Noncoercive quasilinear elliptic operators with singular lower order terms

We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.

The feature of Problem (1.6) is the lack of coercivity for the operator (1.4) and the singularity in the lower order term due to property of b. It is well known that, if the operator in (1.4)-(1.5) is coercive, then a solution to problem (1.6) exists. It can for instance be shown by monotone operator theory [27,6,7,4].
On the other hand, the existence of a bounded solution can be expected when Φ and b are sufficiently smooth. For example, in the model case and even for the corresponding linear case, a solution to Problem (1.6) is bounded whenever Φ and b are in W −1,p ′ (Ω) and L p ′ (Ω, R N ), respectively, with p ′ > N/(p − 1) (see [32,17]).
The space L N,∞ (Ω) is slightly larger than L N (Ω). Nevertheless, there are essential differences between the case b ∈ L N (Ω) ( [5,10]), or even b ∈ L N,q (Ω) ( [29]) with N q < ∞, and the case b ∈ L N,∞ (Ω). In L N,∞ (Ω) bounded functions are not dense. Furthermore, in L N,∞ (Ω) the norm is not absolutely continuous, namely a function can have large norm even if restricted to a set with small measure.
Our first result is the following then Problem (1.6) admits a solution.
Here S N,p denotes the Sobolev constant of Theorem 2.1 below. As an illustration, we state the following immediate consequence. We denote by L N,∞ 0 (Ω) the closure of L ∞ (Ω) in L N,∞ (Ω). (Ω) contains for example all Lorentz spaces L N,q (Ω), for 1 < q < +∞, see Subsection 2.1.
It is not clear if the bound in (1.9) is sharp. However, existence of a solution could fail if no restriction on the distance is imposed, even in the liner case, as the Example 1 in Subsection 2.3 shows (see also [19]). Notice that condition (1.9) does not imply the smallness of the norm of b in L N,∞ (Ω) (see [19]).
In the case p = 2 existence results analogous to that of Theorem 1.1 have been proved in [12,19,33] and in [9,31,34] when the principal part has a coefficient bound in BMO (i.e. the space of functions of bounded mean oscillation). We explicitly remark that in this context the operator (1.4) has the same integrability properties of the principal part (see also [23]). The evolutionary counterpart of Problem (1.6) has been studied in [11]. Other related results can be found in [1,24,28].
An additional difficulty in proving Theorem 1.1 lies in the lack of compactness that the operator exhibits in the case b ∈ L N,∞ (Ω), in contrast with the case b ∈ L N (Ω) (see Example 3 in Subsection 2.3). In order to overcome this issue, first we consider the case in which b ∈ L ∞ (Ω). Under this assumption, we deduce the existence of a solution to Problem (1.6) by means of Leray-Schauder fixed point theorem. The a priori estimate required follows from a lemma that could be interesting in itself (see Lemma 3.1 below).
In order to reduce the general case b ∈ L N,∞ (Ω) to the previous one, we consider a sequence of approximating problems, defined essentially by truncating the vector field A = A(x, u, ξ) in the u-variable. A bound on the sequence of the solutions is achieved due to the assumption (1.9).
We emphasize that our result is also new when b ∈ L N (Ω), in the sense that our approach allows us to treat the general nonlinear operator in (1.6).
Finally, by testing the problems with a suitable admissible test functions, we show that the sequence of solutions to the approximating problems is compact and its limit is a solution to the original problem (1.6).
In Section 5, we show that our approach is robust enough to handle also the corresponding obstacle problem. We prove an existence result in the same spirit of [18] (where the case p = 2 is taken into account).
In Section 6 we present a regularity result. When b ∈ L N (Ω), the study of the higher integrability of a solution to (1.6) has been developed in [13,14] by using the theory of quasiminima. Local summability properties have been recently considered in [8,22] in the linear case. Here, following [17], we prove higher summability of a solution u to (1.6) according to that of the data Φ and ϕ.

Notation and function spaces.
Let Ω be a bounded domain in R N . Given 1 < p < ∞ and 1 q < ∞, the Lorentz space L p,q (Ω) consists of all measurable functions f defined on Ω for which the quantity is finite, where Ω t = {x ∈ Ω : |f (x)| > t} and |Ω t | is the Lebesgue measure of Ω t , that is, λ f (t) = |Ω t | is the distribution function of f . Note that · p,q is equivalent to a norm and L p,q becomes a Banach space when endowed with it (see [30,3,16]). For p = q, the Lorentz space L p,p (Ω) reduces to the Lebesgue space L p (Ω). For q = ∞, the class L p,∞ (Ω) consists of all measurable functions f defined on Ω such that and it coincides with the Marcinkiewicz class, weak-L p (Ω). For Lorentz spaces the following inclusions hold As it is well known, L ∞ (Ω) is not dense in L p,∞ (Ω). For a function f ∈ L p,∞ (Ω) we define In order to characterize the distance in (2.2), we introduce for all k > 0 the truncation operator It is easy to verify that We denote by L p,∞ 0 (Ω) the closure of L ∞ (Ω). We have (see [21, Clearly, for 1 q < ∞ we have L p,q (Ω) ⊂ L p,∞ 0 (Ω), that is, any function in L p,q (Ω) has vanishing distance zero to L ∞ (Ω). Indeed, L ∞ (Ω) is dense in L p,q (Ω), the latter being continuously embedded into L p,∞ (Ω). Actually, the inclusion also follows from (2.4), since λ f (t) = |Ω t | is decreasing and hence the convergence of the integral at (2.1) implies the condition on the right of (2.4).
Assuming the origin 0 ∈ Ω, a typical element of L N,∞ (Ω) is b(x) = B/|x|, with B a positive constant. An elementary calculation shows that where ω N stands for the Lebesgue measure of the unit ball of R N .
The Sobolev embedding theorem in Lorentz spaces reads as 30,2,20]). Let us assume that 1 < p < N, 1 q p, then every function g ∈ W 1,1 0 (Ω) verifying |∇g| ∈ L p,q (Ω) actually belongs to L p * ,q (Ω), where p * = N p N −p and g p * ,q S N,p ∇g p,q where S N,p is the Sobolev constant.

2.2.
A version of the Leray-Schauder fixed point theorem. We shall use the well known Leray-Schauder fixed point theorem in the following form (see [15,Theorem 11.3 pg. 280]). A continuous mapping between two Banach spaces is called compact if the images of bounded sets are precompact.
Theorem 2.2. Let F be a compact mapping of a Banach space X into itself, and suppose there exists a constant M such that x X < M for all x ∈ X and t ∈ [0, 1] satisfying x = tF (x). Then, F has a fixed point.

Critical examples.
Our first example shows that the only assumption that b ∈ L N,∞ (Ω) does not guarantee the existence of a solution to Problem (1.6).

Example 1.
Let Ω be the unit ball. For N 2 < γ + 1 N, the problem does not admit a solution. Assume to the contrary that u is a solution of (2.6). In the right hand side of the equation we recognize that which readly implies v ≡ 0 in Ω, which is clearly not the case.
Next example shows that for the complete operator in general we do not have existence, even in the linear case.  We define a sequence {u n } n∈N setting for x ∈ Ω Observe that u n ∈ W 1,p 0 (Ω) since where ω N denotes the measure of the unit ball of R N . In particular, ∇u n p L p (Ω) is independent of n. On the other hand, a direct calculation shows that Hence, we see that the norm of (b|u n |) p−1 in L p ′ (Ω) is independent of n as well and strictly positive. On the other hand, (b|u n |) p−1 → 0 pointwise in Ω and this readily implies that there is no subsequence of {(b|u n |) p−1 } n∈N strongly converging in L p ′ (Ω).

An elementary lemma.
Lemma 2.1. Assume f n → f a.e. Moreover, let g n , n ∈ N, and g in L q , 1 q < +∞, verify g n → g a.e., |f n | g n a.e., ∀n ∈ N, and Then f n , f ∈ L q and f n → f in L q .
It suffices to apply Fatou lemma to the sequence of nonnegative functions

A weak compactness result
The aim of this section is to establish a weak compactness criterion in the space W 1,p 0 (Ω) that has an interest by itself. Proof. We argue by contradiction and assume B unbounded. Then we construct a sequence {u k } k in B such that as k → ∞. By (3.1) we get, for any k ∈ N and ε > 0 thus ∇T ε u k u k = 0 on the set {x ∈ Ω : |v k (x)| ε}. Dividing (3.3) by u k p we have Now, we let k → +∞. To this end, we note that T ε v k ⇀ T ε v weakly in W 1,p 0 (Ω) and T ε v k → T ε v strongly in L p (Ω). In the left hand side of (3.4), we use semicontinuity of the norm with respect to weak convergence, while in the right hand side we observe that u k −1 → 0. Moreover, if Using Poincaré inequality in the left hand side, this yields ε p |{x : |v| ε}| C ε p |{x : 0 < |v| < ε}| .
Passing to the limit as ε ↓ 0 (assuming (3.5)), we deduce |{x : |v| > 0}| = 0 , that is, v(x) = 0 a.e. Once we know that v k ⇀ 0 weakly in W 1,p 0 (Ω), the above argument (formally with ε = +∞, i.e. without truncating v k ) actually shows that v k → 0 strongly in W 1,p 0 (Ω), compare with (3.6), and this is not possible, as v k = 1, for all k. A which satisfies similar conditions as A, namely Hence, we can consider a quasilinear elliptic operator similar to (1.4) for any w ∈ W 1,p 0 (Ω). The operator at (4.5) is invertible. Indeed, Moreover, the mapping Proof. Existence of a solution is classical, see e.g. [27], [ For the sake of completeness, we prove continuity of the map (4.8). Given v n → v in L p (Ω) and Φ n → Φ in W −1,p ′ (Ω), let u n ∈ W 1,p 0 (Ω) solve The sequence {u n } n is clearly bounded, hence we may assume u n ⇀ u weakly in W 1,p 0 (Ω). Moreover, testing equation (4.9) with u n − u, we have On the other hand, we easily see that A(x, v n , ∇u) → A(x, v, ∇u) strongly in L p ′ (Ω, R N ) and thus (4.10) implies The integrands in (4. By coercivity condition (1.1), we deduce α|∇u n | p A(x, v n , ∇u n )∇u n + (b|v n |) p + ϕ p Trivially Ω (b|v n |) p dx converges to Ω (b|v|) p dx. In view of (4.12), by Lemma 2.1 we get u n → u strongly in W 1,p 0 (Ω), and u solves the equa- In view of Rellich Theorem, we have    6) has a solution u ∈ W 1,p 0 (Ω). Proof. If F is the operator defined in Corollary 4.1, clearly a fixed point of F is a solution to Problem (1.6). To apply Leray-Schauder theorem, we need an a priori estimate on the solution u ∈ W 1,p 0 (Ω) of the equation as t ∈ ]0, 1] varies. By using T σ u with σ > 0 as a test function in (4.14) we get Therefore, using the point-wise condition (4.2) we get (4.16) As 0 < t 1, by Young inequality (4.16) yields The conclusion follows by Lemma 3.1.

4.2.
The approximating problems. For each n ∈ N, we set a.e. x ∈ Ω , and define the vector field The vector field A n has similar properties as A, with b replaced by T n b. More precisely, Applying Proposition 4.2 with A n in place of A, fixed Φ ∈ W −1,p ′ (Ω), we find u n ∈ W 1,p 0 (Ω) such that (4.24) − div A n (x, u n , ∇u n ) = Φ .
Notice that we have, for σ > 0 (4.25) Our next step consists in showing that the sequence {u n } n is bounded in W 1,p 0 (Ω). Let m be a positive integer to be chosen later. For n m we have and hence (4.27) Using Hölder and Sobolev inequalities we get Then (4.26) and (4.27) give (4.28) By our assumption (1.9), the level m can be chosen large enough so that Then, by absorbing in (4.28) the latter term of the right hand side in the left hand side, we get (4.29) for a positive constant C which is independent of n. Now, it is clear that (4.29), via Young inequality, allows us to apply Lemma 3.1, then for a constant M independent of n.
In the model case (1.8), it is easy to show that the operator F defined in (4.13) is compact, also for b ∈ L N (Ω) (see Remark 4.1 below). In the general case, in which b ∈ L N,∞ (Ω) we need more work.
We are going to use Lemma 2.1. To this end, by (4.22) we deduce that for a positive constant C = C(p, β). Hence, we can pass to the limit if 1 < p 2. For p > 2 we choose s satisfying so that ps ′ < N, and we conclude also in this case, further estimating with the aid of Young inequality Now, from (4.33) and (4.34) we get As the integrand is nonnegative, we have (up to a subsequence) a.e. in Ω. Moreover, since γ ′ (u n − u) → 1 a.e. in Ω, the above in turn implies Remark 4.1. We discuss briefly the particular case in which the operator has the form [5]). We can easily show that the operator F defined in (4.13) is compact, also for b ∈ L N (Ω). Indeed, equation (4.7) in this case becomes Defined ϑ n as in (4.18), each mapping where E n = {x ∈ Ω : |b(x)| > n} . Therefore, as n → +∞ we have the convergence being uniform when v varies in a bounded subset of W 1,p 0 (Ω), and compactness is preserved for the limit mapping. An a priori bound for solutions of equation can be easily obtained as above, splitting b ∈ L N (Ω) as for a sufficiently large m. Therefore, in this particular case the existence result of Theorem 1.1 follows simply applying Leray-Schauder fixed point theorem.
In fact, if g ∈ K ψ (Ω), then one can consider the operator defined by the vector fieldÃ (x, u, ξ) := A(x, u + g(x), ξ + ∇g(x)) , satisfying conditions similar to (1.1), (1.2) and (1.3). Now it is clear that, if functionũ ∈ K ψ−g (Ω) satisfies the following variational inequality correspondingly u =ũ+g is a solution to (5.2). Notice that the obstacle function for problem (5.4) is nonpositive, as we are assuming for the original problem. Proof. We follow closely the arguments of Section 4. For each n ∈ N, we consider the function ϑ n as in (4.18) and define the vector fields A n = A n (x, u, ξ) as in (4.20). We consider a sequence of obstacle problems provided by The existence of a solution u n ∈ K ψ (Ω) to (5.5) is proven applying [26, Théorème 8.2, pg. 247] to the operator for a fixed v ∈ W 1,p 0 (Ω), and then using Leray-Schauder Theorem, arguing as in Subsection 4.1. Due to (5.3), for every k > 0 the function is a test function for (5.5). Arguing as in Section 4.2 we obtain u n M with M independent of n (as in (4.30)). Therefore (4.31) holds for some u ∈ W 1,p 0 (Ω). It is clear from (4.31) itself that (5.6) u ∈ K ψ (Ω) As for Theorem 1.1, we shall prove that u is a solution to the original problem (5.2). We proceed as follows. We use in (5.5), where γ(s) = λ arctan(s/λ), for λ > 0, and v ∈ K ψ (Ω) is arbitrary. Note that this is a legitimate test function, that is w ∈ K ψ (Ω). Indeed, on the set where u n v we have γ(u n − v) u n − v and so w v; on the other hand, on the set where u n v we have γ(u n − v) 0 and so w u n . Therefore, from (5.5) we get Following the lines of the proof of Theorem 1.1 (where λ = 1), we get in turn (4.36), (4.38) and finally (4.39). To pass to the limit for fixed general λ > 0 in (5.8), we rewrite it as follows: In the left hand side we use Fatou lemma, as by monotonicity condition (1.3) the integrand is nonnegative. In the right hand side, we note compare with (4.35) where we did not use that u n → u. Hence, we deduce from (5.9) Now we let λ → ∞ in (5.10), noting that γ(u − v) → u − v strongly in W 1,p 0 (Ω). Therefore, we get for all v ∈ K ψ (Ω), which means exactly that u is a solution to our obstacle problem.
Remark 5.1. Clearly, Theorem 5.1 is more general than Theorem 1.1 since we are allowed to choose ψ ≡ −∞. Indeed, in this case, the obstacle problem (5.2) reduces to (1.6).

Regularity of the solution
In this Section, following [17] we study regularity of the problem (1.6).
For fixed k > 0, we use v := u − T k u as a test function in (1.7) to get (6.3) α where Ω k denotes the superlevel set {|u| > k}. For 0 < ε < α, by Young inequality we get with C = C(p, ε) > 0. We let (6.5) λ = r * p * − 1 and multiply both sides of (6.4) by k pλ−1 and integrate w.r.t. k over the interval [0, K], for K > 0 fixed. By Fubini theorem we have (6.6) (α −ε) Ω |∇u| p |T K u| pλ dx Ω (C |F | p + b p |u| p + ϕ p ) |T K u| pλ dx which implies (6.7) (α − ε) 1 p ∇u |T K u| λ p C F |T K u| λ p + b u |T K u| λ p + ϕ |T K u| λ p For M > 0 we write (6.8) b u |T K u| λ p (b − T M b) u |T K u| λ p + M u |T K u| λ p and the proof is concluded. In the complementary case r > p * , we use a bootstrap approach. Precisely, we repeat the previous argument replacing r with p * to get u ∈ L p * * (Ω). Using this information, if r p * * , there is nothing left to prove. Otherwise we repeat previous argument again. In a finite number of similar steps we can conclude our proof. Remark 6.1. In view of (6.12), we may take δ = α 1 p S N,p p * r * in (6.1). Since r → r * is increasing, a similar condition to (6.12) will hold in all intermediate steps, in case we need the bootstrap argument. Note that δ reduces to the distance in (1.9), for r = p.