
Minimax Theory and its Applications 07 (2022), No. 2, 185206 Copyright Heldermann Verlag 2022 Multiplicity of Solutions on a Nehari Set in an Invariant Cone Francesca Colasuonno Dip. di Matematica, Università di Bologna, Italy francesca.colasuonno@unibo.it Benedetta Noris Dip. di Matematica, Politecnico di Milano, Italy benedetta.noris@polimi.it Gianmaria Verzini Dipartimento di Matematica, Politecnico di Milano, Italy gianmaria.verzini@polimi.it [Abstractpdf] For $1 < p < 2$ and $q$ large, we prove the existence of two positive, nonconstant, radial and radially nondecreasing solutions of the supercritical equation \[ \Delta_p u+u^{p1}=u^{q1} \] under Neumann boundary conditions, in the unit ball of $\mathbb R^N$. We use a variational approach in an invariant cone. We distinguish the two solutions upon their energy: one is a ground state inside a Neharitype subset of the cone, the other is obtained via a mountain pass argument inside the Nehari set.\\[1mm] As a byproduct of our proofs, we detect the limit profile of the low energy solution as $q\to\infty$ and show that the constant solution 1 is a local minimizer on the Nehari set. This marks a strong difference with the case $p\ge 2$. Keywords: Quasilinear elliptic equations, Sobolevsupercritical nonlinearities, Neumann boundary conditions, Radial solutions. MSC: 35J92, 35J20, 35B09, 35B45. [ Fulltextpdf (205 KB)] for subscribers only. 