Using the estimates of the gradient vector field \((\partial J)\) obtained in Section 3, we characterize in this section the critical points at infinity associated with problem (1.1) under \((f)_{\beta}\)-condition. First, we rule out the existence of critical points at infinity in \(V(p, \varepsilon)\), \(p\geq2\).
Theorem 4.1
Let
H
be a positive
\(C^{1}\)-function on
\(S^{n-1}\), \(n\geq3\), satisfying
\((f)_{\beta}\)-condition. There exists
\(\eta>0\)
such that if
$$n-2< \beta< (n-1)+\eta, $$
then the potential sets
\(V(p, \varepsilon)\), \(p\geq2\), do not contain any critical points at infinity.
Proof
The proof is an immediate consequence of the following proposition. □
Proposition 4.2
Let
H
be a positive
\(C^{1}\)-function on
\(S^{n-1}\), \(n\geq3\), satisfying
\((f)_{\beta}\)-condition. There exists
\(\eta>0\)
such that if
\(n-2<\beta<(n-1)+\eta\), then there exists a pseudo-gradient
\(W_{1}\)
in
\(V(p,\varepsilon)\), \(p\geq2\), such that, for any
\(u= \sum_{i=1}^{p}\alpha_{i} \delta_{i}\in V(p, \varepsilon)\), we have:
$$\begin{aligned}& \mathrm{(i)} \quad \bigl\langle \partial J(u),W_{1}(u) \bigr\rangle \leq-c \Biggl( \sum_{i=1}^{p} \frac{1}{\lambda_{i}^{\beta}}+ \sum_{i=1}^{p} \frac{\vert \nabla H(a_{i})\vert }{\lambda_{i}} +\sum_{j \neq i} \varepsilon_{ij} \Biggr), \\& \mathrm{(ii)} \quad \biggl\langle \partial J(u+\overline{v}), W_{1}(u)+ \frac{\partial\overline{v}}{\partial(\alpha_{i},a_{i},\lambda _{i})} \bigl(W_{1}(u) \bigr) \biggr\rangle \leq-c \Biggl( \sum_{i=1}^{p} \frac{1}{\lambda_{i}^{\beta}}+ \sum_{i=1}^{p} \frac{\vert \nabla H(a_{i})\vert }{\lambda_{i}} +\sum_{j \neq i} \varepsilon_{ij} \Biggr). \end{aligned}$$
Here
c
is a positive constant independent of
u. Moreover, \(\vert W_{1} \vert \)
is bounded, and the maximum of
\(\lambda_{i}\), \(1 \leq i \leq p\), decreases along the flow-lines of
\(W_{1}\).
Proof
Let \(u= \sum_{i=1}^{p}\alpha_{i} \delta_{i}\in V(p, \varepsilon)\), \(p\geq 2\). We order the \(\lambda_{i}\). Without loss of generality, we can assume that \(\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{p}\) and \(a_{i}\in B(y_{\ell_{i}}, \rho)\), \(y_{\ell_{i}}\in\mathcal{K}\), \(\forall i=1, \ldots, p\). For each index i, we denote by \(Z_{i}(u)\) and \(X_{i}(u)\) the vector fields
$$Z_{i}(u)= \lambda_{i} \frac{ \partial\delta_{i}}{\partial\lambda_{i}} \quad \mbox{and} \quad X_{i}(u) = \sum_{k=1}^{n-1} b_{k} \operatorname {sign}(a_{i} - y_{\ell_{i}})_{k} \frac{1}{\lambda_{i}}\frac{\partial {\delta}_{i}}{\partial(a_{i})_{k}}. $$
We then have the following lemmas.
Lemma 4.3
For any
\(i=2, \ldots, p\),
$$\bigl\langle \partial J(u), Z_{i}(u) \bigr\rangle = -2c_{2}J(u)\sum_{j\neq i}\alpha_{j} \lambda_{i}\frac{\partial\varepsilon _{ij}}{\partial\lambda_{i}} + o \biggl(\sum _{j\neq i}\varepsilon _{ij} \biggr)+ o \biggl( \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}}{\lambda _{i}} \biggr). $$
Proof
Using the expansions of Propositions 3.1, 3.3, and 3.5, for all \(i = 2, \ldots, p\) and any \(\beta>n-2\), we have
$$ \frac{1}{\lambda_{i}^{\beta}}= o (\varepsilon_{1i} ) \quad \mbox{as } \lambda_{i}\rightarrow+\infty. $$
(4.1)
Indeed,
$$\frac{ 1}{\lambda_{i}^{\beta}} \varepsilon_{1i}^{-1}= \frac{ 1}{\lambda_{i}^{\beta}} \biggl(\frac{\lambda_{i}}{\lambda_{1}}+ \frac {\lambda_{1}}{\lambda_{i}} + \lambda_{1}\lambda_{i}\vert a_{i}-a_{1} \vert ^{2} \biggr)^{\frac {n-2}{2} }\leq c \frac{ \lambda_{i}^{n-2}}{\lambda_{i}^{\beta}}. $$
Concerning \(\frac{ \log\lambda_{i}}{\lambda_{i}^{n-1}}\), which appears in the case \(\beta\geq n-1\), we have
$$ \frac{ \log\lambda_{i}}{\lambda_{i}^{n-1}}= o (\varepsilon _{1i} ) \quad \mbox{as } \lambda_{i}\rightarrow+\infty. $$
(4.2)
Indeed,
$$\frac{ \log\lambda_{i}}{\lambda_{i}^{n-1}} \varepsilon_{1i}^{-1}\leq c \frac{ \log\lambda_{i}}{\lambda_{i}}. $$
Last, we discuss the term \(O (\sum_{j\geq2}\frac {\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-j}}{\lambda_{i}^{j}} )\), which appears in all the cases of \(\beta>1\), in three cases.
-
If \(\beta>n-1\) and \(\lambda_{i}^{n-1}\vert a_{i}-y_{\ell_{i}}\vert ^{\beta }\geq{\delta}\), then we have
$$ O \Biggl(\sum_{j=2}^{n-2} \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-j}}{\lambda _{i}^{j}} \Biggr)= o \biggl(\frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}}{\lambda _{i}} \biggr) \quad \mbox{as } \lambda\rightarrow+\infty. $$
(4.3)
Indeed,
$$\begin{aligned} \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-j}}{\lambda_{i}^{j}}\frac{\lambda _{i}}{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}} =& \frac{1}{(\lambda_{i}\vert a_{i}-y_{\ell _{i}}\vert )^{j-1}} \\ \leq & \biggl(\frac{1}{\delta} \biggr)^{\frac{j-1}{\beta}} \frac {1}{(\lambda_{i})^{(j-1)(\beta-(n-1))}}. \end{aligned}$$
-
If \(\beta=n-1\) and \(\lambda_{i}\vert a_{i}-y_{\ell_{i}}\vert \geq\frac {1}{\delta}\), then we have
$$ O \Biggl(\sum_{j=2}^{n-2} \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-j}}{\lambda _{i}^{j}} \Biggr)= o \biggl(\frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}}{\lambda _{i}} \biggr) \quad \mbox{ as } \delta \mbox{ is small}. $$
(4.4)
-
If \(\beta< n-1\) and \(\lambda_{i}\vert a_{i}-y_{\ell_{i}}\vert \geq{\delta}\), then it is easy to see that if \(\lambda_{i}\vert a_{i}-y_{\ell_{i}}\vert \geq\frac {1}{\delta}\), then we have
$$ O \Biggl(\sum_{j=2}^{[\beta]} \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta -j}}{\lambda_{i}^{j}} \Biggr)= o \biggl(\frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta -1}}{\lambda_{i}} \biggr) \quad \mbox{as } \delta \mbox{ is small}, $$
(4.5)
and if \(\delta\leq\lambda_{i}\vert a_{i}-y_{\ell_{i}}\vert \leq\frac{1}{\delta }\), then we have
$$ O \Biggl(\sum_{j=2}^{[\beta]} \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta -j}}{\lambda_{i}^{j}} \Biggr)= O \biggl(\frac{1}{\lambda_{i}^{\beta}} \biggr)= o ( \varepsilon_{1i} )\quad \mbox{by (4.1)}. $$
(4.6)
This concludes the proof of Lemma 4.3. □
Lemma 4.4
For any
\(i= 1, \ldots, p\),
$$\bigl\langle \partial J(u), X_{i}(u) \bigr\rangle \leq -c \frac {\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}}{\lambda_{i}}+ o \biggl(\frac{1}{\lambda _{i}^{n-1}} \biggr) + O \biggl(\sum _{j\neq i}\varepsilon_{ij} \biggr). $$
Proof
It follows from the expansions of Propositions 3.2, 3.4, and 3.6 and from estimates (4.1)-(4.6). □
Lemma 4.5
Let
\(m>0\)
be a small constant. Then
$$\Biggl\langle \partial J(u), \sum_{i=2}^{p} \bigl(-2^{i} Z_{i}(u) + m X_{i}(u) \bigr) \Biggr\rangle \leq -c \Biggl( \sum_{i=2}^{p} \frac{1}{\lambda_{i}^{n-1}}+ \sum_{i=2}^{p} \frac{\vert \nabla H(a_{i})\vert }{\lambda_{i}} +\sum_{j \neq i} \varepsilon_{ij} \Biggr). $$
Proof
Using Lemmas 4.3 and 4.4 and (4.1), we get
$$\Biggl\langle \partial J(u), \sum_{i=2}^{p} \bigl(-2^{i} Z_{i}(u) + m X_{i}(u) \bigr) \Biggr\rangle \leq c \Biggl[ \sum_{i=2}^{p}\sum _{j\neq i}2^{i} \lambda _{i} \frac{\partial \varepsilon_{ij}}{\partial\lambda_{i}} - \sum_{i=2}^{p} \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}}{\lambda_{i}} \Biggr] + o \biggl(\sum_{j \neq i} \varepsilon_{ij} \biggr), $$
taking m small enough. Moreover, for \(1\leq i< j \leq p\), we have
$$2^{i}\lambda_{i} \frac{\partial \varepsilon_{ij}}{\partial\lambda_{i}} + 2^{j} \lambda_{j} \frac{\partial \varepsilon_{ij}}{\partial\lambda _{j}}\leq-c\varepsilon_{ij}. $$
Therefore,
$$\Biggl\langle \partial J(u), \sum_{i=2}^{p} \bigl(-2^{i} Z_{i}(u) + m X_{i}(u) \bigr) \Biggr\rangle \leq -c \Biggl(\sum_{j \neq i} \varepsilon_{ij}+ \sum_{i=2}^{p} \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta -1}}{\lambda_{i}} \Biggr). $$
Now, using (4.1), we can replace \(-\sum_{j \neq i} \varepsilon_{ij}\) by \(-\sum_{i=2}^{p}\frac{1}{\lambda _{i}^{\beta}}\). This concludes the proof of Lemma 4.5 since
$$ \bigl\vert \nabla H(a_{i}) \bigr\vert \sim \vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}. $$
(4.7)
□
Now, we must add the index 1. Let ψ be the following cut-off function:
$$\begin{aligned}& \psi: \mathbb{R} \longrightarrow \mathbb{R} \\& t \longmapsto \psi(t)= \left \{ \textstyle\begin{array}{l@{\quad}l} 1 & \hbox{if } \vert t\vert < \frac{\delta}{2},\\ 0 & \hbox{if } \vert t\vert \geq{\delta}. \end{array}\displaystyle \right . \end{aligned}$$
Lemma 4.6
There exists
\(\eta>0\)
such that, for any
\(i=1, \ldots, p\)
satisfying
\(a_{i}\in B(y_{\ell_{i}}, \rho)\), \(y_{\ell_{i}}\in\mathcal{K}\)
with
\(n-1<\beta<n-1+\eta\), we have
$$\begin{aligned}& \Biggl\langle \partial J(u), \psi \bigl(\lambda_{i}^{n-1} \vert a_{i}-y_{\ell _{i}}\vert ^{\beta} \bigr) \Biggl(- \sum_{k=1}^{n-1}b_{k} \Biggr) Z_{i}(u) + X_{i}(u) \Biggr\rangle \\& \quad \leq -c \biggl( \frac{1}{\lambda_{i}^{\beta}}+ \frac{\vert \nabla H(a_{i})\vert }{\lambda_{i}} \biggr) + O \biggl( \sum_{j \neq i} \varepsilon_{ij} \biggr). \end{aligned}$$
Proof
If \(\lambda_{i}^{n-1}\vert a_{i}-y_{\ell_{i}}\vert ^{\beta}\leq\frac{\delta }{2}\), then in the second expansion of Proposition 3.1, we have
$$O \biggl(\frac{1}{\lambda_{i}^{n-1}} \biggr)= o \biggl(\frac{\sum_{k=1}^{n-1}b_{k}}{(\beta-(n-1))\lambda_{i}^{n-1}} \biggr) $$
by taking \(0< (\beta-(n-1))< \eta\) with η small enough. Therefore, we get
$$ \Biggl\langle \partial J(u), \Biggl(-\sum _{k=1}^{n-1}b_{k} \Biggr) Z_{i}(u) \Biggr\rangle \leq - \frac{c}{\lambda_{i}^{n-1}}+ O \biggl(\sum _{j \neq i} \varepsilon_{ij} \biggr). $$
(4.8)
Hence, Lemma 4.6 follows in this case from Lemma 4.4 and from (4.7) and (4.8).
In the case where \(\lambda_{i}^{n-1}\vert a_{i}-y_{\ell_{i}}\vert ^{\beta }\geq\frac{\delta}{2}\), using the expansion of Proposition 3.2 and (4.3), we obtain
$$ \bigl\langle \partial J(u), X_{i}(u) \bigr\rangle \leq -c \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}}{\lambda_{i}} + o \biggl(\frac {1}{\lambda_{i}^{\gamma}} \biggr)+ O \biggl(\sum _{j \neq i} \varepsilon_{ij} \biggr), $$
(4.9)
where γ is any real in \((n-1, \min\{\beta, n\})\).
Choosing γ in \((\frac{n\beta-(n-1)}{\beta}, \min\{\beta , n\})\), we then have
$$\frac{1}{\lambda_{i}^{\gamma}} \frac{\lambda_{i}}{\vert a_{i}-y_{\ell _{i}}\vert ^{\beta-1}}\leq c \frac{1}{\lambda_{i}^{\frac{\beta(\gamma-n)+ (n-1)}{\beta}}}= o(1). $$
Thus,
$$\begin{aligned} \bigl\langle \partial J(u), X_{i}(u) \bigr\rangle \leq& -c \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}}{\lambda_{i}}+ O \biggl(\sum_{j \neq i} \varepsilon_{ij} \biggr) \\ \leq& -\frac{c}{2} \biggl( \frac{\vert a_{i}-y_{\ell_{i}}\vert ^{\beta-1}}{\lambda_{i}} + \frac {1}{\lambda_{i}^{\beta}} \biggr)+ O \biggl(\sum_{j \neq i} \varepsilon_{ij} \biggr) \end{aligned}$$
(4.10)
since
$$\frac{1}{\lambda_{i}^{\beta}}= o \biggl( \frac{\vert a_{i}-y_{\ell _{i}}\vert ^{\beta-1}}{\lambda_{i}} \biggr). $$
Hence, Lemma 4.6 follows from the first expansion of Proposition 3.1 and from (4.3), (4.7), and (4.10). □
Lemma 4.7
For any
\(i=1, \ldots, p\)
such that
\(a_{i}\in B(y_{\ell_{i}}, \rho)\), \(y_{\ell_{i}}\in\mathcal{K}\)
with
\(1<\beta\leq n-1\), we have
$$\Biggl\langle \partial J(u), \psi \bigl(\lambda_{i}\vert a_{i}-y_{\ell_{i}}\vert \bigr) \Biggl(-\sum _{k=1}^{n-1}b_{k} \Biggr) Z_{i}(u) + X_{i}(u) \Biggr\rangle \leq -c \biggl( \frac{1}{\lambda_{i}^{\beta}}+ \frac{\vert \nabla H(a_{i})\vert }{\lambda_{i}} \biggr) + O \biggl(\sum_{j \neq i} \varepsilon_{ij} \biggr). $$
Proof
We refer the reader to the proof of identity (4.3) in [10]. □
Corollary 4.8
For any
\(i=1, \ldots, p\)
such that
\(a_{i}\in B(y_{\ell_{i}}, \rho)\), \(y_{\ell_{i}}\in\mathcal{K}\)
with
\(1<\beta(y_{\ell_{i}}) < n-1+\eta\), denote
$$\begin{aligned}& Y_{i}(u)= \psi\bigl(\lambda_{i}^{n-1}\vert a_{i}-y_{\ell_{i}}\vert ^{\beta}\bigr) \Biggl(-\sum _{k=1}^{n-1}b_{k}\Biggr) Z_{i}(u) + X_{i}(u)\quad \textit{if }\beta(y_{\ell_{i}}) \in(n-1, n-1+\eta), \\& Y_{i}(u)= \psi\bigl(\lambda_{i}\vert a_{i}-y_{\ell_{i}} \vert \bigr) \Biggl(-\sum_{k=1}^{n-1}b_{k} \Biggr) Z_{i}(u) + X_{i}(u)\quad \textit{if } \beta(y_{\ell_{i}})\in(1, n-1]. \end{aligned}$$
Then we have:
$$\bigl\langle \partial J(u), Y_{i}(u) \bigr\rangle \leq -c \biggl( \frac{1}{\lambda_{i}^{\beta}}+ \frac{\vert \nabla H(a_{i})\vert }{\lambda_{i}} \biggr) + O \biggl(\sum _{j \neq i} \varepsilon_{ij} \biggr). $$
Now, if \(\lambda_{1}<<\lambda_{2}\), then let
$$W_{1}(u)= \sum_{i=2}^{p} \bigl( Z_{i}(u) + m X_{i}(u) \bigr) +m \bigl(Y_{1}(u) \bigr). $$
By Lemmas 4.5 and Corollary 4.8, for m small enough, we obtain
$$\bigl\langle \partial J(u), W_{1}(u) \bigr\rangle \leq-c \Biggl( \sum _{i=1}^{p} \frac{1}{\lambda_{i}^{\beta}}+ \sum _{i=1}^{p} \frac{\vert \nabla H(a_{i})\vert }{\lambda_{i}} +\sum _{j \neq i} \varepsilon_{ij} \Biggr). $$
If \(\lambda_{1}\sim\lambda_{2}\), then let
$$W_{1}(u)= \sum_{i=2}^{p} \bigl( Z_{i}(u) + m X_{i}(u) \bigr) + m X_{1}(u). $$
By Lemma 4.4, Lemma 4.5, and (4.7) we get
$$\bigl\langle \partial J(u), W_{1}(u) \bigr\rangle \leq-c \Biggl( \sum _{i=1}^{p} \frac{1}{\lambda_{i}^{\beta}}+ \sum _{i=1}^{p} \frac{\vert \nabla H(a_{i})\vert }{\lambda_{i}} +\sum _{j \neq i} \varepsilon_{ij} \Biggr). $$
This concludes the proof of claim (i) of Proposition 4.2. By the construction, \(W_{1}\) is bounded, and the maximum of \(\lambda _{i}(s)\), \(i=1, \ldots, p\), decreases along the flow lines of \(W_{1}\). Claim (ii) of Proposition 4.2 follows (as in the Appendix 2 of [14]) from (i) and the fact that \(\Vert \bar{v}\Vert ^{2}\) is small with respect to the absolute value of the upper bound of claim (i) (see Prop. 2.4 of [10], which is valid for any \(\beta>1\)). This completes the proof of Proposition 4.2. □
In the following, we characterize the critical point at infinity in \(V(1, \varepsilon)\).
Theorem 4.9
Let
H
be a positive
\(C^{1}\)-function on
\(S^{n-1}\), \(n\geq3\), satisfying
\((f)_{\beta}\)-condition. There exists
\(\eta>0\)
such that if
$$1< \beta< (n-1)+\eta, $$
then the only critical points at infinity of
J
in
\(V(1, \varepsilon )\)
are
$$(y)_{\infty}:=\frac{1}{H(y)^{\frac{n-2}{2}}} {\delta}_{(y, \infty)}, \quad y \in \mathcal {K}^{+}. $$
The Morse index of
\((y)_{\infty}\)
is equal to
\(i(y)_{\infty} :=(n-1)-\widetilde{i}(y)\).
Proof
Let \(u= \alpha_{1}\delta_{(a_{1}, \lambda_{1})}\in V(1, \varepsilon)\). We may assume that \(a_{1} \in B(y_{\ell_{1}}, \rho)\), \(y_{\ell_{1}}\in\mathcal {K}\), \(\rho>0\). Using the notation and the result of Corollary 4.8, we obtain
$$\begin{aligned}& \mathrm{(i)} \quad \bigl\langle \partial J(u), Y_{1}(u) \bigr\rangle \leq-c \biggl( \frac{1}{\lambda_{1}^{\beta}}+ \frac{\vert \nabla H(a_{1})\vert }{\lambda_{1}} \biggr), \\& \mathrm{(ii)}\quad \biggl\langle \partial J(u+\overline{v}), Y_{1}(u)+ \frac{\partial\overline{v}}{\partial(\alpha,a,\lambda)} \bigl(W_{2}(u) \bigr) \biggr\rangle \leq -c \biggl( \frac{1}{\lambda_{1}^{\beta}}+ \frac{\vert \nabla H(a_{1})\vert }{\lambda_{1}} \biggr). \end{aligned}$$
In addition, from the construction of \(Y_{1}\) we observe that the Palais-Smale condition is satisfied along each flow line of \(Y_{1}\), until the concentration point of the flow \(a_{1}(s)\) does not enter some neighborhood of y such that \(y\in\mathcal {K}^{+}\) since \(\lambda _{1}(s)\) decreases on the flow line in this set. On the other hand, if \(a_{1}(s)\) is near \(y_{\ell_{1}}, y_{\ell_{1}}\in\mathcal{K}^{+}\), then we observe that \(\lambda_{1}(s)\) increases and goes to +∞. Thus, we obtain a critical point at infinity. In this region, the functional J can be expanded after a suitable change of variables as
$$\begin{aligned} J(\alpha_{1} \delta_{(a_{1}, \lambda_{1})} + \bar{v}) =& J( \widetilde{\alpha_{1} }\delta_{(\widetilde{a_{1}}, \widetilde {\lambda_{1}})}) \\ =& \frac{S_{n}}{\widetilde{\alpha_{1}}^{\frac {4}{n-2}}H(\widetilde{a_{1}})^{\frac{n-2}{2}}} \biggl(1+ \frac{(-\sum_{k=1}^{n-1}b_{k})}{\widetilde{\lambda_{1}}^{\beta}} \biggr). \end{aligned}$$
Thus, the index of such critical point at infinity is \(n-1-\widetilde{i}(y)\). Since J behaves in this region as \(\frac {1}{H^{\frac{n-2}{2}}}\), this finishes the proof of Theorem 4.9. □
The next proposition is extracted from [10], Lemma 4.4. As mentioned in [10], it is still correct for any \(\beta>\frac{n-2}{2}\).
Proposition 4.10
Let
w
be a solution of (1.1). Assume that the function
H
satisfies condition
\((f)_{\beta}\)
with
\(\beta>\frac{n-2}{2}\). Then, for each
\(p \in\mathbb{N}^{\star}\), there is no critical point at infinity in
\(V(p, \varepsilon, w)\).