General uniqueness results for large solutions

Abstract

We give a series of very general sufficient conditions in order to ensure the uniqueness of positive large solutions for $-\Delta u+f(x,u)=0$ in a bounded domain $\Omega$ where $f:{{\overline{\Omega }}}\times {\mathbb {R}}\mapsto {\mathbb {R}}_+$ is a continuous function, such that $f(x,0)=0$ for $x\in {{\overline{\Omega }}}$, and $f(x,r)>0$ for x in a neighborhood of $\partial \Omega$ and all $r>0$.

Appendix

1.1 On the Keller–Osserman condition

The next result shows how imposing the Keller–Osserman condition on the associated function g is stronger than imposing it on f.

Proposition 4.1

There are increasing functions f that satisfy (KO) and such that the corresponding function g does not.

Proof

To construct such an example, one can consider any function f such that

$$\begin{aligned} u^2\le f(u)\le u^3 \quad \hbox {and}\quad f(u)=f(\min I_n) \quad \hbox {for all}\;\; u\in I_n, \end{aligned}$$

where $I_n$, $n\ge 1$, is an arbitrary sequence of intervals such that

$$\begin{aligned} \lim _{n\rightarrow +\infty } (\max I_n-\min I_n) =+\infty \quad \hbox {and}\quad \max I_n <\min I_{n+1}\quad \hbox {for all}\;\; n\in \mathbb {N}. \end{aligned}$$

By the properties of $u^2$ and $u^3$, such a sequence of intervals exists. For this choice we have that, for any given $\ell >0$ and $u\in I_n$, $[u,\ell +u]\subset I_n$ for sufficiently large $n>0$ and hence,

$$\begin{aligned} f(\ell +u)-f(u)=0. \end{aligned}$$

Thus, $g(\ell )=0$. Therefore, $g\equiv 0$, which does not satisfy (KO). $\square$

1.2 On the strong barrier property

The general problem of finding conditions so that the strong barrier property occurs is open. We give below some cases where it holds and a case where it does not. They all deal with nonlinearity of the form

$$\begin{aligned} g(x,r)={{\mathfrak {a}}}(x){\tilde{g}}(r) \end{aligned}$$

(4.1)

where ${{\mathfrak {a}}}\in C({{\overline{\Omega }}})$ is nonnegative and positive near $\partial \Omega$ and ${\tilde{g}}:{\mathbb {R}}_+\mapsto {\mathbb {R}}_+$ is continuous and nondecreasing, vanishes at 0 and satisfies (KO).

1. If ${{\mathfrak {a}}}>0$ on $\partial \Omega$, then the Keller–Osserman condition holds in ${{\overline{{{\mathcal {V}}}}}}$, where ${{\mathcal {V}}}$ is a neighborhood of $\partial \Omega$, because the function ${{\mathfrak {a}}}$ can be extended to $\Omega ^c$ as a continuous and positive function by Whitney embedding theorem (see e.g., [9]). It is a completely open problem to find out sufficient conditions in the case where ${{\mathfrak {a}}}>0$ vanishes on the boundary.

2. If $\partial \Omega$ is $C^2$ and, for some $\alpha >0$,

$$\begin{aligned} g(x,r)\ge d^\alpha (x)u^p \end{aligned}$$

it is proved in [24] that the strong barrier property holds. When $\partial \Omega$ is Lipschitz, the distance function loses its intrinsic interest and has often to be replaced by the first eigenfunction $\phi _1$ of $-\Delta$ in $H^{1}_0(\Omega )$. In such case, we conjecture that the strong barrier property holds if

$$\begin{aligned} g(x,r)\ge \phi _1^\alpha (x)u^p \end{aligned}$$

for some $\alpha >0$.

3. If $\partial \Omega$ is $C^2$ and

$$\begin{aligned} g(x,r)\le \mathrm{e}^{-\frac{\kappa }{d(x)}}r^p \end{aligned}$$

with $\kappa >0$ and $p>1$, then the strong barrier property does not hold. Indeed, it is proved in [26] that, for every $a\in \partial \Omega$ and $k>0$, the problem

$$\begin{aligned} \begin{array} {lll} -\Delta u+\mathrm{e}^{-\frac{\kappa }{d(x)}}u^p=0&{}\quad \text {in }\;\Omega ,\\ u=k\delta _a &{}\quad { \text {on }}\;\partial \Omega , \end{array} \end{aligned}$$

(4.2)

admits a unique positive solution, $v_{a,k}$. Furthermore, the nonlinearity $r\mapsto r^p$ satisfies the Keller–Osserman condition. Hence, the equation

$$\begin{aligned} -\Delta u+\mathrm{e}^{-\frac{\kappa }{d(x)}}u^p=0\quad \text {in }\;\Omega \end{aligned}$$

(4.3)

admits a minimal, $u^\mathrm{min}$, and a maximal, $u^\mathrm{max}$, large solution (probably they are equal). However, $v_{a,k}\uparrow u^\mathrm{min}$ when $k\rightarrow \infty$. Arguing by contradiction, assume that the equation satisfies the strong barrier property at $z\in \partial \Omega$. Then, there exists $r>0$ such that the solution $u:=u_n$ of the problem

$$\begin{aligned} \begin{array} {lll} -\Delta u+\mathrm{e}^{-\frac{\kappa }{d(x)}}u^p=0 &{}\quad \text {in }\;B_r(z)\cap \Omega , \\ u=n &{}\quad \text {on }\;\Omega \cap \partial B_r(z),\\ u=0 &{}\quad \text {on }\;\partial \Omega \cap B_r(z), \end{array} \end{aligned}$$

converges, as $n\rightarrow \infty$, to a barrier function $u_{r,z}\in C({{\overline{\Omega }}}\cap B_r(z))$ satisfying

$$\begin{aligned} \begin{array}{lll} -\Delta u+\mathrm{e}^{-\frac{\kappa }{d(x)}}u^p=0 &{}\quad \text {in }\;B_r(z)\cap \Omega , \\ u=\infty &{}\quad \text {on }\;\Omega \cap \partial B_r(z). \end{array} \end{aligned}$$

Taking a point $a\in \partial \Omega \cap B^c_{2r}(z)$, for any $k>0$ there exists $n=n(k)$ such that $v_{a,k}\le n(k)$ on $\Omega \cap \partial B_r(z)$. Since $v_{a,k}=0$ on $\partial \Omega \cap B_r(z)$, it follows that $v_{a,k}\le u_n$. Thus, letting $k\rightarrow \infty$ yields $u^\mathrm{min}\le u_{r,z}$, which is a contradiction.

4. If $\partial \Omega$ is $C^2$ and

$$\begin{aligned} g(x,r)= \mathrm{e}^{-\frac{1}{d^\alpha (x)}}r^p, \end{aligned}$$

with $0<\alpha <1$ and $p>1$, it is proved in [33] that the limit when $k\rightarrow \infty$ of the solutions $v_{a,k}$ of

$$\begin{aligned} \begin{array}{ll} -\Delta u+\mathrm{e}^{-\frac{1}{d^\alpha (x)}}u^p=0 &{}\quad \text {in }\;\Omega \\ u=k\delta _a &{}\quad \text {on }\;\partial \Omega , \end{array} \end{aligned}$$

(4.4)

is a solution of

$$\begin{aligned} -\Delta u+\mathrm{e}^{-\frac{1}{d^\alpha (x)}}u^p=0\quad \text {in }\;\Omega \end{aligned}$$

which vanishes on $\partial \Omega {\setminus }\{a\}$ and blows up at a. We conjecture that the strong barrier property holds if

$$\begin{aligned} g(x,r)\ge \mathrm{e}^{-\frac{1}{d^\alpha (x)}}r^p. \end{aligned}$$