A "nonlinear duality" approach to $W_0^{1, 1}$ solutions in elliptic systems related to the Keller-Segel model

1.   Background: results by Guido Stampacchia and Neil Trudinger

The following results about the summability of the solutions of Dirichlet problems for equations with discontinuous coefficients are nowadays classical, since the papers ^[19] and ^[20] by Guido Stampacchia and Neil Trudinger ^[12].

If $f\in L^{m}(\Omega)$, $m\geq\frac{2N}{N+2}$, thanks to Lax-Milgram Theorem and Sobolev embedding, there exist a weak solution $u\in{W_0^{1, 2}(\Omega)}$ of

where $\Omega$ is a bounded, open subset of ${\mathbb{R}}^{N}$, $N \geq 2$; the matrix $M(x)$ is symmetric, uniformly elliptic and bounded: there exist ${\alpha} > 0$ and $\beta > 0$ such that

for every $\xi$ in ${\mathbb{R}}^{N}$, and for almost every $x$ in $\Omega$.

Moreover:

$1)$ If $f\in L^{m}(\Omega)$, $\frac{2N}{N+2}\leq m\leq\frac N2$, the summability of $u$ (which belongs to $L^{{m^{**}}}(\Omega)$, $m^{**} = \frac{mN}{N-2m}$, if $\frac{2N}{N+2}\leq m < \frac N2$ and it has exponential summability if $m = \frac N2$) was proved in ^[19] (see also ^[5,13,20,21]).

$2)$ If $f\in L^{m}(\Omega)$, $m > \frac N2$, the boundedness of $u$, was proved in ^[19] (see also ^[13,20]).

$3)$ If $f\in L^{m}(\Omega)$, $1 < m < \frac{2N}{N+2}$, in ^[8], is proved a (nonlinear) Calderon-Zygmund theory for operators with discontinuous coefficients, showing the existence of a distributional solution $u\in{W_0^{1, m^*}(\Omega)}$, $m^{*} = \frac{mN}{N-m}$. Note that, in this case $m^*\in(1^*, 2)$ and, thanks to the Sobolev embedding, $u\in L^{{m^{**}}}(\Omega)$ as in (1).

$4)$ The existence in $W_0^{1, 1^*}(\Omega)$ is proved, in ^[8], if ${\int_{\Omega}}|f|\log(1+|f|) < \infty$.

Then a question arises: is it possible to prove (as in (3)) that

In ^[4], it is proved that the above statement is false if $\frac N2 < m < N$.

The results recalled in (1), (2), (3) and (4) are crucial in the next proofs.

2.   A stationary system weakly related to the Keller-Segel model

In this paper, we prove existence of distributional solutions of the following nonlinear elliptic boundary value problem:

with

Of course, $f(x)$ needs not to be identically null. On the power $\sigma$ we suppose $1 < \sigma < \frac{2N-2}{N-2}$, but it is preferable (in the following proofs) to split the assumption as

or

or

There are many theoretical models for chemotaxis; one of the most important is the Keller-Segel one (see ^[14,16,17] and also ^[3,10]). Following ^[15] (see also ^[14]), one of the possible models is the "chemical signal driven logistic model" which leads, in the stationary case, to the system above. Note that the equation for $u$ includes a chemotaxis term with nonlinear flux limitation having a kind of logarithmic dependence: $\frac{{\nabla}\psi}{1 + \psi} = {\nabla}\log(1+\psi)$.

The original model of chemotaxis presents a linear dependence of the gradient of the concentration of the chemical substance $\psi$ in the equation of $u$ in the form $- {\rm{div}}(\chi\, {\rm{u}}\, \nabla \psi),$ for a positive given constant $\chi$.

2.1.   Approximate problems

Note that, with our assumption, we have $\sigma-1 > 0$. We define ${f_{n}} (x) = \frac{f(x)}{1+\frac1n\, f(x)}$ and we consider the following approximate Dirichlet problem

where, $\forall\, k\in{\mathbb{R}}^+$,

Here we enumerate some properties of the solutions ${u_{n}}$, ${\psi_{n}}$.

${\bf{A}}$ The existence of $({u_{n}}, {\psi_{n}})$ is a consequence of Proposition 3.1 of ^[11] (with minimal changes).

${\bf{B}}$ The positivity of $f(x)$ gives, in the first equation, ${u_{n}}\geq0$ (see ^[3]), which, in the second equation, implies ${\psi_{n}}\geq0$.

${\bf{C}}$ In the first equation, $0\leq f_n(x)\leq n$ and the modulus of the function in the divergence term is less than $n^2$, so that, with the boundedness theorem by G. Stampacchia (^[19], see also ^[20]), we deduce $\left\|u_n\right\|_{L^{\infty}(\Omega)} \leq C_0 n^2$. Thus, in the second equation, we observe that $T_{(n^3)}({u_{n}}) = {u_{n}}$ (for $n > n_0$) and we can rewrite the above system as

3.   Nonlinear duality method

In the following lemma, in spite of the nonlinearity of the problem, we use a kind of duality, which will be advantageous to prove a priori estimates.

Lemma 3.1. We assume (1.2), (2.2), $1 < \sigma < \frac{2N-2}{N-2}$. Let $a\in(0, 1)$. Then the following "nonlinear dual" inequality holds

Proof. In the above system, we use $\log(1+{\psi_{n}})$ as test function in the first equation, $\dfrac{-\, {{u_{n}} } }{(1+{\psi_{n}})}$ as test function in the second equation and we have

Then, after simplifications (we use $0\leq\dfrac{T_n({u_{n}})}{1+\frac1n\, |{\nabla}{\psi_{n}}|} \leq{u_{n}}$), we deduce that

and, dropping a positive term, we prove the inequality

Now we use the inequality $0\leq\log(1+{\psi_{n}})\leq \frac1a({\psi_{n}})^a$, $a\in(0, 1)$, and we have (3.1). □

Lemma 3.2. We assume (1.2), (2.2). Then the sequence $\{{{u_{n}} } \}$ is bounded in

Proof. First part: $\frac{N}{N-2} < \sigma < \frac{2N-2}{N-2}$ - Let $q < \sigma$. Then (we use Hölder inequality with exponents $\frac{\sigma}{q}$ and $\frac{\sigma}{\sigma-q}$) we have, using (3.1),

that is

Define $q = \frac{N}{N-2}$ and $p = \frac{N}{(\sigma-1)(N-2)}$; we note that $p > 1$ since $\sigma < \frac{2N-2}{N-2}$.

Then we use Calderon-Zygmund type estimates for Dirichlet problems with infinite energy solutions, proved in ^[8,19] (see (1) and (3)) and we have

We note that $p(\sigma-1) = q$ and we rewrite the last inequality as

Thus for $a > 0$ close to zero, we have proved the following estimate, where $\rho > 1$ is close to one,

Second part: $\sigma = \frac{N}{N-2}$ - There is only a slight change with respect to the previous case: $p(\sigma-1) < q$.

Third part: $1\leq \sigma < \frac{N}{N-2}$ - Recall the following $L^\infty$ estimate (proved in ^[19], see also ^[20]), concerning the second equation,

Then we deduce directly from (3.2)

and

Let $p = \sigma'$ (which implies $\sigma < \frac{N}{N-2}$). Then

□

Corollary 3.3. We assume (1.2), (2.2). As a consequence of the previous lemma, the sequence $\{({{u_{n}} })^{\sigma-1}\}$ is bounded in

Thus the right hand side of the second equation is bounded in $L^{1}(\Omega)$ if $\frac{N}{(N-2)(\sigma-1)}\geq1$; that is, if $\sigma\leq \frac{2N-2}{N-2}$.

Corollary 3.4. If, in the second equation of (2.6), we take as test function $\dfrac{{\psi_n} }{1+{\psi_n} }$, $($following ^[2,3]$)$, we have

Corollary 3.5. The sequence $\{{\psi_{n}}\}$ is bounded in ${W_0^{1, 2}(\Omega)}$ if the right hand side of the second equation is bounded in $L^{{\frac{2N}{N+2}}}(\Omega)$ that is if

Corollary 3.6. If in the first equation of (2.6) we take as test function $\dfrac{{{u_{n}} } }{1+{{u_{n}} } }$, following ^[2,3], and we use Young inequality, we have

That is, the sequence $\{\frac{|{\nabla} {{u_{n}} } |}{(1 +{{u_{n}} })}\}$ is bounded in $L^{2}(\Omega)$; with this boundedness, in ^[3], is proved that there exists a measurable function $u(x)$ such that

Corollary 3.7. If in the first equation of (2.6) we take as test function $T_k({{u_{n}} })$, following ^[2,3], we deduce

so that we can add to (3.6) the following weak convergence

Corollary 3.8. If $1 < \sigma < \frac{2N-2}{N-2}$, the sequence $\{({{u_{n}} })^{\sigma-1}\}$ is bounded in $L^{{\nu}}(\Omega)$, $\nu > 1$ (and more: in $L^{{\sigma'}}(\Omega)$ if $1 < \sigma < \frac{N}{N-2}$). Then the above a.e. convergence (3.6) and the Vitali theorem say that the sequence $\{({{u_{n}} })^{\sigma-1}\}$ converges in $L^{1}(\Omega)$ to $\{u^{\sigma-1}\}$.

Then (see ^[7,8]) the sequence $\{{\psi_{n}}\}$ is compact in ${W^{1, q}_{0}(\Omega)}$, $q < \frac{N}{N-1}$, at least; in Corollary 3.5 is proved a stronger result for a smaller subset of exponents $\sigma$. Define $\psi$ a cluster point of $\{{\psi_{n}}\}$ in ${W^{1, q}_{0}(\Omega)}$.

Corollary 3.9. A result by Leone-Porretta (^[18]) states that the sequence $\{{\nabla} T_k(\psi_{n})\}$ is $L^2$ compact, because the right hand side of the second equation in (2.6) is $L^1$ compact (Corollary 3.8).

Lemma 3.10. The sequence

Proof. If in the second equation of (2.6) we take $\Big[\dfrac{{\psi_{n}}}{1+{\psi_{n}}}-\dfrac{k}{1+k}\Big]^+$ as test function and we use Hölder inequality, we have (recall (3.3))

Now we use this inequality to prove the $L^1$ equi-integrability of the sequence $\Big\{\dfrac{|{\nabla}\psi_{n}|^2}{(1 + \psi_{n})^{2} }\Big\}$. Indeed, for every measurable subset $E\subset\Omega$, we have

Now Corollary 3.9 says that, for every $k\in{\mathbb{R}}^+$, the last integral is small (uniformly with respect to $n$) if $|E|$ is small. Here $|E|$ denotes the measure of the subset $E$.

Moreover $|\{k < {\psi_{n}}\}\big|$ is small (uniformly with respect to $n$) for $k$ large enough. Thus the last two sentences prove that

Furthermore a result proved in ^[8] implies that the sequences $\{{\nabla}\psi_{n}(x)\}$ and $\{\psi_{n}(x)\}$ converge almost everywhere, so that these a.e. convergences, (3.10) and Vitali theorem yield (3.8). □

Corollary 3.11. In the first equation of (2.6) we take as test function $\Big[\dfrac{{u_{n}} }{1+{u_{n}} }-\dfrac{k}{1+k}\Big]^+$, $k\in{\mathbb{R}}^+$, (following ^[2,3]) we use the Young inequality and we have

Moreover, there is a second important consequence of (3.10): the a priori estimates on the sequence $\{{{u_{n}} } \}$ imply that $|\{k < {{u_{n}} } \}\big|$ is small for $k$ large (uniformly with respect to $n$), so that, in (3.11), the term ${\int_{\{k < {u_{n}} \}}}\frac{|{\nabla}\psi_{n}|^2}{(1 + \psi_{n})^{2} }$ is small (uniformly with respect to $n$) if $k$ is large enough and then the term

3.1.   Entropy solutions

Following ^[3] and ^[1] we recall the definition of entropy solution, useful in cases (as here) of very singular framework, where the definition of distributional solution is meaningless.

Note that, if $N > 4$, $u\not\in L^{2}(\Omega)$, so that the term $u\frac{{\nabla}\psi}{1+\psi}$ does not belong to $L^1$.

Definition 3.12. A measurable function $u$ is an entropy solution of the first equation of our system if

Thanks to (3.6), Corollary 3.8, (3.8), we can use the above definition for our problem, we can repeat the proof of Theorem 3.9 of ^[3] and we prove the following result.

Theorem 3.13. Assume (1.2), (2.2), $1 < \sigma < \frac{2N-2}{N-2}$. Then there exists an entropy solution $u\geq0$ of the first equation in the sense of Definition 3.12. Moreover there exists a weak solution $0\leq\psi\in{W_0^{1, 2}(\Omega)}$ of the second equation, if $\sigma\leq\frac{3(N+2)}{2(N-2)} \hbox{ and } \frac{N}{N-2} < \sigma < \frac{2N-2}{N-2}$, or a distributional solution $0\leq\psi\in{W^{1, q}_{0}(\Omega)}$, in the other range of value of $\sigma$.

Remark 3.14. Note that we have not proved that $u$, entropy solution of the first equation, belongs to some Sobolev space; we only have, from (3.5), that $\log(1+u)$ belongs to ${W_0^{1, 2}(\Omega)}$.

3.2.   Distributional solutions

In this subsection we study a case of distributional solutions $u$, that is a case of ${\nabla} u\in L^1$.

Observe that Lemma 3.2 says that the sequence $\{{u_{n}}\}$ is bounded in $L^{2}(\Omega)$ if

Lemma 3.15. Assume (3.14). Then the sequence $\{{\nabla}{u_{n}}\}$ is equi-integrable and the sequence $\{{u_{n}}\}$ is $L^1$ compact.

Proof. Here we follow an approach of ^[9] (see also ^[6]). Since we observed that the sequence $\{{u_{n}}\}$ is bounded in $L^{2}(\Omega)$, we use the Hölder inequality, (3.11) and we have

In (3.12) is proved that $\omega_k$ is small (uniformly with respect to $n$) if $k$ is large enough. Then, for every measurable subset $E\subset\Omega$, we deduce that

which implies (recall Corollary 3.7)

that is the equi-integrability.

The above inequalities, with $k = 0$, give the $L^1$ boundedness of of the sequence $\{{\nabla}{u_{n}}\}$. Then the $L^1$ compactness of the sequence $\{{u_{n}}\}$ is a consequence of the Rellich theorem.

Thus we improved (3.6):

Now we can state the existence of distributional solutions. □

Theorem 3.16. Under the assumptions of Theorem 3.13, let assume (3.14). Then there exist distributional solutions $0\leq u\in{W_0^{1, 1}(\Omega)}$ and $0\leq\psi\in{W^{1, q}_{0}(\Omega)}$, $q < \frac{N}{N-1}$, of system (2.1); that is, we have that

for every $v$ in $C^{1}_{0}(\Omega)$, and

for every ${\varphi}$ in $C^{1}_{0}(\Omega)$.

3.3.   A direct approach to the boundedness of the sequence $\{{\psi_{n}}\}$

In this subsection, we assume (1.2), $f\in L^{1}(\Omega)$, $1 < \sigma < \frac32+\frac1N$.

Following ^[3], we prove the following a priori estimate

Indeed, if we take $\; { }\frac{u_n}{{h}+|u_n|}\;$ as test function in the first equation, we have (thanks to the Young inequality)

which implies, dropping a positive term and letting ${h}\to 0$, the estimate (3.16).

Thus, for the right hand side of the second equation we have the estimate

and, if $\frac1{\sigma-1} > \frac N2$ (that is $\sigma-1 < \frac2N$), the right hand side of the second equation is bounded in $L^{s}(\Omega)$, $s > \frac N2$, which implies that the sequence of the solutions $\{\psi_n\}$ is bounded in $L^{s}(\Omega)$; if $\frac1{\sigma-1}\geq\frac{2N}{N+2}$ (that is $\sigma-1\leq\frac12+\frac1N$), the right hand side of the second equation is bounded in $L^{{\frac{2N}{N+2}}}(\Omega)$, which implies that the sequence of the solutions $\{\psi_n\}$ is bounded in ${W_0^{1, 2}(\Omega)}$.

Summarizing, with this approach,

with the use of the estimate (3.16).

3.4.   General nonlinearities

It is possible to adapt our approach (nonlinear duality) to the case of the system

with $\gamma\in{\mathbb{R}}^+$. A possible approach (which we only sketch here) is

● define an approximate system (as in (2.6));

● use as test functions $\Big(g({\psi_{n}}), \dfrac{{{u_{n}} } }{h({\psi_{n}})}\Big)$

with

Conflict of interest

The author declares no conflict of interest.