Phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball

Abstract

We consider the following problem (0.1){Δpu+λu+f(u,r)=0u>0in B,andu=0on ∂B where B is the unitary ball in Rn. Merle and Peletier considered the classical Laplace case p=2, and proved the existence of a unique value λ0∗ for which a radial singular positive solution exists, assuming f(u,r)=uq−1 and q>2∗≔2nn−2. Then Dolbeault and Flores proved that, if q>2∗ but q is smaller than the Joseph–Lundgren exponent σ∗, then there is an unbounded sequence of radial positive classical solutions for (0.1), which accumulate at λ=λ0∗, again for p=2.We extend both Merle–Peletier and Dolbeault–Flores results to the p-Laplace setting with the technical restriction 1<p≤2, and to more general nonlinearities f, which may have more complicated dependence on u and may be spatially non-homogeneous. Then we reproduce the results also for similar bifurcation problems where the linear term λu is replaced by a superlinear and subcritical term of the form λrηu|u|Q−2. Our analysis relies on a generalized Fowler transformation and profits of invariant manifold theory, and it allows to discuss radial nodal solutions too.