Location of solutions for quasi-linear elliptic equations with general gradient dependence

Existence and location of solutions to a Dirichlet problem driven by (p, q)Laplacian and containing a (convection) term fully depending on the solution and its gradient are established through the method of subsolution-supersolution. Here we substantially improve the growth condition used in preceding works. The abstract theorem is applied to get a new result for existence of positive solutions with a priori estimates.


Introduction
The aim of this paper is to study the following nonlinear elliptic boundary value problem −∆ p u − µ∆ q u = f (x, u, ∇u) in Ω u = 0 on ∂Ω (P µ ) by means of the method of subsolution-supersolution on a bounded domain Ω ⊂ R N .For regularity reasons we assume that the boundary ∂Ω is of class C 2 .In order to simplify the presentation we suppose that N ≥ 3. The lower dimensional cases N = 1, 2 are simpler and can be treated by slightly modified arguments.
In the statement of problem (P µ ), there are given real numbers µ ≥ 0 and 1 < q < p.The leading differential operator in (P µ ) is described by the p-Laplacian and q-Laplacian, namely ∆ p u = div(|∇u| p−2 ∇u) and ∆ q u = div(|∇u| q−2 ∇u).Hence if µ = 0, problem (P µ ) is governed by the p-Laplacian ∆ p , whereas if µ = 1, it is driven by the (p, q)-Laplacian ∆ p + ∆ q , which is an essentially different type of nonlinear operator.
is continuous for a.e.x ∈ Ω.We emphasize that the term f (x, u, ∇u) (often called convection term) depends not only on the solution u, but also on its gradient ∇u.This fact produces serious difficulties of treatment mainly because the convection term generally prevents to have a variational structure for problem (P µ ), so the variational methods are not applicable.
Existence results for problem (P µ ) or for systems of equations of this form have been obtained in [1,[4][5][6][7][10][11][12].Location of solutions through the method of subsolution-supersolution in the case of systems involving p-Laplacian operators has been investigated in [3].Here, in the case of an equation possibly involving the (p, q)-Laplacian, we focus on the location of solutions within ordered intervals determined by pairs of subsolution-supersolution of problem (P µ ) under a much more general growth condition on the right-hand side f (x, u, ∇u) (see hypothesis (H) below).We also provide a new result guaranteeing the existence of positive solutions to (P µ ).
The functional space associated to problem (P µ ) is the Sobolev space W Its dual space is W −1,p (Ω), with p = p/(p − 1), and the corresponding duality pairing is denoted •, • .
A solution of problem (P µ ) is understood in the weak sense, that is any function Our study of problem (P µ ) is based on the method of subsolution-supersolution. We refer to [2,9] for details related to this method.We recall that a function u ∈ W 1,p (Ω) is a supersolution for problem (P µ ) if u ≥ 0 on ∂Ω and In the sequel we suppose that N > p (if N ≤ p the treatment is easier).Then the critical Sobolev exponent is p * = N p N−p .Given a subsolution u ∈ W 1,p (Ω) and a supersolution u ∈ W 1,p (Ω) for problem (P µ ) with u ≤ u a.e. in Ω, we assume that f : Ω × R × R N → R satisfies the growth condition: (H) There exist a function σ ∈ L γ (Ω) for γ = γ γ−1 with γ ∈ (1, p * ) and constants a > 0 and β ∈ 0, p (p * ) such that Notice that, under assumption (H), the integrals in the definitions of the subsolution u and the supersolution u exist.Our main goal is to obtain a solution u ∈ W 1,p 0 (Ω) of problem (P µ ) with the location property u ≤ u ≤ u a.e. in Ω.This is done through an auxiliary truncated problem termed (T λ,µ ) depending on a positive parameter λ (for any fixed µ ≥ 0).It is shown in Theorem 2.1 that whenever λ > 0 is sufficiently large, problem (T λ,µ ) is solvable.The next principal step is performed in Theorem 3.1, where it is proven by adequate comparison that every solution u ∈ W 1,p 0 (Ω) of problem (T λ,µ ) is within the ordered interval [u, u] determined by the subsolution- supersolution, that is u ≤ u ≤ u a.e. in Ω.Then the expression of the equation in (T λ,µ ) enables us to conclude that u is actually a solution of the original problem (P µ ) verifying the location property u ≤ u ≤ u a.e. in Ω.We emphasize that Theorem 2.1 improves all the growth conditions for the convection term f (x, u, ∇u) considered in the preceding works.Finally, in Theorem 4.1, the procedure to construct solutions located in ordered intervals [u, u] is conducted to guarantee the existence of a positive solution to problem (P µ ).It is also worth mentioning that this result provides a priori estimates for the obtained solution.

Auxiliary truncated problem
This section is devoted to the study of an auxiliary problem related to problem (P µ ).We start with some notation.The Euclidean norm on R N is denoted by | • | and the Lebesgue measure on R N by | • | N .For every r ∈ R, we set r + = max{r, 0}, r − = max{−r, 0}, and if r > 1, r = r r−1 .Let u and u be a subsolution and a supersolution for problem (P µ ), respectively, with u ≤ u a.e. in Ω such that hypothesis (H) is satisfied.We consider the truncation operator which is known to be continuous and bounded.By means of the constant β in hypothesis (H) we introduce the cut-off function π : (2.2) We observe that π satisfies the growth condition with a constant c > 0 and a function ∈ L p β (Ω).Here it is used that u, u ∈ W 1,p (Ω) ⊂ L p * (Ω) and β < p (p * ) .By (2.3), the fact that β < p (p * ) and Rellich-Kondrachov compactness embedding theorem, it follows that the Nemytskij operator Π : W with positive constants r 1 and r 2 .
Next we consider the Nemytskij operator N : [u, u] → W −1,p (Ω) determined by the function f in (P µ ), that is which is well defined by virtue of hypothesis (H).
With the data above, for any λ > 0 let the auxiliary truncated problem associated to (P µ ) be formulated as follows For problem (T λ,µ ) we have the following result.
Theorem 2.1.Let u and u be a subsolution and a supersolution of problem (P µ ), respectively, with u ≤ u a.e. in Ω such that hypothesis (H) is fulfilled.Then there exists λ 0 > 0 such that whenever λ ≥ λ 0 there is a solution u ∈ W 1,p 0 (Ω) of the auxiliary problem (T λ,µ ).
Proof.For every λ > 0 we introduce the nonlinear operator Due to (2.3) and (H), the operator A λ is bounded.We claim that A λ in (2.5) is a pseudomonotone operator.In order to show this, let a sequence {u n } ⊂ W Recalling from (H) that σ ∈ L γ (Ω) with γ < p * , by Hölder's inequality, (2.6) and the Rellich-Kondrachov compact embedding theorem we get Let us show that The definition of the truncation operator T : W Using Hölder's inequality, (2.6) and the Rellich-Kondrachov compact embedding theorem, as well as the inequality p p−β < p * , enables us to find that Therefore (2.9) holds true.

Main result
We state our main abstract result on problem (P µ ).Theorem 3.1.Let u and u be a subsolution and a supersolution of problem (P µ ), respectively, with u ≤ u a.e. in Ω such that hypothesis (H) is fulfilled.Then problem (P µ ) possesses a solution u ∈ W 1,p 0 (Ω) satisfying the location property u ≤ u ≤ u a.e. in Ω.
Proof.Theorem 2.1 guarantees the existence of a solution of the truncated auxiliary problem (T λ,µ ) provided λ > 0 is sufficiently large.Fix such a constant λ and let u ∈ W 1,p 0 (Ω) be a solution of (T λ,µ ).
We prove that u ≤ u a.e. in Ω. Acting with (u − u) + ∈ W 1,p 0 (Ω) as a test function in the definition of the supersolution u of (P µ ) and in the definition of the solution u for the auxiliary truncated problem (T λ,µ ) results in and From (3.1), (3.2) and (2.1) we derive we are able to derive from (2.2) and (3.3) that It follows that u ≤ u a.e in Ω.
In an analogous way, by suitable comparison we can show that u ≤ u a.e in Ω.Consequently, the solution u of the auxiliary truncated problem (T λ,µ ) satisfies Tu = u and Π(u) = 0 (see (2.1) and (2.2)), so it becomes a solution of the original problem (P µ ), which completes the proof.
Our result on the existence of positive solutions for problem (P µ ) is as follows.
Proof.With the notation in hypothesis (H1), consider the following auxiliary problem −∆ p u − µ∆ q u + b|u| p−2 u = a 0 (u + ) r in Ω, u = 0 on ∂Ω.To this end, we consider the Euler functional associated to (4.5), that is the C 1 -function I : W 1,p 0 (Ω) → R defined by From the assumption on r in hypothesis (H1) and Sobolev embedding theorem, it is easy to prove that I is coercive.Since I is also sequentially weakly lower semicontinuous, there exists u ∈ W 1,p 0 (Ω) such that On the basis of the conditions r < p − 1 if µ = 0 and r < q − 1 if µ > 0 (see hypothesis (H1)), it is seen that for any positive function v ∈ W 1,p 0 (Ω) and with a sufficiently small t > 0, there holds I(tv) < 0, so inf u∈W 1,p 0 (Ω) I(u) < 0. This enables us to deduce that u is a nontrivial solution of (4.4).Testing equation (4.4) with −u − yields u ≥ 0. By the nonlinear regularity theory and strong maximum principle we obtain that u ∈ C 1 0 (Ω) and u > 0 in Ω.According to the latter properties, we can utilize u α+1 , with any α > 0, as a test function in (4.4).Through Hölder's inequality and because r + 1 < p, this leads to We claim that u is a subsolution for problem (P µ ).Specifically, due to (4.1) and (4.5), we can insert s = u(x) and ξ = ∇u(x) in (4.2), which in conjunction with (4.4) for u = u reads as Thereby the claim is proven.Now we notice that hypothesis (H2) guarantees that u = s 0 is a supersolution of problem (P µ ).Indeed, in view of (4.3), we obtain in Ω.We point out from assumption (H2) that s 0 > δ, which in conjunction with (4.1) and (4.5), entails that u < u in Ω.
We also note that hypothesis (H) holds true for the constructed subsolution-supersolution (u, u) of problem (P µ ).Therefore Theorem 3.1 applies ensuring the existence of a solution u ∈ W 1,p 0 (Ω) to problem (P µ ), which satisfies the enclosure property u ≤ u ≤ u a.e. in Ω. Taking into account that u > 0, we conclude that the solution u is positive.Moreover, the regularity up to the boundary invoked for problem (P µ ) renders u ∈ C 1 0 (Ω), whereas the inequality u ≤ u implies the estimate u(x) ≤ s 0 for all x ∈ Ω.This completes the proof.

Remark 4 . 2 .Example 4 . 3 .
Proceeding symmetrically, a counterpart of Theorem 4.1 for negative solutions can be established.We illustrate the applicability of Theorem 4.1 by a simple example.Let f :Ω × R × R N → R be defined by f (x, s, ξ) = |s| r − |s| p−1 + (2 p−r p−r−1 − s)|ξ| β for all (x, s, ξ) ∈ Ω × R × R N ,where the constants r, p, β are as in conditions (H) and (H1).For simplicity, we have dropped the dependence with respect to x ∈ Ω. Hypothesis (H1) is verified by taking for instancea 0 = b = 1 and δ = 2p−r p−r−1 (see (4.1) and (4.2)).Hypothesis (H2) is fulfilled for every s 0 > δ = 2 p−r p−r−1 .It is also clear that the growth condition for f on Ω × [0, s 0 ] × R N required in the statement of Theorem 4.1 is satisfied, too.Consequently, Theorem 4.1 applies to problem (P µ ) with the chosen function f (x, s, ξ) giving rise to a positive solution belonging to C 1 0 (Ω).