An elliptic problem in dimension N with a varying drift term bounded in L N

Abstract

The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in $L^{N} (Ω)$ , with N the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space $H_{0}^{1} (Ω)$ . However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in (J. Differential Equations 258 (2015) 2290–2314), and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.

Keywords

Asymptotic behavior elliptic problem drift term varying coefficients

1. Introduction

For a bounded open set $Ω \subset R^{N}$ , we are interested in passing to the limit in a sequence on elliptic equations with a varying drift term, whose coefficients are just bounded in $L^{N} {(Ω)}^{N}$ , ( $L^{p} {(Ω)}^{N}$ , $p > 2$ if $N = 2$ ). The problem is written as $\begin{matrix} (1.1) & \{\begin{array}{l} - div (A \nabla u_{n} + E_{n} u_{n}) = f_{n} in Ω \\ u_{n} \in H_{0}^{1} (Ω), \end{array} \end{matrix}$ with $A \in L^{\infty} {(Ω)}^{N \times N}$ satisfying the usual uniform ellipticity condition, and $f_{n}$ bounded in $H^{- 1} (Ω)$ . Thanks to Sobolev’s inequality, the integrability assumption on $E_{n}$ is the weaker one to get the first order term well defined in $H^{- 1} (Ω)$ . It is known that for every $n \in N$ problem (1.1) has a unique solution ([3,4,15]), however, the problem is known to be not coercive and thus, there is no estimate for $u_{n}$ in $H_{0}^{1} (Ω)$ . We recall some of these results in Section 2.

L. Boccardo found in [2] an estimate for $ln (1 + | u_{n} |)$ in $H_{0}^{1} (Ω)$ depending only on $‖ f_{n} ‖_{H^{- 1} (Ω)}$ and $‖ E_{n} ‖_{L^{2} {(Ω)}^{N}}$ . As a consequence of this result we get that the measure of the sets ${| u_{n} | > k}$ tends to zero when k tends to infinity uniformly in n. Thanks to this result L. Boccardo proved in [5] that $f_{n}$ converging weakly to some f in $H^{- 1} (Ω)$ , $E_{n}$ converging weakly to $E_{0}$ in $L^{N} (Ω)$ and $| E_{n} |^{N}$ equi-integrable, imply that the solution $u_{n}$ of (1.1) converges weakly in $H_{0}^{1} (Ω)$ to the solution $u_{0}$ of $\begin{matrix} \{\begin{array}{l} - div (A \nabla u_{0} + E_{0} u_{0}) = f in Ω \\ u_{0} \in H_{0}^{1} (Ω) . \end{array} \end{matrix}$ In the case where $| E_{n} |^{N}$ is not equi-integrable, we do not have an estimate for $u_{n}$ in $H_{0}^{1} (Ω)$ and then the proof is more involved. The ideas in [2] imply (see [5]) that ${ln}^{q} (1 + | u_{n} |)$ is bounded in $H_{0}^{1} (Ω)$ for any $q > 1$ . Using also the approximation of $u_{n}$ given by the solution of a perturbation of (1.1) by a nonlinear zero order term (see (3.12) below), we still manage to pass to the limit in (1.1) but, instead of the weak convergence in $H_{0}^{1} (Ω)$ , we only get $\begin{matrix} {ln}^{q} (1 + | u_{n} |) sgn (u_{n}) ⇀ {ln}^{q} (1 + | u_{0} |) sgn (u_{0}) in H_{0}^{1} (Ω), \forall q ⩾ 1 . \end{matrix}$ This is the main result of the paper, which is proved in Theorem 3.1.

Another problem related to (1.1) is given by its adjoint formulation $\begin{matrix} (1.2) & \{\begin{array}{l} - div (A^{T} \nabla u_{n}) + E_{n} \cdot \nabla u_{n} = f_{n} in Ω \\ u_{n} \in H_{0}^{1} (Ω) . \end{array} \end{matrix}$ Some results about the asymptotic behavior of the solutions of this problem have been obtained in [5]. Namely, assuming the equi-integrability condition on $| E_{n} |$ and reasoning by duality, we deduce from the results stated above for problem (1.1) that the solutions of (1.2) are bounded in $H_{0}^{1} (Ω)$ . Moreover, the results in [9] prove that $u_{n}$ is compact in $W_{0}^{1, q} (Ω)$ for $1 ⩽ q < 2$ . Then, it is immediate to pass to the limit in this problem. In the case where $| E_{n} |^{N}$ is not equi-integrable, we do not have any estimate for $u_{n}$ and then we are not able to passing to the limit in (1.2). However, adding a sequence of zero order terms $a_{n} u_{n}$ with $a_{n}$ satisfying $\begin{matrix} a_{n} ⩾ γ a.e. in Ω, \end{matrix}$ for some $γ > 0$ , and assuming $f_{n}$ bounded in $L^{\infty} (Ω)$ , it is proved in [5] that $u_{n}$ is bounded in $H_{0}^{1} (Ω) \cap L^{\infty} (Ω)$ . This allows us to pass to the limit in (1.2).

The homogenization of a sequence of elliptic PDE’s with a singular term of first order has also been carried out in other papers. In this sense we refer to [6] where it is considered problem (1.1) with the matrix function A also depending on n, the sequence $E_{n}$ converging weakly in $L^{2} {(Ω)}^{N}$ , and $div E_{n}$ converging strongly in $H^{- 1} (Ω)$ . The definition of solution in this case is related to the definition of entropy or renormalized solution (see e.g. [1,8,14]). In [10] and [13] it is considered the case of a first order term of the form $\begin{matrix} E_{n} \nabla u_{n} + div (E_{n} u_{n}) . \end{matrix}$ Since this term is skew-symmetric we can obtain an estimate in $H_{0}^{1} (Ω)$ which is independent of $E_{n}$ (we refer to [7] for a related existence result). Assuming $E_{n}$ just bounded in $L^{2} {(Ω)}^{N}$ it is proved that the limit problem contains a new term of zero order. This is related to the results obtained in [16] for the Stokes equation with an oscillating Coriolis force. We also refer to [11,12] for related results in the case of the evolutive elastic system submitted to an oscillating magnetic field. Now, the limit problem is nonlocal in general.

2. Some reminders about elliptic problems with a drift or convection term

For a bounded open set $Ω \subset R^{N}$ , $N ⩾ 2$ , a matrix function $A \in L^{\infty} {(Ω)}^{N \times N}$ , such that there exists $α > 0$ satisfying $\begin{matrix} (2.1) & A (x) ξ \cdot ξ ⩾ α | ξ |^{2}, \forall ξ \in R^{N}, a.e. x \in Ω, \end{matrix}$ two measurable functions $E : Ω \to R^{N}$ , $a : Ω \to R$ , $a ⩾ 0$ a.e. in Ω, and a distribution $f \in H^{- 1} (Ω)$ , we recall in this section, some results about the existence and uniqueness of solution for problems $\begin{matrix} (2.2) & \{\begin{array}{l} A u = f in Ω \\ u \in H_{0}^{1} (Ω), \end{array} \{\begin{array}{l} A^{*} u = f in Ω \\ u \in H_{0}^{1} (Ω), \end{array} \end{matrix}$ with $\begin{matrix} A u = - div (A \nabla u + E u) + a u, A^{*} u = - div (A^{T} \nabla u) + E \cdot \nabla u + a u . \end{matrix}$ Observe that due to Sobolev imbedding theorem, in order to have the terms $div (E u)$ , $E \cdot \nabla u$ and $a u$ in $H^{- 1} (Ω)$ , when u is in $H_{0}^{1} (Ω)$ , we need to assume $\begin{matrix} (2.3) & \{\begin{array}{ll} E \in L^{N} {(Ω)}^{N} & if N > 2 \\ E \in L^{p} {(Ω)}^{N}, p > 2 & if N = 2, \end{array} \{\begin{array}{ll} a \in L^{\frac{N}{2}} (Ω) & if N > 2 \\ a \in L^{q} {(Ω)}^{N}, q > 1 & if N = 2 . \end{array} \end{matrix}$ In this case, $A$ and $A^{*}$ are continuous linear operators from $H_{0}^{1} (Ω)$ into $H^{- 1} (Ω)$ and $A^{*}$ is the adjoint operator of $A$ .

Assuming the further assumption $\begin{matrix} (2.4) & E \in L^{p} {(Ω)}^{N}, p > N, \end{matrix}$ the compact imbedding of $H_{0}^{1} (Ω)$ into $L^{\frac{2 p}{p - 2}} (Ω)$ proves that for every $ε > 0$ , there exists $C_{ε} > 0$ such that $\begin{matrix} (2.5) & ‖ u ‖_{L^{\frac{2 p}{p - 2}} (Ω)} ⩽ α ε ‖ \nabla u ‖_{L^{2} {(Ω)}^{N}} + C_{ε} ‖ u ‖_{L^{2} (Ω)}, \forall u \in H_{0}^{1} (Ω) . \end{matrix}$ Thus, Hölder’s and Young’s inequalities imply $\begin{aligned} {⟨ A u, u ⟩}_{H^{- 1} (Ω), H_{0}^{1} (Ω)} & = {⟨ A^{*}, u ⟩}_{H^{- 1} (Ω), H_{0}^{1} (Ω)} = \int_{Ω} (A \nabla u \cdot \nabla u d x + u E \cdot \nabla u + a u^{2}) d x \\ ⩾ α ‖ \nabla u ‖_{L^{2} {(Ω)}^{N}}^{2} - ‖ E ‖_{L^{p} {(Ω)}^{N}} ‖ \nabla u ‖_{L^{2} {(Ω)}^{N}} ‖ u ‖_{L^{\frac{2 p}{p - 2}} (Ω)} \\ ⩾ α (\frac{1}{2} - ε ‖ E ‖_{L^{p} {(Ω)}^{N}}) ‖ \nabla u ‖_{L^{2} {(Ω)}^{N}}^{2} - \frac{C_{ε}^{2} ‖ E ‖_{L^{p} {(Ω)}^{N}}^{2}}{2 α} ‖ u ‖_{L^{2} (Ω)}^{2} . \end{aligned}$ Therefore, taking $ε ‖ E ‖_{L^{p} {(Ω)}^{N}} < 1 / 2$ , we deduce from Lax–Milgram theorem that replacing a by $a + γ$ , with $\begin{matrix} γ ⩾ \frac{C_{ε}^{2} ‖ E ‖_{L^{p} {(Ω)}^{N}}^{2}}{2 α}, \end{matrix}$ there exists a unique solution for both problems in (2.2). The compactness of ${(A + γ I)}^{- 1}$ , considered as an operator in $L^{2} (Ω)$ , allows then to use Fredholm theory to deduce that the existence and uniqueness of solutions for both problems in (2.2) and every $f \in H^{- 1} (Ω)$ is equivalent to the uniqueness of solutions for one of them.

In [15], Theorem 8.1, it is proved that problem $\begin{matrix} \{\begin{array}{l} A^{*} u = f in Ω \\ u - ϕ \in H_{0}^{1} (Ω), \end{array} \end{matrix}$ with $ϕ \in H^{1} (Ω)$ satisfies the weak maximum principle so that the second problem in (2.2) has at most one solution (in [15] it is assumed $E \in L^{\infty} {(Ω)}^{N}$ , but it is immediate to check that the same proof works for E just satisfying (2.3)).

We observe however that although the above result implies the existence and continuity of the operators $A^{- 1}$ and ${(A^{*})}^{- 1}$ , it does not provide any estimate for the norm of these operators and then on the solutions of both problems in (2.2).

If we assume $N > 2$ and $E \in L^{N} {(Ω)}^{N}$ , the above reasoning fails because (2.5) with $p = N$ does not hold in general. The existence of solutions in this case can be obtained from the following result due to L. Boccardo ([2,5]). We recall that the truncate function at height $k > 0$ is defined as $\begin{matrix} (2.6) & T_{k} (s) = \{\begin{array}{ll} - k & if s < - k \\ s & if - k ⩽ s ⩽ k \\ k & if s > k . \end{array} \end{matrix}$

Theorem 2.1.
For every $f \in H^{- 1} (Ω)$ , $E \in L^{2} {(Ω)}^{N}$ , and $a \in L^{1} (Ω)$ , $a ⩾ 0$ a.e. in Ω, there exists an entropy solution of $\begin{matrix} (2.7) & \{\begin{array}{l} A u = f in Ω \\ u = 0 on \partial Ω, \end{array} \end{matrix}$ in the following sense $\begin{matrix} (2.8) & \{\begin{array}{l} T_{k} (u) \in H_{0}^{1} (Ω), \forall k > 0, log (1 + | u |) \in H_{0}^{1} (Ω) \\ \int_{Ω} ((A \nabla u + E u) \cdot \nabla T_{k} (u - φ) + a u T_{k} (u - φ)) d x ⩽ {⟨ f, T_{k} (u - φ) ⟩}_{H^{- 1} (Ω), H_{0}^{1} (Ω)}, \\ \forall φ \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω) . \end{array} \end{matrix}$ Moreover, there exists $C > 0$ , independent of f and E such that $\begin{matrix} (2.9) & {‖ log (1 + | u |) ‖}_{H_{0}^{1} (Ω)} ⩽ C (‖ f ‖_{H^{- 1} (Ω)} + ‖ E ‖_{L^{2} {(Ω)}^{N}}) . \end{matrix}$ If there exists $γ > 0$ such that $\begin{matrix} (2.10) & a ⩾ γ a.e. in Ω, \end{matrix}$ and f belongs to $L^{1} (Ω)$ , such solution is also in $L^{1} (Ω)$ and satisfies $\begin{matrix} (2.11) & ‖ u ‖_{L^{1} (Ω)} ⩽ \frac{1}{γ} ‖ f ‖_{L^{1} (Ω)} . \end{matrix}$

Let us prove that the function u given by the previous theorem is in fact a distributional solution of the first problem in (2.2) when E and a satisfy (2.3). This is given by the following theorem
Theorem 2.2.
Assume that the functions E and a in Theorem 2.1 satisfy ( 2.3 ), then the solution u of ( 2.7 ) is in $H_{0}^{1} (Ω)$ and satisfies the elliptic equation in the distributional sense. Moreover, if there exists $γ > 0$ such that ( 2.10 ) holds, then this solution is unique.
Proof.
For $k, m > 0$ , we take $φ = T_{m} (u)$ in (2.8). This gives $\begin{matrix} \int_{{m < | u | < k + m}} ((A \nabla u + E u) \cdot \nabla u) d x + \int_{Ω} a u T_{k} (u - T_{m} (u)) d x ⩽ {⟨ f, T_{k} (u - T_{m} (u)) ⟩}_{H^{- 1} (Ω), H_{0}^{1} (Ω)} . \end{matrix}$ Using in this equality that $\begin{matrix} | \int_{{m < | u | < k + m}} E u \cdot \nabla u d x | ⩽ \int_{{m < | u | < k + m}} | E | (m + | T_{k} (u - T_{m} (u)) |) | \nabla u | d x, \end{matrix}$ combined with (2.1), Sobolev’s imbedding theorem and Hölder’s inequality we deduce the existence of $C_{S} > 0$ , depending only on N (and $| Ω |$ if $N = 2$ ), such that $\begin{aligned} {‖ T_{k} (u - T_{m} (u)) ‖}_{H_{0}^{1} (Ω)}^{2} (α - C_{S} ‖ E ‖_{L^{r} {({m < | u |})}^{N}}) \\ (2.12) & ⩽ (‖ f ‖_{H^{- 1} (Ω)} + m C_{S} ‖ E ‖_{L^{r} {(Ω)}^{N}}) {‖ T_{k} (u - T_{m} (u)) ‖}_{H_{0}^{1} (Ω)}, \end{aligned}$ with $r = p$ if $N = 2$ , $r = N$ if $N > 2$ .

By (2.9), we can take m sufficiently large to get $\begin{matrix} C_{S} ‖ E ‖_{L^{r} ({m < | u |})}^{N} < α . \end{matrix}$ Letting m fixed satisfying this inequality and taking k tending to infinity, we deduce that $u - T_{m} (u)$ belongs to $H_{0}^{1} (Ω)$ . Since $T_{m} (u)$ also belongs to $H_{0}^{1} (Ω)$ , then u belongs to $H_{0}^{1} (Ω)$ .

Once we know that u belongs to $H_{0}^{1} (Ω)$ , we can take k tending to infinity in (2.8) to deduce $\begin{matrix} \int_{Ω} ((A \nabla u + E u) \cdot \nabla (u - φ) + a u (u - φ)) d x ⩽ {⟨ f, u - φ ⟩}_{H^{- 1} (Ω), H_{0}^{1} (Ω)}, \end{matrix}$ for every $φ \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω)$ . Since $H_{0}^{1} (Ω) \cap L^{\infty} (Ω)$ is dense in $H_{0}^{1} (Ω)$ , the result holds for φ just in $H_{0}^{1} (Ω)$ . Replacing φ by $u + φ$ we deduce that u is a distributional solution of $A u = f$ in Ω.

The fact that the renormalized solutions are distributional solutions for E satisfying (2.3) allows us to reason by linearity to deduce that they are unique if and only if the unique entropy solution of (2.7) with $f = 0$ is the null function. When (2.10) holds this follows from (2.11). □

Theorem 2.1 proves that the first problem in (2.2) always has at least a solution and it is unique if a satisfies (2.10). As above, this allows us to use Fredholm theory to deduce Corollary 2.3.
Assume that E satisfies ( 2.3 ), then for every $f \in H^{- 1} (Ω)$ both problems in ( 2.2 ) have a unique solution. Moreover this solution depends continuously on f.

3. Passing to the limit with a varying drift term

In this section, for a bounded open set $Ω \subset R^{N}$ , $N ⩾ 2$ , a sequence of vector measurable functions $E_{n} : Ω \to R^{N}$ such that there exists $E_{0} : Ω \to R^{N}$ satisfying $\begin{matrix} (3.1) & \{\begin{array}{ll} E_{n} ⇀ E_{0} in L^{N} {(Ω)}^{N} & if N > 2 \\ E_{n} ⇀ E_{0} in L^{p} {(Ω)}^{N}, for some p > 2 & if N = 2, \end{array} \end{matrix}$ we are interested in the asymptotic behavior of the solutions of $\begin{matrix} (3.2) & \{\begin{array}{l} - div (A \nabla u_{n} + E_{n} u_{n}) = f_{n} in Ω \\ u_{n} \in H_{0}^{1} (Ω), \end{array} \end{matrix}$ where $A \in L^{\infty} {(Ω)}^{N \times N}$ satisfies (2.1) and $f_{n} \in H^{- 1} (Ω)$ is such that there exists $f \in H^{- 1} (Ω)$ satisfying $\begin{matrix} (3.3) & f_{n} ⇀ f in H^{- 1} (Ω) . \end{matrix}$ As we recalled in the previous section, problem (3.2) has a unique solution but we do not know if it is bounded in $H_{0}^{1} (Ω)$ . Using the estimates for the solutions of this problem given in [2] let us prove

Theorem 3.1.

Assume Ω a bounded open set of $R^{N}$ , $N ⩾ 2$ , and $E_{n}$ a sequence of vector measurable functions in Ω such that there exists $E_{0}$ satisfying ( 3.1 ). Then, for every sequence $f_{n}$ which satisfies ( 3.3 ), the sequence of solutions $u_{n}$ of ( 3.2 ) satisfies $\begin{array}{c} (3.4) & T_{m} (u_{n}) ⇀ T_{m} (u_{0}) in H_{0}^{1} (Ω), \forall m > 0, \\ (3.5) & {ln}^{q} (1 + | u_{n} |) ⇀ {ln}^{q} (1 + | u_{0} |) in H_{0}^{1} (Ω), \forall q > 0, \end{array}$ with u the unique solution of $\begin{matrix} (3.6) & \{\begin{array}{l} - div (A \nabla u_{0} + E_{0} u_{0}) = f in Ω \\ u_{0} \in H_{0}^{1} (Ω) . \end{array} \end{matrix}$ Moreover, if one of the following assumptions hold:

$N = 2$ .

$N > 2$ , $| E_{n} |^{N}$ is equi-integrable,

then the convergence holds in the weak topology of

H_{0}^{1} (Ω)

Proof.

Step 1. We start getting some estimates for the solutions of (3.2). They are based on [2].

For $t ⩾ 2$ , we use as test function in (3.2) $v = ϕ (u_{n})$ , with ϕ defined by $\begin{matrix} ϕ (s) = \int_{0}^{s} \frac{{ln}^{t} (1 + | ρ |)}{{(1 + | ρ |)}^{2}} d ρ, \forall s \in R . \end{matrix}$ We get $\begin{matrix} \int_{Ω} (A \nabla u_{n} + E_{n} u_{n}) \cdot \nabla u_{n} \frac{{ln}^{t} (1 + | u_{n} |)}{{(1 + | u_{n} |)}^{2}} d x = ⟨ f_{n}, ϕ (u_{n}) ⟩ . \end{matrix}$ Using (2.1), Young’s and Hölder’s inequalities, and $ϕ^{'} \in L^{\infty} (R)$ , we deduce the existence of $C > 0$ depending on α, t, N (α, t, p, $| Ω |$ if $N = 2$ ) such that $\begin{aligned} \int_{Ω} {| \nabla ({ln}^{\frac{t + 2}{2}} (1 + | u_{n} |) sgn (u_{n})) |}^{2} d x & = \frac{{(t + 2)}^{2}}{4} \int_{Ω} | \nabla u_{n} |^{2} \frac{{ln}^{t} (1 + | u_{n} |)}{{(1 + | u_{n} |)}^{2}} d x \\ ⩽ C \int_{Ω} | E_{n} |^{2} | u_{n} |^{2} \frac{{ln}^{t} (1 + | u_{n} |)}{{(1 + | u_{n} |)}^{2}} d x + C ‖ f_{n} ‖_{H^{- 1} (Ω)}^{2} \\ ⩽ C \int_{Ω} | E_{n} |^{2} {ln}^{t} (1 + | u_{n} |) d x + C ‖ f_{n} ‖_{H^{- 1} (Ω)}^{2} \\ ⩽ C {(\int_{Ω} | E_{n} |^{r} d x)}^{\frac{2}{r}} {(\int_{Ω} {ln}^{\frac{t r}{r - 2}} (1 + | u_{n} |) d x)}^{\frac{r - 2}{r}} \\ (3.7) & + C ‖ f_{n} ‖_{H^{- 1} (Ω)}^{2}, \end{aligned}$ with $\begin{matrix} (3.8) & r = \{\begin{array}{ll} p & if N = 2 \\ r = N & if N > 2 . \end{array} \end{matrix}$

By Sobolev’s inequality, we also have $\begin{matrix} {(\int_{Ω} {ln}^{\frac{(t + 2) μ}{2}} (1 + | u_{n} |) d x)}^{\frac{2}{μ}} ⩽ C_{S} \int_{Ω} {| \nabla ({ln}^{\frac{t + 2}{2}} (1 + | u_{n} |) sgn (u_{n})) |}^{2} d x, \end{matrix}$ with $1 ⩽ μ$ , if $N = 2$ and $1 ⩽ μ ⩽ 2 N / (N - 2)$ , if $N > 2$ . Choosing $\begin{matrix} 1 < μ : = \frac{2 t r}{(t + 2) (r - 2)} < \frac{2 r}{r - 2} \end{matrix}$ and using Young’s inequality, we deduce (for another constant $C > 0$ ), after simplifying equal terms, $\begin{matrix} {(\int_{Ω} {ln}^{\frac{t r}{r - 2}} (1 + | u |) d x)}^{\frac{(t + 2) (r - 2)}{t r}} ⩽ C {(\int_{Ω} | E_{n} |^{r} d x)}^{\frac{t + 2}{r}} + C ‖ f_{n} ‖_{H^{- 1} (Ω)}^{2} . \end{matrix}$ Replacing this estimate in the right-hand side of (3.7), we finally get $\begin{matrix} (3.9) & \int_{Ω} {| \nabla ({ln}^{\frac{t + 2}{2}} (1 + | u_{n} |) sgn (u_{n})) |}^{2} d x ⩽ C {(\int_{Ω} | E_{n} |^{r} d x)}^{\frac{t + 2}{r}} + C ‖ f_{n} ‖_{H^{- 1} (Ω)}^{2} . \end{matrix}$ Since t can be taken as large as we want, we conclude $\begin{matrix} (3.10) & {ln}^{q} (1 + | u_{n} |) sgn (u_{n}) bounded in H_{0}^{1} (Ω), \forall q > 0 . \end{matrix}$ In particular, $T_{m} (u_{n})$ is bounded in $H_{0}^{1} (Ω)$ , for every $m \in N$ . These estimates and the Rellich–Kondrachov’s compactness theorem prove the existence of a subsequence of n, still denoted by n, and a measurable function u such that for every $m, q > 0$ , we have $\begin{matrix} (3.11) & T_{m} (u_{n}) ⇀ T_{m} (u) in H_{0}^{1} (Ω), {ln}^{q} (1 + | u_{n} |) ⇀ {ln}^{q} (1 + | u |) in H_{0}^{1} (Ω) . \end{matrix}$

Now, we have to prove that $u = u_{0}$ .

Step 2. Let us first consider the case $N = 2$ or $N ⩾ 3$ , $| E_{n} |^{N}$ equi-integrable. It has been first carried out in [5]. Indeed, since for $N = 2$ , p in (3.1) is any number bigger than 2, the problem reduces to assume $| E_{n} |^{r}$ equi-integrable with r given by (3.8). Taking as test function in (3.2) $u_{n} - T_{m} (u_{n})$ , we deduce $\begin{matrix} \int_{{m < | u_{n} |}} (A \nabla u_{n} + E_{n} u_{n}) \cdot \nabla u_{n} d x = {⟨ f_{n}, u_{n} - T_{m} (u_{n}) ⟩}_{H^{- 1} (Ω), H_{0}^{1} (Ω)}, \end{matrix}$ which similarly to (2.12) implies $\begin{aligned} {‖ u_{n} - T_{m} (u_{n}) ‖}_{H_{0}^{1} (Ω)}^{2} (α - C_{S} ‖ E_{n} ‖_{L^{r} {({m < | u_{n} |})}^{N}}) \\ ⩽ (‖ f_{n} ‖_{H^{- 1} (Ω)} + m C_{S} ‖ E_{n} ‖_{L^{2} {(Ω)}^{N}}) {‖ u_{n} - T_{m} (u_{n}) ‖}_{H_{0}^{1} (Ω)} . \end{aligned}$ Thanks to (3.10), we have $\begin{matrix} lim_{m \to \infty} sup_{n \in N} | {| u_{n} | > m} | = 0, \end{matrix}$ which combined with the equi-integrability of $| E_{n} |^{r}$ implies the existence of $m \in N$ such that $‖ E_{n} ‖_{L^{r} {({m < | u_{n} |})}^{N}} < α / (2 C_{S})$ , for every $n \in N$ . Therefore $u_{n} - T_{m} (u_{n})$ is bounded in $H_{0}^{1} (Ω)$ . Since $T_{m} (u_{n})$ is bounded in $H_{0}^{1} (Ω)$ too, we get $u_{n}$ bounded in $H_{0}^{1} (Ω)$ . Taking into account (3.4), (3.5) we have that $u_{n}$ converges weakly to u in $H_{0}^{1} (Ω)$ . By the Rellich–Kondrachov’s compactness theorem, we can now easily pass to the limit in (3.2) in the distributional sense to deduce that $u = u_{0}$ , the solution of (3.6).

Step 3. In this and the following step we assume $N ⩾ 3$ , $| E_{n} |^{N}$ not necessarily equi-integrable.

For $δ > 0$ , we define $w_{n, δ}$ as the solution of $\begin{matrix} (3.12) & \{\begin{array}{l} - div (A \nabla w_{n, δ} + E_{n} w_{n, δ}) + δ | w_{n, δ} |^{\frac{4}{N - 2}} w_{n, δ} = f_{n} in Ω \\ w_{n, δ} \in H_{0}^{1} (Ω) . \end{array} \end{matrix}$ Using $w_{n, δ}$ as test function in (3.12), we have $\begin{matrix} (3.13) & \int_{Ω} A \nabla w_{n, δ} \cdot \nabla w_{n, δ} d x + \int_{Ω} w_{n, δ} E_{n} \cdot \nabla w_{n, δ} d x + δ \int_{Ω} | w_{n, δ} |^{\frac{2 N}{N - 2}} d x = ⟨ f_{n}, w_{n, δ} ⟩ . \end{matrix}$

In the second term of this equality we use Young’s inequality with exponents $2 N / (N - 2)$ , N and 2, to get $\begin{aligned} | \int_{Ω} w_{n, δ} E_{n} \cdot \nabla w_{n, δ} d x | & = | \int_{Ω} (δ^{\frac{N - 2}{2 N}} w_{n, δ}) (\frac{2^{\frac{1}{2}}}{α^{\frac{1}{2}} δ^{\frac{N - 2}{2 N}}} E_{n}) \cdot (\frac{α^{\frac{1}{2}}}{2^{\frac{1}{2}}} \nabla w_{n, δ}) d x | \\ ⩽ \frac{N - 2}{2 N} δ \int_{Ω} | w_{n, δ} |^{\frac{2 N}{N - 2}} d x + \frac{2^{\frac{N}{2}}}{N α^{\frac{N}{2}} δ^{\frac{N - 2}{2}}} \int_{Ω} | E_{n} |^{N} d x \\ + \frac{α}{4} \int_{Ω} | \nabla w_{n, δ} |^{2} d x . \end{aligned}$

In the last term, Young’s inequality also gives $\begin{matrix} | ⟨ f_{n}, w_{n, δ} ⟩ | ⩽ ‖ f_{n} ‖_{H^{- 1} (Ω)} ‖ w_{n, δ} ‖_{H^{- 1} (Ω)} ⩽ \frac{1}{α} ‖ f_{n} ‖_{H^{- 1} (Ω)}^{2} + \frac{α}{4} \int_{Ω} | \nabla w_{n, δ} |^{2} d x . \end{matrix}$ Using also (2.1) in the first term, we deduce from (3.13) $\begin{matrix} (3.14) & \frac{α}{2} \int_{Ω} | \nabla w_{n, δ} |^{2} d x + \frac{N + 2}{2 N} δ \int_{Ω} | w_{n, δ} |^{\frac{2 N}{N - 2}} d x ⩽ \frac{2^{\frac{N}{2}}}{N α^{\frac{N}{2}} δ^{\frac{N - 2}{2}}} \int_{Ω} | E_{n} |^{N} d x + \frac{1}{α} ‖ f_{n} ‖_{H^{- 1} (Ω)}^{2} . \end{matrix}$ Thus, for every $δ > 0$ , $w_{n, δ}$ is bounded in $H_{0}^{1} (Ω)$ . Thanks to Rellich–Kondrakov’s compactness theorem this allows us to pass to the limit in (3.12) in the distributional sense, to deduce $\begin{matrix} (3.15) & w_{n, δ} ⇀ w_{*, δ} in H_{0}^{1} (Ω), \end{matrix}$ with $w_{*, δ}$ the solution of $\begin{matrix} (3.16) & \{\begin{array}{l} - div (A \nabla w_{*, δ} + E_{0} w_{*, δ}) + δ | w_{*, δ} |^{\frac{4}{N - 2}} w_{*, δ} = f in Ω \\ w_{*, δ} \in H_{0}^{1} (Ω) . \end{array} \end{matrix}$

We take $u_{0}$ the solution of (3.6). Taking $T_{ρ} (w_{*, δ} - u_{0})$ , with $ρ > 0$ as test function in the difference of (3.16) and (3.6), we have $\begin{aligned} \int_{{| w_{*, δ} - u_{0} | < ρ}} (A \nabla (w_{*, δ} - u_{0}) + E (w_{*, δ} - u_{0})) \cdot \nabla (w_{*, δ} - u_{0}) d x \\ + δ \int_{Ω} (| w_{*, δ} |^{\frac{4}{N - 2}} w_{*, δ} - | u_{0} |^{\frac{4}{N - 2}} u_{0}) T_{ρ} (w_{*, δ} - u_{0}) d x = - δ \int_{Ω} | u_{0} |^{\frac{4}{N - 2}} u_{0} T_{ρ} (w_{*, δ} - u_{0}) d x . \end{aligned}$ Using Young’s inequality in the second term of the first integral and $\begin{matrix} \exists c_{N} > 0 : c_{N} | | x | + | y | |^{\frac{4}{N - 2}} | x - y |^{2} ⩽ (| x |^{\frac{4}{N - 2}} x - | y |^{\frac{4}{N - 2}} y) (x - y), \forall x, y \in R, \end{matrix}$ in the second integral, we get $\begin{aligned} \frac{α}{2} \int_{{| w_{*, δ} - u_{0} | < ρ}} {| \nabla (w_{*, δ} - u_{0}) |}^{2} d x + c_{N} δ \int_{Ω} {(| w_{*, δ} | + | u_{0} |)}^{\frac{4}{N - 2}} | w_{*, δ} - u_{0} | | T_{ρ} (w_{*, δ} - u_{0}) | d x \\ (3.17) & ⩽ \frac{1}{2 α} \int_{{| w_{*, δ} - u_{0} | < ρ}} | E |^{2} | w_{*, δ} - u_{0} |^{2} d x + δ \int_{Ω} | u_{0} |^{\frac{N + 2}{N - 2}} | T_{ρ} (w_{*, δ} - u_{0}) | d x . \end{aligned}$ This proves that for every $ρ > 0$ , $T_{ρ} (w_{*, δ} - u_{0})$ is bounded in $H_{0}^{1} (Ω)$ . Dividing by ρ and taking the limit when ρ tends to zero, thanks to the Lebesgue dominated convergence theorem, we get $\begin{matrix} \int_{Ω} {(| w_{*, δ} | + | u_{0} |)}^{\frac{4}{N - 2}} | w_{*, δ} - u_{0} | d x ⩽ \frac{1}{c_{N}} \int_{Ω} | u_{0} |^{\frac{N + 2}{N - 2}} d x, \end{matrix}$ and thus $w_{*, δ}$ is bounded in $L^{\frac{N + 2}{N - 2}} (Ω)$ . Hence, for a subsequence of δ which converges to zero, still denoted by δ, there exits $w_{*}$ such that $\begin{matrix} (3.18) & \{\begin{array}{l} w_{*, δ} \to w_{*} a.e. in Ω, w_{*, δ} ⇀ w_{*} in L^{\frac{N + 2}{N - 2}} (Ω) \\ T_{ρ} (w_{*, δ} - u_{0}) ⇀ T_{ρ} (w_{*} - u_{0}) in H_{0}^{1} (Ω), \forall ρ > 0 . \end{array} \end{matrix}$ By the lower semicontinuity of the norm in $H_{0}^{1} (Ω)$ , this allows us to pass to the limit when δ tends to zero in (3.17) to deduce $\begin{matrix} \int_{{| w_{*} - u_{0} | < ρ}} {| \nabla (w_{*} - u_{0}) |}^{2} d x ⩽ \frac{1}{α^{2}} \int_{{| w_{*} - u_{0} | < ρ}} | E |^{2} | w_{*} - u_{0} |^{2} d x . \end{matrix}$ Dividing by $ρ^{2}$ and taking the limit when ρ tends to zero, this proves $\begin{matrix} \frac{1}{ρ} T_{ρ} (w_{*} - u_{0}) \to 0 in H_{0}^{1} (Ω) when ρ \to 0 . \end{matrix}$ Combined with $\begin{matrix} \frac{1}{ρ} T_{ρ} (w_{*} - u_{0}) ⇀ sgn (w_{*} - u_{0}) a.e. in Ω, \end{matrix}$ we get $\begin{matrix} (3.19) & w_{*} = u_{0} a.e. in Ω . \end{matrix}$

Step 4. For $ρ > 0$ we take, similarly to the Step 3, $T_{ρ} (w_{n, δ} - u_{n})$ as test function in the difference of (3.12) and (3.2). We get $\begin{aligned} \frac{α}{2} \int_{{| w_{n, δ} - u_{n} | < ρ}} {| \nabla (w_{n, δ} - u_{n}) |}^{2} d x + δ \int_{Ω} | w_{n, δ} |^{\frac{4}{N - 2}} w_{n, δ} T_{ρ} (w_{n, δ} - u_{n}) d x \\ ⩽ \frac{1}{2 α} \int_{{| w_{n, δ} - u_{n} | < ρ}} | E_{n} |^{2} | w_{n, δ} - u_{n} |^{2} d x . \end{aligned}$ Taking into account (3.11) and (3.15), and defining $E$ by (it exists for a subsequence) $\begin{matrix} | E_{n} |^{2} ⇀ E in L^{\frac{N}{2}} (Ω), \end{matrix}$ we can pass to the limit in n in this inequality by semicontinuity and the Rellich–Kondrachov’s compactnes theorem to deduce $\begin{aligned} \frac{α}{2} \int_{{| w_{*, δ} - u | < ρ}} {| \nabla (w_{*, δ} - u) |}^{2} d x + δ \int_{Ω} | w_{*, δ} |^{\frac{4}{N - 2}} w_{*, δ} T_{ρ} (w_{*, δ} - u) d x \\ ⩽ \frac{1}{2 α} \int_{{| w_{*, δ} - u | ⩽ ρ}} E | w_{*, δ} - u |^{2} d x . \end{aligned}$

Now we pass to the limit when δ tends to zero thanks to (3.18) and (3.19), to get $\begin{matrix} \int_{{| u_{0} - u | < ρ}} {| \nabla (u_{0} - u) |}^{2} d x ⩽ \frac{1}{α^{2}} \int_{{| u_{0} - u | ⩽ ρ}} E | u_{0} - u |^{2} d x \end{matrix}$ Dividing by $ρ^{2}$ and passing to the limit when ρ tends to zero we deduce as at the end of Step 3 that $u = u_{0}$ . This finishes the proof. □

Remark 3.2.

One of the applications of Theorem 3.1 is the existence of solutions for some control problems in the coefficients. In this way, combined with Fatou’s Lemma it immediately proves the existence of solution for $\begin{aligned} min {\int_{Ω} (G (x, u) + μ | E |^{p}) d x : E \in L^{p} {(Ω)}^{N}} \\ \{\begin{array}{l} - div (A \nabla u + E u) = f in Ω \\ u \in H_{0}^{1} (Ω), \end{array} \end{aligned}$ with $p > 2$ if $N = 2$ , $p ⩾ N$ if $N > 2$ , $f \in H^{- 1} (Ω)$ and $G : Ω \times R \to R$ such that $\begin{array}{c} G (\cdot, s) measurable, \forall s \in R, G (x, \cdot) continuous for a.e. x \in Ω, \\ \exists a \in R, b ⩾ 0, such that G (x, s) ⩾ a - b | s |, \forall s \in R, a.e. x \in Ω . \end{array}$

Footnotes

Acknowledgement

This work has been partially supported by the project PID2020-116809GB-I00 of the Ministerio de Ciencia e Innovación of the government of Spain.

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