Global bifurcation in some degenerate quasilinear elliptic equations by a variational inequality approach (Q5949364)

The author considers boundary value problems of the form NEWLINE\[NEWLINE-\text{div}(\varphi(|\nabla u|)\nabla u)=F(x,u,\lambda), \quad\text{in }\Omega,\qquad u=0, \quad\text{on }\partial\Omega,NEWLINE\]NEWLINE which have the weak formulation NEWLINE\[NEWLINE \int_\Omega \varphi(|\nabla u|)\nabla u\cdot \nabla v=\int_\Omega F(x,u,\lambda)v, \qquad v\in W^{1,\gamma}_0(\Omega),\;u\in W^{1,\gamma}_0(\Omega).\tag{*}NEWLINE\]NEWLINE Here, \(W^{1,\gamma}_0(\Omega)\) is an appropriate Sobolev space determined by the growth of \(\varphi\). The goal of the paper is to study the Rabinowitz alternative of bifurcation branches of (*). The author successfully shows that at the principal eigenvalue of some appropriate \(p\)-Laplacian, the bifurcation branch of (*) is either unbounded or contains another bifurcation point. This is an extremely well-written paper.