Quasilinear Dirichlet Problems with Degenerated p -Laplacian and Convection Term

: The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p -Laplacian and containing a convection term. The presence of the degenerated operator forces a substantial change to the functional setting of previous works. The existence and location of solutions through a sub-supersolution is established. The abstract result is applied to ﬁnd nontrivial, nonnegative and bounded solutions.


Introduction
In this paper, we study the following quasilinear elliptic problem      −div(a(x)|∇u| p−2 ∇u) = f (x, u, ∇u) in Ω u = 0 on ∂Ω (P) on a bounded domain Ω ⊂ R N with N ≥ 2 and p ∈ (1, N).We assume that the boundary ∂Ω of Ω is locally Lipschitzian, i.e., each point of ∂Ω has a neighborhood whose intersection with ∂Ω is the graph of a Lipschitz continuous function.Throughout the text we denote by | • | and • the standard Euclidean norm and scalar product on R N , respectively.A main feature of the present work is that the leading part of the equation in (P) is the differential operator in divergence form div(a(x)|∇u| p−2 ∇u) known as the degenerated p-Laplacian with the weight a ∈ L 1 loc (Ω).It is supposed that the function a be positive almost everywhere in Ω and that the following condition holds In the case where a(x) ≡ 1 we recover the ordinary p-Laplacian.Various examples of useful weights meeting the requirement (1) are given in [1].For instance, it is obvious that defining a(x) = dist(x, S) for x ∈ Ω, with a nonempty closed subset S of ∂Ω, one obtains a function a on Ω for which (1) holds true with any listed s.
The natural space associated with problem (P) is W 1,p 0 (a, Ω) that is the closure of C ∞ 0 (Ω) in the weighted Sobolev space W 1,p (a, Ω).In Section 2 we briefly survey the spaces W 1,p (a, Ω) and W Corresponding to the constant s in (1) we set p s = ps s + 1 and the Sobolev critical exponent p * s = N p s N−p s (we note that 1 ≤ p s < N).There is a continuous embedding W 1,p (a, Ω) → L p * s (Ω), so a continuous embedding L (p * s ) (Ω) → (W 1,p 0 (a, Ω)) * , where (p * s ) stands for the Hölder conjugate of p * s , i.e., (p * s ) = p * s p * s −1 .In order to handle problem (P) the idea is to arrange that the right-hand side f (x, u, ∇u) become an element of L (p * s ) (Ω), which basically will be achieved through an adequate growth condition (see assumption (H)).We emphasize that the nonlinearity f (x, u, ∇u) depends on the solution u and on its gradient ∇u, which generally makes the variational methods be ineffective.Such a term f (x, u, ∇u) is often called convection.It is expressed by means of a function f : The goal of our work is to build a systematical approach to problem (P) via the method of sub-supersolution.It is for the first time when the method of sub-supersolution is implemented for problem (P) involving the degenerated p-Laplacian and related convection.In this respect, the functional setting is adapted to the novel situation of degenerated operators relying in an essential way on the associated exponent p s .For results on the method of sub-supersolution applied to problems exhibiting convection terms but not driven by degenerated differential operators we refer to [2][3][4][5][6].
By a (weak) solution to problem (P) we mean a function u ∈ W A function u ∈ W 1,p (a, Ω) is called a subsolution for problem (P) if u ≤ 0 on ∂Ω (in the sense of traces), Corresponding to a subsolution u and a supersolution u with u ≤ u a.e. in Ω we can consider the ordered interval The following hypothesis for f (x, s, ξ) is adapted to an ordered sub-supersolution u ≤ u.Hypothesis 1.Given an ordered sub-supersolution u ≤ u for problem (P), the Carathéodory function f : with a function σ ∈ L ps r (Ω) and constants b > 0 and r ∈ (0, p s (p * s ) ).
According to Hypothesis 1 we have thus the integrals in the definitions above exist since Under Hypothesis 1, our main result establishes the existence of a weak solution to problem (P) with the additional location property u ∈ [u, u].We stress that this location property represents a significant qualitative information for the solution giving actually a priori estimates for it.As an application we prove the existence of a nontrivial nonnegative solution for a class of problems of type (P).The applicability of the stated result is demonstrated by an example.

Preliminary Material
The notation |Ω| stands for the Lebesgue measure of the bounded domain Ω in R N .In this section we discuss a few facts about the degenerated p-Laplacian entering problem (P).More details can be found in [1].
We note that (1) implies Indeed, it is seen that Ω a(x) since according to (1) one has s ≥ 1 p−1 and a −s ∈ L 1 (Ω).The weighted Sobolev space W 1,p (a, Ω) consists of all the functions u ∈ L p (Ω) for which a 1 p |∇u| ∈ L p (Ω).It is endowed with the norm becoming a uniformly convex Banach space (due to the preceding property of the weight a(x), see ( [1], [Theorem 1.3])), thus reflexive, that contains C ∞ 0 (Ω).The space W 1,p 0 (a, Ω) is the closure of C ∞ 0 (Ω) with respect to the norm • 1,p,a .There is an extensive literature devoted to the weighted Sobolev spaces including embeddings and traces related to different boundary value problems (see, e.g., [1,7,8]).The results depend strongly on what type of weight is used, generally attempting reduction to nonweighted spaces.As described below, under assumption (1), we can embed the space W 1,p (a, Ω) into the ordinary Sobolev space W 1,p s (Ω), hence automatically having the trace (note the boundary ∂Ω is Lipschitz).This fact is needed in the definition of the sub-supersolution.
From (1) it is known that s ≥ 1 p−1 , so one has p s ≥ 1 and the continuous embedding which is relation (1.22) in [1].More precisely, observing that p > p s , through Holder's inequality and (1) we get for all u ∈ W 1,p (a, Ω).As a consequence of the above inequality, we can endow W for which it holds The Sobolev embedding theorem ensures the continuous embedding W Hence there exists a constant T 0 > 0 such that The best embedding constant T 0 has been estimated by Talenti [9] as follows , where Γ is the Euler function Moreover, by the Rellich-Kondrachov compact embedding theorem, if 1 ≤ r < p * s then the embedding W 1,p s 0 (Ω) → L r (Ω) is compact.By (7) and Hölder's inequality we infer that for every u ∈ W 6) and ( 8) we arrive at for all u ∈ W 1,p 0 (a, Ω) and r ∈ [1, p * s ], with the constant We readily check that the operator A in (10) is well defined noticing by means of Hölder's inequality that for all u, v ∈ W Important properties of the operator A introduced in (10) are listed in the statement below.Proposition 1. Assume that the measurable function a : Ω → R satisfies condition (1).Then the (negative) degenerated p-Laplacian A : W 1,p 0 (a, Ω) → (W 1,p 0 (a, Ω)) * defined by (10) has the following properties: A is a bounded operator in the sense that it maps bounded sets to bounded sets; (ii) A is a coercive operator, i.e., (iv) A has the S + property meaning that any sequence is strongly convergent.
Proof.(i) From ( 10) and ( 11) we infer that We obtain whence A is bounded.
(ii) By (10) we have that Taking into account that p > 1, it follows that the operator A is coercive.
(iii) In view of the strict monotonicity of the mapping ξ → |ξ| p−2 ξ on R N , it turns out 0 (a, Ω) and (12).Using the monotonicity of the operator A and (12) we have Through Hölder's inequality we obtain from which we find that lim n→+∞ u n = u .Due to the uniform convexity of W We also need the first eigenvalue λ 1 of the operator A : W admits a nontrivial solution called eigenfunction corresponding to the first eigenvalue λ 1 .

Main Results
Our main abstract result provides the existence of a solution to problem (P) and its location within the ordered interval determined by a sub-supersolution.Theorem 1.Let the weight a ∈ L 1 loc (Ω) fulfill the requirement (1) and assume that the condition (H) for a subsolution u and a supersolution u with u ≤ u a.e. is satisfied.Then problem (P) possesses at least a solution u ∈ W 1,p 0 (a, Ω) with the location property u ≤ u ≤ u for a.e.x ∈ Ω.
Proof.By means of the given sub-supersolution u ≤ u for problem (P), we introduce some related mappings.The cut-off function π : where s and r are the constants given in (1)  (Ω).Moreover, proceeding as in [4], we can establish that (Ω).Therefore, the mapping Π : W s owing to the assumption r ∈ (0, p s (p * s ) ) in (H)).Hypothesis (H) and ( 5) imply that the Nemytskij operator We also make use of the truncation operator T : W 1,p 0 (a, Ω) → W 1,p (a, Ω) given by for all u ∈ W 1,p 0 (a, Ω) and a.e.x ∈ Ω.It is a continuous and bounded mapping (in the sense that it maps bounded sets to bounded sets).Notice that its range lies in [u, u], so T can be composed with the operator N f .Now we consider for every λ > 0 the operator Explicitly, it reads as From Proposition 1(i) it is known that the operator A : W 1,p 0 (a, Ω) → (W 1,p 0 (a, Ω)) * is bounded, while the above comments demonstrate that the operators Π, N f and T are all of them bounded.Therefore from (18) we infer that the operator The sequence {Π(u Consequently, complying with (18), we see that (20) reduces to (12).This, in conjunction with the weak convergence u n u, enables us to apply Proposition 1(iv) ensuring that the strong convergence u n → u in W 1,p 0 (a, Ω) holds.From the strong convergence a(•) This amounts to saying that Au n Au in (W Again, from the strong convergence a(•) as n → ∞.Taking into account the continuity of the mappings Π and N f • T, we have and as n → ∞, for every λ > 0. We can conclude that A λ : W The next step in the proof is to show that the operator 16), (19) and Hypothesis 1 that for all u ∈ W 1,p 0 (a, Ω).Now we estimate the last term in (21) based on the fact that by (5) we know that ∇u ∈ (L p s (Ω)) N , and so ∇(Tu) ∈ (L p s (Ω)) N .Using the definition of Tu in (17), Hölder's inequality and the continuous embedding in (9) it turns out that with a constant c 1 > 0. We can insert the preceding inequality in (21) to derive with a constant c 2 > 0. The Hölder's and Young's inequalities in conjunction with embedding (5) imply , with constants c 3 > 0 and c 4 > 0. Then (22) entails for all u ∈ W 1,p 0 (a, Ω).Recalling from (16) that b 1 > 0, we can choose λ > 0 so large to have λb 1 > bc 4 .Hence due to p > p s ≥ 1 (see (1)), (23) yields the coercivity of A λ , i.e., lim We have shown that the nonlinear operator A λ : W 1,p 0 (a, Ω) → (W 1,p 0 (a, Ω)) * is bounded, pseudomonotone and coercive provided λ > 0 is sufficiently large.Therefore, for such an A λ we can apply the main theorem of pseudomonotone operators (see, e.g., ([2], Theorem 2.99)) ensuring that there exists a solution u ∈ W 1,p 0 (a, Ω) to the equation Fix an admissible λ > 0 as pointed out above.We are going to prove that u ∈ W 1,p 0 (a, Ω) resolving ( 24) is a weak solution of the original problem (P), which means that (2) is satisfied.To this end, notice that (19) and (24) yield We proceed by comparing u with the subsolution u and supersolution u postulated in assumption (H).We claim that u ≤ u a.e. in Ω. Towards this, it can be readily checked that (u − u) + = max{u − u, 0} ∈ W 1,p 0 (a, Ω), where the condition u ≥ 0 on ∂Ω in the sense of traces is essentially used.Thus, we can insert v = (u − u) + in (25) and (4) which gives From ( 26) and ( 27), by subtraction we are led to By ( 14), (17), and the preceding inequality we get Since the function a(x) is positive almost everywhere in Ω and the mapping ξ → |ξ| p−2 ξ on R N is monotone, we arrive at Therefore, the Lebesgue measure of the set {u > u} is zero, i.e., u ≤ u a.e. in Ω.
Similarly, we can prove that u ≤ u a.e. in Ω.Specifically, relying on the condition u ≤ 0 on ∂Ω (in the sense of traces), it holds (u − u) + = max{u − u, 0} ∈ W and Arguing as before, we deduce from (28), (29), (14), and (17) the following estimate At this point, the positivity of the function a(x) on Ω and the monotonicity of the mapping ξ → |ξ| p−2 ξ on R N confirm that {u>u} (u(x) − u(x)) ps ps −r dx ≤ 0, from which we can readily derive that u ≤ u a.e in Ω.
Based on the enclosure property u ≤ u ≤ u a.e. in Ω, it follows through (17) that T(u) = u and through (14) that Π(u) = 0.As a result, (25) takes the form of (2), thus the proof is complete.Now we present an application of Theorem 1 describing how the existence of a nontrivial nonnegative solution can be established by effectively determining a sub-supersolution.In the sequel, by λ 1 we denote the first eigenvalue of problem (13) (see Section 2).Theorem 2. Let the weight a ∈ L 1 loc (Ω) fulfill the requirement (1).Assume that the Carathéodory function f : Ω × R × R N → R satisfies the conditions: (j) there is a constant µ > 0 such that (jj) there is a constant C > 0 such that f (x, C, 0) ≤ 0 for a.e.x ∈ Ω; (jjj) there are a function σ ∈ L ps r (Ω) and constants b > 0 and r ∈ (0, p s Then problem (P) has a nondegenerate, nonnegative and bounded weak solution u ∈ W 1,p 0 (a, Ω) satisfying the estimate u ≤ C.
Let us fix an ε > 0 for which (30) and (32) are fulfilled.We claim that u = εu 1 is a subsolution to problem (P).Indeed, by (13) with u 1 in place of u and (31) we note that for all v ∈ W 1,p 0 (a, Ω), v ≥ 0 a.e. in Ω, which proves the claim.It is clear from (32) that u(x) ≤ u(x) for a.e. in Ω. Assumption (jjj) ensures that the growth condition required in Hypothesis 1 of Theorem 1 holds true.Therefore, all the hypotheses of Theorem 1 are verified, which permits the conclusion that there exists a solution u ∈ W 1,p 0 (a, Ω) of problem (P) within the ordered interval [u, u].Since the function u = εu 1 is nontrivial and nonnegative, and u ≥ u, we have that u is nontrivial and nonnegative, whereas u ∈ [u, u] renders the boundedness of u and the a priori estimate u ≤ C. The proof is complete.
We end the paper with a simple example for which Theorem 2 applies.
Example 1. Fix a positive weight a ∈ L 1 loc (Ω) with the property (1).Let the function f : Ω × R × R N → R be defined by with some r ∈ [1, p s (p * s ) ) and ρ ∈ L ∞ (Ω) satisfying ρ(x) ≥ λ 1 for a.e.x ∈ Ω.It follows that f is a Carathéodory function for which conditions (j) − (jjj) in Theorem 2 are verified.Precisely, condition (j) holds with µ = 1 because ρ(x) ≥ λ 1 , condition (jj) holds with C = 2, and condition (jjj) is fulfilled with the given r.Hence Theorem 2 applies to problem (P) whose equation has the right-hand side expressed with the function f (x, t, ξ) given above.