Some existence theorems for the sublinear Neumann boundary value problem
The author studies the Neumann boundary value problem in the cases of both \(k= 1\) and \(k> 1\): \[ -\Delta u= \mu_k u+ g(x,u)\quad\text{in }\Omega,\quad {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega,\tag{1} \] where \(\Omega\subset \mathbb{R}^N\) \((N> 1)\) is a bounded domain with smooth boundary and outward normal \(n(x)\) and \({\partial u\over\partial n}= n(x)\cdot\nabla u\), \(\mu_k\) is the \(k\)th distinct eigenvalue of the eigenvalue problem \[ -\Delta u= \mu u\quad\text{in }\Omega,\quad {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega, \] \(\mu_1= 0\), \(g:\overline\Omega\times \mathbb{R}\to\mathbb{R}\) is a Carathéodory function. Under some assumptions on the function \(g(x,t)\), using the minimax method in critical point theory and saddle theorem, the author obtains some existence theorems for the solution of problem (1), i.e. problem (1) has at least one solution in the Sobolev space \(H^1(\Omega)\).
scientific article; zbMATH DE number 1741841
scientific article