Quasilinear Dirichlet problems with competing operators and convection

Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.


Introduction
The object of the paper is to study the following quasilinear problem with homogeneous Dirichlet boundary condition of the negative p-Laplacian Δ p and of the q-Laplacian Δ q with < < < +∞ q p 1 . The operator − + Δ Δ p q has a completely different behavior in comparison to the operator − − Δ Δ p q , which is the (negative) ( ) p q , -Laplacian. Note that in − + Δ Δ p q there is competition between −Δ p and −Δ q taking their difference and thus destroying the ellipticity in contrast to what happens in the case of − − Δ Δ p q . The right-hand side ( ∇ ) f x u u , , of equation (1) is a so-called convection term meaning that it depends on the point ∈ x Ω in the domain, on the solution u and on its gradient ∇u. The convection term is expressed through the Nemytskii operator associated with a Carathéodory function × × → f : Ω N , i.e., ( ) f x s ξ , , is measurable in ∈ x Ω for all ( ) ∈ × s ξ , N and is continuous in ( ) ∈ × s ξ , N for a.e. ∈ x Ω. The function f will be subject to appropriate growth conditions (H1)-(H2) in Section 2.
Such a problem without any available ellipticity but with a variational structure that prevents to have convection was studied for the first time in [1]. Specifically, in [1] the following particular variational version of (1) was investigated:  with a Carathéodory function × → g : Ω . In order to highlight the core of the problem we have skipped the nonsmooth formulation in [1]. The difference between problems (1) and (2) consists in the fact that the reaction term ( ∇ ) f x u u , , of (1) depends on the gradient ∇u which is excluded in (2). This is an essential feature because the (somewhat) variational approach in [1] for (2) cannot be implemented for (1).
We briefly discuss some major characteristics of problem (1). Here we continue the study in [1] setting forth a nonvariational counterpart. The main aspect for the relevance of this work in comparison with the existing literature is the lack of ellipticity for the operator − + Δ Δ p q in the principal part of (1). As pointed out in [1], taking a nonzero ∈ ( ) u W Ω . It represents a real challenge with respect to [1] treating (2) because any variational method is inapplicable to (1). In [1] it was possible to build for (2) a variational approach through Ekeland's variational principle on finite dimensional spaces despite the lack of ellipticity. This is not anymore possible for (1), so here we proceed totally different using nonvariational arguments. A systematic study of nonvariational methods applied to elliptic problems can be found in [3]. However, such methods cannot be directly implemented in the case of problem (1) taking into account the lack of needed ellipticity. We overcome this difficulty by resolving finite dimensional approximated problems and then passing to the limit in an appropriate sense.
Moreover, in problem (1) there is a lack of any monotonicity property for the driving operator − + Δ Δ p q . This is the reason why the surjectivity theorem for pseudomonotone operators (see, e.g., is not pseudomonotone. For any sequence that holds thanks to the weak lower semicontinuity of the norm. Besides, , which entails the strong convergence → u u n since the space ( ) H Ω 0 1 is uniformly convex. Thus, we reach the contradiction that every weakly convergent sequence is strongly convergent, which proves the claim.
Due to the deficit of ellipticity, monotonicity and variational structure, there are no available techniques to handle problem (1). A fundamental idea of the paper is to seek a solution to (1) as a limit of finite dimensional approximations. To this end, we develop a finite dimensional fixed point approach and then generate a passing to the limit process to get generalized solutions. Our assumptions on the convection term are general and verifiable comprising solely conditions (H1)-(H2). Under a stronger assumption instead of (H1) and with (H2) as it is we are able to prove the existence of a generalized solution in a stronger sense. Finally, we observe that the same procedure applied to a problem driven by the ordinary ( ) p q , -Laplacian − − Δ Δ p q and under the same hypotheses leads to the existence of a weak solution. The rest of the paper consists of sections regarding mathematical background and hypotheses, approximate solutions and existence of generalized solutions to problem (1).

Mathematical background and hypotheses
In a Banach space, the strong convergence is denoted by → and the weak convergence by ⇀ . The Euclidean norm on the Euclidean space m for any ≥ m 1 is denoted by |⋅|, while the standard scalar product is denoted by ⋅ . For every real number > r 1, we set ′ = /( − ) r r r 1 (the Hölder conjugate of r). In particular, for < < < +∞ It is a strictly monotone and continuous operator, so pseudomonotone. The only linear case is when = p 2 giving rise to the ordinary Laplacian. Similarly, we have the negative q-Laplacian − ( ) . Due to the assumption < < < +∞ q p 1 there is a continuous embedding Next, we turn to the nonlinear term ( ∇ ) f x u u , , in the right-hand side of equation (1). Such a term depending on the function u and on its gradient ∇u is often called convection. It prevents to settle a variational structure for problem (1). Our hypotheses on the convection term are as follows: for a.e. Ω, all , .
1 for a.e. Ω, all , . The next lemma will be useful in the sequel.
Proof. Assumption ( ) H1 and Hölder's inequality lead to Now it suffices to invoke the Sobolev embedding theorem for obtaining the stated conclusion. □ We introduce the notion of solution to problem (1) whose existence can be established under hypotheses (H1)-(H2).
is said to be a generalized solution to problem (1) if there exists a sequence { } ≥ u n n 1 in , , Ω , is well defined. Moreover, again by Lemma 2.2, there exists a constant > C 0 such that the following estimate holds We focus a bit more on the integral term ∫ ( , d n n n Ω strengthening the growth condition in (H1).
and a nonnegative function With ( )′ H1 in place of ( ) H1 we consider the existence of a solution to problem (1) in a stronger sense. We end this section with the following consequence of Brouwer's fixed point theorem that will be an essential tool in our approach. For a proof we refer to [5, p. 37  Proof. For each ≥ n 1 we introduce the mapping → A X X :