Unbounded mass radial solutions for the Keller–Segel equation in the disk

Abstract

We consider the boundary value problem

$$\begin{aligned}\left\{ \begin{array}{rcll} -\Delta u+ u -\lambda e^u&{}=&{}0,\ u>0 &{} \mathrm {in}\ B_1(0)\\ \partial _\nu u&{}=&{}0&{}\mathrm {on}\ \partial B_1(0), \end{array}\right. \end{aligned}$$

whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here $B_1(0)$ is the unit disk, $\nu$ the outer normal to $\partial B_1(0)$, and $\lambda >0$ is a parameter. We show that, provided $\lambda$ is sufficiently small, there exists a family of radial solutions $u_\lambda$ to this system which blow up at the origin and concentrate on $\partial B_1(0)$, as $\lambda \rightarrow 0$. These solutions satisfy

$$\begin{aligned}\lim _{\lambda \rightarrow 0} \frac{u_\lambda (0)}{|\ln \lambda |}=0\quad \text{ and }\quad 0<\lim _{\lambda \rightarrow 0} \frac{1}{|\ln \lambda |}\int _{B_1(0)}\lambda e^{u_\lambda (x)}dx<\infty , \end{aligned}$$

having in particular unbounded mass, as $\lambda \rightarrow 0$.

Appendix A: An elliptic estimate

We show a very rough elliptic estimate which is needed in the proof of Lemma 3.2.

Lemma A.1

Let $R>0$ and $u\in H^1 (B_R (0))$ be a radial solution to

$$\begin{aligned} \left\{ \begin{array}{rcll}-\Delta u +u &{}=&{} f&{} \mathrm {in}\ B_R(0)\\ u^\prime (R)&{}=&{}g,&{} \end{array}\right. \end{aligned}$$

for some $f\in L^q (B_R (0))$, with $q>2$. Then, we have

$$\begin{aligned} \Vert u\Vert _{L^\infty (B_R (0))} \le C\left[ \left( \dfrac{1}{R}+R\right) R^{1-2/q} \Vert f\Vert _{L^q (B_R (0))}+\left( \dfrac{ 1}{R}+ R^2 \right) \Vert g\Vert _{L^\infty (\partial B_R (0))} \right] \end{aligned}$$

and

$$\begin{aligned}\Vert u' \Vert _{L^\infty (B_R (0))}\le C\left[ R^{1-2/q} \Vert f\Vert _{L^q (B_R (0))}+( 1+ R ) \Vert g\Vert _{L^\infty (\partial B_R (0))} \right] \end{aligned}$$

for some constant C not depending on R.

Proof

Multiplying the equation by u and integrating by parts, we get

$$\begin{aligned} \Vert u\Vert _{H^1 (B_R )}^2 \le \Vert f\Vert _{L^2 (B_R )} \Vert u\Vert _{H^1 (B_R )}+R |u' (R)| |u(R)| . \end{aligned}$$

(A.1)

Since $u(R)-u(r)=\int _r^R u' (s) ds$, one deduces that

$$\begin{aligned}|u(R)|^2 \le C \left[ |u(r)|^2 +\Vert u'\Vert _{L^2 (B_R )}^2 \ln \dfrac{R}{r} \right] , \end{aligned}$$

where throughout the proof C denotes a constant not depending on R. Multiplying the previous inequality by r and integrating, we find

$$\begin{aligned}R^2 |u(R)|^2 \le C[\Vert u\Vert _{L^2 (B_R )}^2+\Vert u'\Vert _{L^2 (B_R) }^2 R^2 ]. \end{aligned}$$

This implies that

$$\begin{aligned} |u(R)|\le C\left( \dfrac{1}{R}+1 \right) \Vert u\Vert _{H_1 (B_R )}. \end{aligned}$$

(A.2)

From (A.1), (A.2), and $u' (R)=g$, we obtain that

$$\begin{aligned} \Vert u\Vert _{H^1 (B_R )}^2 \le \Vert f\Vert _{L^2 (B_R )} \Vert u\Vert _{H^1 (B_R)}+C( 1+ R ) \Vert g\Vert _{L^\infty (\partial B_R )} \Vert u \Vert _{H^1 (B_R)} . \end{aligned}$$

Thanks to Hölder inequality, we find that

$$\begin{aligned} \Vert u\Vert _{H^1 (B_R)} \le C[ R^{1-2/q} \Vert f\Vert _{L^q (B_R)} +(1+R) \Vert g\Vert _{L^\infty (\partial B_R)} ] .\end{aligned}$$

(A.3)

Next, observe that for any $s\in (0,R)$ we can rewrite the equation as

$$\begin{aligned}u' (s) s= \int _0^s (u-f) r dr. \end{aligned}$$

From Hölder inequality, we obtain that

$$\begin{aligned}|u'(s)|\le C \Vert u-f \Vert _{L^2 (B_R ) } \le C (\Vert u\Vert _{L^2 (B_R )}+ R^{1-2/q} \Vert f\Vert _{L^q (B_R)} ). \end{aligned}$$

From (A.3), we deduce that

$$\begin{aligned}\Vert u'\Vert _{L^\infty (B_R )}\le C (R^{1-2/q} \Vert f\Vert _{L^q (B_R )} +(1+R ) \Vert g\Vert _{L^\infty (\partial B_R )} ). \end{aligned}$$

By noting that

$$\begin{aligned}u(R)-u({{\tilde{s}}})= \int _{{{\tilde{s}}}}^R u' (r) dr, \end{aligned}$$

we get from (A.2) that

$$\begin{aligned} \Vert u\Vert _{L^\infty (B_R)}&\le C\left[ \left( \dfrac{1}{R}+1 \right) \Vert u\Vert _{H^1 (B_R)}+ R \Vert u'\Vert _{L^\infty (B_R)}\right] \\&\le C \left[ \left( \dfrac{1}{R}+1+R\right) R^{1-2/q} \Vert f\Vert _{L^q (B_R)} + \left( \dfrac{1}{R}+R^2\right) \Vert g\Vert _{L^\infty (\partial B_R)}\right] . \end{aligned}$$

This concludes the proof. $\square$