Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem. (English) Zbl 1150.35006
The authors investigate the problem: \(\varepsilon^2\Delta v-v-\gamma_1Vv+f(v)=0\), \(\Delta V+\gamma_2v^2=0\) with \(v, V>0\) in \(\Omega\), and \(v=V=0\) on \(\partial\Omega,\) where \(\Omega\) is a smooth bounded domain in \(\subset \mathbb R^3\), \(\varepsilon,\;\gamma_1,\;\gamma_2\) are positive constants. The function \(f\in C^2(\mathbb R)\) is subject to the following conditions: \(f(t)\) vanishes for \(t\leq 0\), \(f(t)/t^3\) is non decreasing for \(t>0\), \(f(t)=O(t^p)\) as \(t\to\infty\) with \(3<p<5\), there exists a constant \(\theta>4\) such that \(0<\theta\int_0^tf(s)\,ds\leq t f(t)\) for \(t>0\), and the problem \(\Delta w-w+f(w)=0\) in \(\mathbb R^3\), \(w(0)=\max_{\mathbb R^3} w(x)\), \(\lim_{|x|\to\infty}w(x)=0\) has a unique positive non-degenerate solution.
This problem has a variational structure. Indeed, for every \(v\in H^1_0(\Omega)\) let \((-\Delta)^{-1}[v^2]\in H^1_0(\Omega)\) be the unique solution of the problem \(\Delta V+v^2=0\) in \(\Omega\), \(V=0\) on \(\partial\Omega\). Then the previous system can be written as \[ \varepsilon^2\Delta v-v-\gamma (-\Delta)^{-1}[v^2]v+f(v)=0,\;\;v>0\;\text{in}\;\Omega,\;v=0\;\text{on}\;\partial\Omega, \] with \(\gamma=\gamma_1\gamma_2\). Associated with this problem is the energy functional \[ J_\varepsilon[v]=\frac{1}{2}\int_\Omega(\varepsilon^2|\nabla v|^2+v^2)\,dx+\frac{\gamma}{4}\int_\Omega(-\Delta)^{-1}[v^2]v^2\,dx-\int_\Omega F(v)\,dx, \] with \(F(t)=\int_0^tf(s)\,ds.\) The main result is the following. For every \(\varepsilon>0\) there exists a least-energy solution \(v_\varepsilon\in H^1_0(\Omega)\). Furthermore, as \(\varepsilon\to 0\), \(v_\varepsilon\) develops a spike near the maxima of the mean curvature of \(\partial\Omega\); more precisely, there exists \(P_\varepsilon\in\Omega\) such that (i) \(v_\varepsilon=w(\frac{x-P_\varepsilon}{\varepsilon}) +o(1)\) uniformly in \(\Omega\), (ii) dist\((P_\varepsilon,\partial\Omega)=(1+o(1))\varepsilon\log\frac{1}{\varepsilon}\), (iii) for every sequence \(\varepsilon_n\to 0^+\), up to a subsequence, \(P_{\varepsilon_n}\rightarrow P_0\in\partial\Omega,\) where \(P_0\) is a point of maximum mean curvature of \(\partial\Omega\). The proof of this interesting result is based on the energy method.
This problem has a variational structure. Indeed, for every \(v\in H^1_0(\Omega)\) let \((-\Delta)^{-1}[v^2]\in H^1_0(\Omega)\) be the unique solution of the problem \(\Delta V+v^2=0\) in \(\Omega\), \(V=0\) on \(\partial\Omega\). Then the previous system can be written as \[ \varepsilon^2\Delta v-v-\gamma (-\Delta)^{-1}[v^2]v+f(v)=0,\;\;v>0\;\text{in}\;\Omega,\;v=0\;\text{on}\;\partial\Omega, \] with \(\gamma=\gamma_1\gamma_2\). Associated with this problem is the energy functional \[ J_\varepsilon[v]=\frac{1}{2}\int_\Omega(\varepsilon^2|\nabla v|^2+v^2)\,dx+\frac{\gamma}{4}\int_\Omega(-\Delta)^{-1}[v^2]v^2\,dx-\int_\Omega F(v)\,dx, \] with \(F(t)=\int_0^tf(s)\,ds.\) The main result is the following. For every \(\varepsilon>0\) there exists a least-energy solution \(v_\varepsilon\in H^1_0(\Omega)\). Furthermore, as \(\varepsilon\to 0\), \(v_\varepsilon\) develops a spike near the maxima of the mean curvature of \(\partial\Omega\); more precisely, there exists \(P_\varepsilon\in\Omega\) such that (i) \(v_\varepsilon=w(\frac{x-P_\varepsilon}{\varepsilon}) +o(1)\) uniformly in \(\Omega\), (ii) dist\((P_\varepsilon,\partial\Omega)=(1+o(1))\varepsilon\log\frac{1}{\varepsilon}\), (iii) for every sequence \(\varepsilon_n\to 0^+\), up to a subsequence, \(P_{\varepsilon_n}\rightarrow P_0\in\partial\Omega,\) where \(P_0\) is a point of maximum mean curvature of \(\partial\Omega\). The proof of this interesting result is based on the energy method.
Reviewer: Giovanni Porru (Cagliari)
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35J60 | Nonlinear elliptic equations |
| 35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |
| 92C15 | Developmental biology, pattern formation |
| 92C40 | Biochemistry, molecular biology |
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