, non-Newtonian fluids and image processing. Because of the broad set of applications, several studies related to the p-Laplacian, or in general the p(x)-Laplacian, operator have been reported (see for instance [4, 6, 16-18, 20-27, 29, 31, 32] and the references therein). One of the approaches to study the existence of solutions of elliptic partial differential equations is the sub-supersolution method. Some problems such as have been studied via the sub-supersolution method (see [1,11,33]). Also, one may refer to [5,7,8,10,[12][13][14]28] for other similar model problems.
Recently, the existence of solutions for nonlocal problems involving the p(x)-Laplacian operator in , wherep(x) u := -div(|∇u| p(x)-2 ∇u), has been studied [14,15] via a new sub-supersolution method. In [14], the problem (1.1) for p(x) ≡ 2 (i.e.,p(x) = -) is considered. They study the existence of a weak solution for three problems (the sublinear problem, the concaveconvex problem and the logistic equation). Their arguments are mainly based on the existence of the first eigenvalue of the Laplacian operator (-, H 1 0 ( )). The p(x)-Laplacian operator, in general, has no first eigenvalue, that is, the infimum of the eigenvalues equals 0 (see [19]).
The lack of the existence of the first eigenvalue implies a considerable difficulty when dealing with boundary value problems involving the p(x)-Laplacian by using the subsupersolution method. Papers that consider such problems by using the mentioned method are rare in the literature. Among such works we mention papers such as [2,3,24,34].
In this paper, we are interested in the nonlocal problem in , where is a bounded domain in R N (N > 1) with C 2 boundary, |.| L m (x) is the norm of the space L m(x) ( ), r, q, s, α, γ : → [0, ∞) are measurable functions and A, f , g : × R → R are continuous functions satisfying certain conditions. To be more specific about the structure of the operator in (1.2), we deal with function a : R + → R + of class C 1 satisfying the following conditions: (a1) There exist constants k 1 , k 2 , k 3 , k 4 ≥ 0, 1 < p ≤ l < N such that (i) If a(t) = 1, we obtain the p-Laplacian that is in , in , with l = p, k 1 + k 2 = 2 and k 3 + k 4 = 1.
with p ≥ 2 we obtain the generalized p-mean curvature operator, that is in , in , with l = p, k 1 + k 2 = 1 and k 3 + k 4 = 2.
The main aim of this paper is to prove the existence of a weak positive solution for (1.2) via the sub-supersolution method.
In the next section we present some preliminaries to construct a function space where the solution of (1.2) makes sense.

Function spaces
To study the solution of problem (1.2), we need to introduce a suitable function space, where the solution makes sense. To do this, we recall some facts about the known spaces L p(x) ( ), W 1,p(x) ( ) and W 1,p(x) 0 ( ) (see [18,30] and the references therein for more details).
For p ∈ L ∞ + ( ), the generalized Lebesgue space and its norm are defined by respectively. It is easy to see that the space (L p(x) ( ), |.| L p(x) ) is a Banach space. Set where m ∈ L ∞ ( ).
For all u, u n ∈ L p(x) ( ), n ∈ N, the following assertions hold: The following statements hold: One can define the generalized Sobolev space

Proposition 2.4 Let ⊂ R N be a bounded domain and consider p, q
The following statements hold: continuous and compact.
Note that u := |∇u| L p(x) defines a norm in W 1,p(x) 0 ( ) that is equivalent to the norm . * (by (i) of Proposition 2.4).
The following result is contained in [ (2.1) Here, C * and C * are positive constants dependent only on p + , p -, N , | | and C 0 , where C 0 is the best constant of the embedding W 1,1 Regarding the function z λ of the previous result, it follows from [17, Theorem 1.2] and [21, Theorem 1] that z λ ∈ C 1 ( ) with z λ > 0 in .

Weak positive solution
In this section we prove the existence of a weak positive solution of problem (1.2), via the sub-supersolution method. In fact, we prove there exists u ∈ [u, u] as the weak solution of (1.2), where u and u are subsolution and supersolution, respectively. To do this, we state the definition of a solution of the problem (1.2).
Definition 3. 2 We say that (u, u) is a sub-supersolution pair for for all w ∈ [u, u].
We will assume that the functions r, p, q, s, α and γ satisfy the following hypotheses: and α, γ ∈ L ∞ ( ) satisfy The main result of this section is to prove the existence of at least one solution of (1.2).

Theorem 3.3
Suppose that r, p, q, s, α and γ satisfy (H 0 ), a : R + → R + is a C 1 function satisfying (a1)-(a3), (u, u) is a pair of sub-supersolution for (1.2) with u > 0 a.e. in , To prove this theorem, we need to prove some facts in the series of lemmas. First, we study the existence and uniqueness of the solution From (a 1 ) the functional (3.4) is well defined and thus I ∈ C 1 (W 1,q 0 ( ), R). Also, I is strictly convex and weakly lower semicontinuous by (a 2 ). Note that (a 1 ), |G(v)| ≤ K 0 and Hölder's inequality imply for some constant C > 0 and all u ∈ W 1,l(x) 0 ( ) with ρ(|∇u|) ≥ 1, which shows that I is coercive. Hence, I has a unique critical point (a global minimizer), which is the unique solution to (3.3).

Lemma 3.5 Under the hypotheses of Theorem 3.3, define the operator T
where u, u ∈ L ∞ ( ) and Tu ∈ [u, u]. Moreover, let the operator H : [u, u] → L p (x) ( ) be defined by

where p (x) = p(x) p(x)-1 and |.| L m(x) denotes the norm of L m(x) ( ). Then, the operators T, H and u → HoT(u) are well defined and u → HoT(u) is continuous.
Proof Similar to [14] one can show the operators H and u → HoT(u) are well defined and u → HoT(u) is continuous.   Thus, the sequence (u n ) is bounded in L p(x) ( ) and by Hölder's inequality, we have where the constant C does not depend on n ∈ N. The continuity of HoT shows which implies the continuity of S.
We recall a special case of the far-reaching Leray-Schauder theorem called Schaefer's Fixed Point Theorem.

Theorem 3.7 Let S be a continuous and compact mapping of a Banach space X into itself, such that the set
is bounded. Then, S has a fixed point. Proof Since we can apply Theorem 3.7, we need to show that there exists R > 0 such that if u = S(u) with ∈ [0, 1], then |u| L p(x) < R. In fact, if = 0, then u = 0. Suppose that = 0. In this case, we have S(u) = u and such an equality implies the identity for all ϕ ∈ W 1,p(x) 0 ( ). Using the test function ϕ = u and by the embedding where C > 0 is a constant that does not depend on u and . If |∇u| L p(x) > 1, by Poincaré's inequality and Proposition 2.1, |u| where C is a constant that does not depend on u and . Now, we can prove Theorem 3.3.
Proof Lemma 3.5 shows the operators H and u → HoT(u) are well defined and u → HoT(u) is continuous.

Applications
The main goal of this section is to apply Theorem 3.3 to some classes of nonlocal problems.

A generalization of the logistic equation
Here, we study a generalization of the classic logistic equation as follows: in , where the function A(x, t) satisfies Note that there is a constant C > 0 such that Then, Thus, for each λ ≥ λ 0 μ 0 and w ∈ [ϕ, θ ], we obtain Since f (θ ) = 0, it follows that (z 0 , θ ) is the sub-supersolution pair for (4.1) and the result is proved.

A sublinear problem
Here, we use Theorem 3.3 to study the nonlocal problem in , (4.2) The above problem in the case p(x) ≡ 2, was considered recently in [14]. The result of this section generalizes [14,Theorem 3] and [15,Theorem 4.1].
Fix k > 0 satisfying (4.8). Let λ > 1 such that (4.3) occurs and where K > 1 is a constant that does not depend on k and λ (see Lemma 2.7). Thus, for all w ∈ [μφ, z λ ], we have in .