Quasilinear Dirichlet problems with competing operators and convection

1 Introduction

The object of the paper is to study the following quasilinear problem with homogeneous Dirichlet boundary condition

on a bounded domain Ω ⊂ ℝ N with the boundary ∂ Ω . In the left-hand side of (1), we have the sum − Δ p + Δ q of the negative p-Laplacian Δ p and of the q-Laplacian Δ q with 1 < q < p < + ∞ . The operator − Δ p + Δ q has a completely different behavior in comparison to the operator − Δ p − Δ q , which is the (negative) ( p , q ) -Laplacian. Note that in − Δ p + Δ 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 p − Δ 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 × ℝ × ℝ N → ℝ , i.e., f ( x , s , ξ ) is measurable in x ∈ Ω for all ( s , ξ ) ∈ ℝ × ℝ N and is continuous in ( s , ξ ) ∈ ℝ × ℝ 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 × ℝ → ℝ . 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 p + Δ q in the principal part of (1). As pointed out in [1], taking a nonzero u 0 ∈ W 0 1 , p ( Ω ) and a number λ > 0 it turns out that

is not of constant sign, being positive for λ sufficiently large and negative otherwise, thus the ellipticity is lost. For this reason we call the operator − Δ p + Δ q the competing ( p , q ) -Laplacian. We also note that the left-hand side of the equation in (1) is in the divergence form − Δ p u + Δ q u = − div ( a ( | ∇ u | ) ) with a ( t ) = t p − 2 − t q − 2 for all t > 0 . A minimal condition of ellipticity for an operator in divergence form − div ( a ( | ∇ u | ) ) (see, e.g., [2]) is to have, among other things, a ( t ) > 0 for all t > 0 , which is not satisfied in the case of a ( t ) = t p − 2 − t q − 2 .

Another important aspect of problem (1) is the presence of the convection term f ( x , u , ∇ u ) . 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 p + Δ q . This is the reason why the surjectivity theorem for pseudomonotone operators (see, e.g., [4, p. 40]) cannot be applied. A striking difference between the operators − Δ p − Δ q and − Δ p + Δ q is that the operator − Δ p − Δ q is strictly monotone and continuous, so pseudomonotone, whereas this fails for − Δ p + Δ q . Let us note that even the linear continuous operator Δ on H 0 1 ( Ω ) = W 0 1 , 2 ( Ω ) is not pseudomonotone. For any sequence u n ⇀ u in H 0 1 ( Ω ) with lim sup n → ∞ 〈 Δ u n , u n − u 〉 ≤ 0 , the pseudomonotonicity of Δ would mean that lim inf n → ∞ 〈 Δ u n , u n − v 〉 ≥ 〈 Δ u , u − v 〉 whenever v ∈ H 0 1 ( Ω ) , in particular (with v = 0 ), lim inf n → ∞ 〈 Δ u n , u n 〉 ≥ 〈 Δ u , u 〉 . Note that lim sup n → ∞ 〈 Δ u n , u n − u 〉 ≤ 0 is always true being equivalent to lim inf n → ∞ ∇ u n L 2 ( Ω ) 2 ≥ ∥ ∇ u ∥ L 2 ( Ω ) 2 that holds thanks to the weak lower semicontinuity of the norm. Besides, lim inf n → ∞ 〈 Δ u n , u n 〉 ≥ 〈 Δ u , u 〉 is equivalent to lim sup n → ∞ ∇ u n L 2 ( Ω ) 2 ≤ ∥ ∇ u ∥ L 2 ( Ω ) 2 , which entails the strong convergence u n → u since the space H 0 1 ( Ω ) 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 f ( x , u , ∇ u ) 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 p − Δ 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).

2 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 ℝ 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 1 < q < p < + ∞ we have p ′ = p / ( p − 1 ) < q ′ = q / ( q − 1 ) .

Given a bounded domain Ω ⊂ ℝ N , the Sobolev spaces W 0 1 , p ( Ω ) and W 0 1, q ( Ω ) are endowed with the norms ∥ ∇ ( ⋅ ) ∥ L p ( Ω ) and ∥ ∇ ( ⋅ ) ∥ L q ( Ω ) , respectively, where ∥ ⋅ ∥ L r ( Ω ) stands for the usual L r -norm. The dual spaces of W 0 1 , p ( Ω ) and W 0 1 , q ( Ω ) are denoted W − 1 , p ′ ( Ω ) and W − 1 , q ′ ( Ω ) , respectively. In order to avoid repetitive arguments, we assume that N > p . The case N ≤ p is simpler and can be handled along the same lines. Under the assumption N > p , the critical exponent is p ⁎ = N p / ( N − p ) with the conjugate ( p ⁎ ) ′ = p ⁎ / ( p ⁎ − 1 ) .

We recall that the negative p-Laplacian − Δ p : W 0 1 , p ( Ω ) → W − 1 , p ′ ( Ω ) is expressed as

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 − Δ q : W 0 1 , q ( Ω ) → W − 1 , q ′ ( Ω ) . Due to the assumption 1 < q < p < + ∞ there is a continuous embedding W 0 1 , p ( Ω ) ↪ W 0 1 , q ( Ω ) . Consequently, the differential operator − Δ p + Δ q in the left-hand side of (1) is well defined on W 0 1 , p ( Ω ) .

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:

• (H1) There exist a nonnegative function σ ∈ L ( p ⁎ ) ′ ( Ω ) and constants b ≥ 0 and c ≥ 0 such that

  f ( x , s , ξ ) ≤ σ ( x ) + b | s | p ⁎ − 1 + c | ξ | p − 1 for a .e . x ∈ Ω , all s ∈ ℝ , ξ ∈ ℝ N .

• (H2) There exist constants c 0 < 1 , c 1 > 0 and α ∈ [1, p ) such that

We provide a simple example of function verifying ( H 1) − ( H 2) .

The next lemma will be useful in the sequel.

We introduce the notion of solution to problem (1) whose existence can be established under hypotheses (H1)–(H2).

3 Finite dimensional approximate solutions

Since the Banach space W 0 1 , p ( Ω ) with 1 < p < + ∞ is separable, there exists a Galerkin basis of W 0 1 , p ( Ω ) , which means a sequence { X n } n ≥ 1 of vector subspaces of W 0 1 , p ( Ω ) satisfying

1. dim ( X n ) < ∞ , ∀ n ;

2. X n ⊂ X n + 1 , ∀ n ;

3. ⋃ n = 1 ∞ X n ¯ = W 0 1 , p ( Ω ) .

Fix a Galerkin basis { X n } n ≥ 1 of W 0 1 , p ( Ω ) . The notation | Ω | stands for the Lebesgue measure of Ω .

4 Existence of generalized solutions

The statement below constitutes our existence result for generalized solutions to problem (1).

Next we state our second existence result.

Finally, we note that contrary to what happens for problem (1), our notion of strong generalized solution in the case of the problem

coincides with the classical notion of weak solution. Specifically, under hypothesis ( H 1 ) ′ , a function u ∈ W 0 1 , p ( Ω ) is by definition a strong generalized solution to problem (16) if there exists a sequence { u n } n ≥ 1 in W 0 1 , p ( Ω ) such that u n ⇀ u in W 0 1 , p ( Ω ) as n → ∞ ,

and

We claim that u ∈ W 0 1 , p ( Ω ) is a strong generalized solution to problem (16) if and only if it is a weak solution to problem (16), that is,

Indeed, if u ∈ W 0 1 , p ( Ω ) is a weak solution to problem (16), then posing u n = u it is clear that u is a strong generalized solution to problem (16). Conversely, assume that u ∈ W 0 1 , p ( Ω ) is a strong generalized solution to problem (16). Hence, there exists a sequence u n ⇀ u in W 0 1 , p ( Ω ) as n → ∞ for which (17) and (18) hold. From (18), the monotonicity of − Δ q and the weak convergence u n ⇀ u in W 0 1 , p ( Ω ) it follows that

As the operator − Δ p on W 0 1 , p ( Ω ) fulfills the ( S + ) -property (see, e.g., [4, p. 45]), we infer the strong convergence u n → u in W 0 1 , p ( Ω ) . Then the continuity of the operators − Δ p : W 0 1 , p ( Ω ) → W − 1 , p ′ ( Ω ) , − Δ q : W 0 1 , q ( Ω ) → W − 1 , q ′ ( Ω ) and N f : W 0 1 , p ( Ω ) → W − 1 , p ′ ( Ω ) enables us to deduce from (17) the validity of (19), thus u is a weak solution to problem (16).