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 \(r>0\).
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Appendix
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
where \(I_n\), \(n\ge 1\), is an arbitrary sequence of intervals such that
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,
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
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 \(\alpha >0\),
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
for some \(\alpha >0\).
3. If \(\partial \Omega \) is \(C^2\) and
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
admits a unique positive solution, \(v_{a,k}\). Furthermore, the nonlinearity \(r\mapsto r^p\) satisfies the Keller–Osserman condition. Hence, the equation
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
converges, as \(n\rightarrow \infty \), to a barrier function \(u_{r,z}\in C({{\overline{\Omega }}}\cap B_r(z))\) satisfying
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
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
is a solution of
which vanishes on \(\partial \Omega {\setminus }\{a\}\) and blows up at a. We conjecture that the strong barrier property holds if
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López-Gómez, J., Maire, L. & Véron, L. General uniqueness results for large solutions. Z. Angew. Math. Phys. 71, 109 (2020). https://doi.org/10.1007/s00033-020-01325-5
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DOI: https://doi.org/10.1007/s00033-020-01325-5
Keywords
- Keller–Osserman condition
- local graph condition
- large solutions
- maximum principle
- strong barrier property