Critical points of a singular perturbation problem via reduced energy and local linking
The authors consider the following Neumann problem: \[ -\varepsilon^2\Delta u+u-f(u)=0\quad \text{in } \Omega, \qquad u'_\nu=0\quad \text{on} \partial\Omega, \] the generalized solution \(u\in H^1(\Omega)\) of which is the critical point of the functional \(I[u]=\frac{1}{2} \int_\Omega[\varepsilon^2|\nabla u|^2+u^2-2F(u)] dx,\) where \(F'=f\in C^{1,1}(\mathbb R)\), \(F(u)=0\) if \(u<0\), \(F(u)=O_{u\to\infty}(u^{p+1})\), \(1<p<(d+2)/(d-2)\), for \(d=2\), \(1<p<\infty,\) (there are also some other assumptions on \(f\), an example is \(F(u)=u^3/3\)); \(\Omega\subset\mathbb R^d\) is a bounded domain with \(C^3\) boundary, \(\varepsilon\) is a small parameter. The authors prove facts about the asymptotics of critical values of the following type: Theorem 1. There exists a critical point \(u_\varepsilon\) of \(I\) with the critical value \[ c_\varepsilon=\varepsilon^d\{I_0[W]- \gamma_0(\xi_0,\varepsilon) \exp(-2\text{ dist}(\xi_0,\partial\Omega)/\varepsilon)\}, \] where \(\xi_0\in\Omega\) is a point at which \(\text{dist}(\xi_0,\partial\Omega)\) attains its maximum, \(\exists C>0\), \(C\varepsilon^{(d-1)/2}\) \(\leq\gamma_0(\xi_0,\varepsilon)\leq C,\) \(I_0[u]=I[u]|_{\varepsilon=1}\), \(W\) is the unique, positive radially symmetric solution to \(-\Delta W+W-f(W)=0\) in \(\mathbb R^d.\)
nonlinear elliptic equation with parameter
scientific article; zbMATH DE number 1417616
scientific article