Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions

Boscaggin, Alberto; Colasuonno, Francesca; Noris, Benedetta

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1625-1644.

For $1<p<$, we consider the following problem $$\begin{array}{c}\hfill {}_{\Delta }pu=f\left(u\right),u>0\mathrm{in}\Omega ,{\partial }_{\nu }u=0\mathrm{on}\partial \Omega \end{array}$$

where $\Omega \subset {ℝ}^N$ is either a ball or an annulus. The nonlinearity $f$ is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity $f\left(s\right)=-s^{p-1}+s^{q-1}$ for every $q>p$. We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution $u\equiv 1$. In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris and T. Weth, Ann. Inst. Henri Poincaré Anal. Non Linéaire 29 (2012) 573−588], that is to say, if $p=2$ and $f\text{'}\left(1\right)>{\lambda }_{k+1}^{\mathrm{rad}}$, with ${\lambda }_{k+1}^{\mathrm{rad}}$ the $(k+1)$-th radial eigenvalue of the Neumann Laplacian, there exists a radial solution of the problem having exactly $k$ intersections with $u\equiv 1$, for a large class of nonlinearities.

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