An asymptotic nondegeneracy result for a biharmonic equation with the nearly critical growth
The author considers the biharmonic equation \[ \begin{cases} \Delta^2 u = c_0u^{p_{\varepsilon}} & \text{in } \Omega,\\ u>0 & \text{in } \Omega,\\ u=\Delta u=0 & \text{on } \partial \Omega \end{cases} \] where \(\Omega\subset \mathbb{R}^N (N\geq 5)\) is a smooth bounded domain, \(c_0=(N-4)(N-2)N(N+2)\), \(\varepsilon>0\), and \(p_{\varepsilon}=p-\varepsilon\) where \(p=(N+4)/(N-4)\) is the critical Sobolev exponent in view of the Sobolev embedding \(H^2(\Omega)\cap H^1_0(\Omega)\hookrightarrow L^{p+1}(\Omega)\). Let \(G(x,z)\) be Green's function of \(\Delta^2\) under the Navier boundary condition \[ \begin{cases} \Delta^2 G(\cdot,z) = \delta_z & \text{in } \Omega,\\ G(\cdot,z)=\Delta G(\cdot,z)=0 & \text{on } \partial \Omega. \end{cases} \] Denote by \(H(x,z)\) the regular part of \(G(x,z)\) and \(R(x)=H(x,x)\) the Robin function of \(G(x,z)\). Then the main result of the paper can be stated as follows: Let \(u_{\varepsilon}\) be a solution to the above equation such that \(\{u_{\varepsilon}\}\) is a minimizing sequence of the Sobolev quotient, and let the blow up point of \(\{u_{\varepsilon}\}\) be a nondegenerate critical point of the Robin function \(R(x)\), then \(u_{\varepsilon}\) is nondegenerate for \(\varepsilon>0\) sufficiently small.
scientific article; zbMATH DE number 5617600