On a quasilinear elliptic eigenvalue problem with constraint
The authors consider a quasilinear elliptic problem with constraint in the following form \[ -\text{div}(| \nabla u| ^{p-2}\nabla u)=\mu g(x,u) \quad\text{in }\Omega, \quad u=0\text{ on }\partial \Omega, \quad\int_{\Omega }| \nabla u| ^{p}\,dx=r^{p},\tag{1} \] where \(r>0,\) \(\Omega \) \(\subset \mathbb{R}^{N}\) is a bounded domain with smooth boundary \(\partial \Omega ,\) \(1<p<N.\) They construct a pseudogradient vector field \(V\) and they show the regularity of the flow associated to \(V\). By the descending flow argument, the authors prove the existence of a positive, a negative and a sign-changing solutions to problem (1).
scientific article; zbMATH DE number 5005684