Quasilinear equations with a multiple bifurcation
Radial solutions of quasilinear boundary value problems of the type \[ -\Delta_p u = \lambda h(u) + g(r,u)\;\text{ for } r=|x|<1,\;u=0\;\text{ for } r=|x|=1 \tag \(\ast\) \] are studied where \(\Delta_p\) is the \(p\)-Laplace operator in \(\mathbb{R}^N,\;\lambda \geq 0,\;h(u) \simeq |u|^{q-2}u,\;1<q<p\) near \(u=0\), \(g\) is of higher order with respect to \(h\) at \(u=0\). The existence of infinitely many different continua of radial solutions of (\(\ast \)) emanating from \([\lambda ,u]=[0,0]\) is proved. Each continuum is characterized by nodal properties. As a consequence, the existence of infinitely many radial solutions follows for any \(\lambda >0\) small enough. The nonlinearities are not supposed to be odd and no growth restrictions are considered. Global behaviour of the bifurcating branches is described under different assumptions on \(g\). Particularly, if \(g(u) \simeq |u|^{\alpha -2}u\) as \(|u|\to \infty \), \(\alpha < p^*\) (the subcritical growth) then each continuum mentioned is bounded for \(\lambda \geq 0\) in the \(L_\infty\) norm and (*) possesses infinitely many pairs of radial solutions for any \(\lambda \geq 0\) small enough. Further, the case \(h(u)=u^{q-1},\;g(u)=u^{\alpha -1}\) for \(u \geq 0\) with \(\alpha \leq p^*\) is considered. In this case the set of all positive radial solutions forms a branch. This branch blows up to infinity as \(\lambda \to 0_+\) if \(\alpha =p^*\) and \(p,q\) satisfy certain additional assumptions.
scientific article; zbMATH DE number 1095820