The Nehari manifold for a semilinear elliptic equation involving a sublinear term
The author discusses the problem of existence and multiplicity of non-negative solutions to the problem \[ \begin{cases} -\Delta u(x)=\lambda u(x)+b(x)| u(x)| ^{\gamma-2}u(x)& \text{for }u\in\Omega\\ u(x)=0& \text{for }x\in\partial\Omega,\end{cases}\eqno(1) \] where \(\Omega\subset \mathbb R^N\) is a smooth bounded domain, \(b:\Omega\to \mathbb R\) a smooth function, \(\lambda\in \mathbb R\) and \(1<\gamma<2\). When \(1<\gamma<2\) the problem \((1)\) is asymptotically linear and the author establishes results on bifurcation from infinity when \(\lambda=\lambda_{1}\), the principal eigenvalue of the linear problem \[ -\Delta u(x)=\lambda u(x)\quad x\in\Omega;\qquad u(x)=0\quad x\in\partial\Omega. \] By exploiting the relationship between the Nehari manifold and the fibering maps (maps of the form \(t\mapsto J(tu)\) where \(J\) is the Euler functional associated to \((1)\)), the author studies how the Nehari manifold changes as \(\lambda\) varies. The bifurcation is then described in terms of the sign of the quantity \(\int_{\Omega}b\phi_{1}^{\gamma}\,dx\) where \(\phi_{1}\) is the positive eigenfunction of the above linear problem corresponding to \(\lambda_{1}\).
scientific article; zbMATH DE number 2161207