Multiple existence of solutions for a semilinear elliptic problem with Neumann boundary condition (Q2577463)

The authors consider semilinear elliptic problem \[ -\Delta_x u-\lambda u+ ae^u=\varepsilon f\quad\text{in }\Omega,\tag{1} \] \[ {\partial u\over\partial\vec n}= 0\quad\text{on }\partial\Omega,\tag{2} \] where \(\lambda\), \(a\) and \(\varepsilon\in \mathbb{R}^+\), \(f\in H^{-1}(\Omega)\), \(\vec n\) is the outernormal vector and \(\Omega\) is a bounded domain in \(\mathbb{R}^d\) \((d\geq 3)\) with smooth boundary \(\partial\Omega\). The multiplicity of solutions for (1)--(2) is investigated.