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arXiv · 2109.14443

Multiplicity of solutions on a Nehari set in an invariant cone

Francesca Colasuonno·Benedetta Noris·Gianmaria Verzini
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Abstract

For $1<p<2$ and $q$ large, we prove the existence of two positive, nonconstant, radial and radially nondreacreasing 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. 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 minimum on the Nehari set.

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Francesca Colasuonno, Benedetta Noris, Gianmaria Verzini. 2021-09-29. Multiplicity of solutions on a Nehari set in an invariant cone. https://arxiv.org/abs/2109.14443

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