In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem.

In this paper, we revisit a problem having a geometric origin. Namely, let

Note that, to solve the problem (

In [

The main contribution of the present work is to generalize certain previous existence results of [

In the first part of this paper, we provide existence and multiplicity results under perturbative hypothesis.

In order to state our results, we introduce the following notations and assumptions.

Through the whole of this paper, we assume that

Let

We introduce the following assumptions:

For each critical point

We then have the following perturbative result.

Under assumptions,

Further, if one assumes that all the solutions of (

We recall here that for a generic function,

We recall also that

The existence result of Theorem

We will provide a more general result than Theorem

We then have the following existence and multiplicity result.

Under the conditions,

Further, if one assumes that all the solutions of (

Theorem

In the second part of this work, we will establish nonperturbative results. For this, denoting by

Under the conditions,

(

If in addition one assumes that all the solutions of (

As a corollary of Theorem

Under the conditions,

Furthermore, if we assume that all the solutions of (

The result of Corollary

In view of such an interpretation, we raise the following question: what happens if the total contribution is trivial, but a subset of

With respect to the above question, Theorem

As pointed out above, our result does not only give existence results but also, under generic conditions, gives a lower bound on the number of solutions of (

In what follows we show a situation where Corollary

Let

The rest of our paper is organized as follows. In Section

In this Section we recall the variational formulation of the problem (

Let

The failure of the Palais-Smale condition can be described (see Proposition 1 in [

Let

Here

For any integers

Thus, we can write any

We also consider the case, where

There exists a

The next proposition characterizes the critical points at infinity of the associated variational problem. We recall that critical points at infinity are the orbits of the gradient flow of

Assume that, for any

If

If

Using Corollary 3 of [

We start the proofs by recalling the following results.

For

Lemma

Let

For

Lemma

Let

The contraction

Using the same arguments of the proof of Theorem

We now claim that

On the other side, using Proposition

Let

Thus the critical points at infinity of our variational problem lie in

The unstable manifolds at infinity, for the pseudogradient

Recall that

The authors declare that there is no conflict of interests regarding the publication of this paper.