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Minimax Theory and its Applications 07 (2022), No. 2, 185--206
Copyright Heldermann Verlag 2022



Multiplicity of Solutions on a Nehari Set in an Invariant Cone

Francesca Colasuonno
Dip. di Matematica, UniversitÓ 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



[Abstract-pdf]

For $1 < p < 2$ and $q$ large, we prove the existence of two positive, nonconstant, radial and radially nondecreasing solutions of the supercritical equation \[ -\Delta_p u+u^{p-1}=u^{q-1} \] 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 Nehari-type 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, Sobolev-supercritical nonlinearities, Neumann boundary conditions, Radial solutions.

MSC: 35J92, 35J20, 35B09, 35B45.

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