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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

Benedetta Noris
Dip. di Matematica, Politecnico di Milano, Italy

Gianmaria Verzini
Dipartimento di Matematica, Politecnico di Milano, Italy


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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