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Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials

Abstract

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an indefinite weight. In particular we are interested in the case where such a weight is positive in a ball and negative outside, or in the reversed situation. We extend the approach to elliptic equations in presence of Hardy potentials, i.e. to

$$\begin{aligned} \varDelta u +\frac{h(|\text {x}|)}{|\text {x}|^2} u+ f(u, |\text {x}|)=0 \end{aligned}$$

where h is not necessarily constant. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.

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Acknowledgements

The authors wish to thank the referee for the useful comments and suggestions which lead to an improvement of the paper.

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Correspondence to Matteo Franca.

Additional information

The authors were partially supported by G.N.A.M.P.A.-I.N.d.A.M. and M.U.R.S.T. (Italy).

Appendix

Appendix

On the Lack of Continuability

In this appendix we first review briefly some well known facts concerning exponential dichotomy, see, e.g. [12]. Then we develop the construction of stable and unstable manifolds for non-autonomous systems, i.e. \(W^u_{l_u}(\tau )\) and \(W^s_{l_s}(\tau )\), when continuability of the trajectories of (S) is ensured, i.e. when hypothesis C holds. Then we extend our discussion to the case where C does not hold.

Denote by \(\mathcal {A}_l(t)=\left( \begin{array}{cc} \alpha _{l} &{} 1 \\ -h(\text {e}^t) &{} \gamma _{l} \end{array} \right) \) the linearization of the right hand side of (S) in the origin, and by \(\mathcal {A}_l(\pm \infty )= \lim _{t \rightarrow \pm \infty } \mathcal {A}_l(t)\). Assume either \(\varvec{Gu}\) or \(\varvec{gu}\): note that \(\mathcal {A}_{l_u}(-\infty )\) has \(\lambda _2<0<\lambda _1\) as eigenvalues where \(\lambda _1:=\alpha _{l_u}-\kappa (\eta )\) and \(\lambda _2:=\alpha _{l_u}+2-n+\kappa (\eta )\). By H, \(\mathcal {A}_{l_u}(t)\) can be seen as an \(L^1\) perturbation of \(\mathcal {A}_{l_u}(-\infty )\), therefore it admits exponential dichotomy in negative semi-lines \((-\infty ,\tau ]\). More precisely let X(t) be the fundamental matrix of

$$\begin{aligned} \dot{x}=\mathcal {A}_{l_u}(t) x \, , \end{aligned}$$
(6.1)

i.e. the matrix solution of (6.1) such that \(X(0)=I\), where I denotes the identity matrix. Then, for any \(\tau \in {\mathbb {R}}\) there is a constant \(K=K(\tau )>1\), exponents \(\bar{\lambda }_2<0<\bar{\lambda }_1\) and a projection \(\mathcal {P}^-\) such that

$$\begin{aligned} \begin{array}{cc} \Vert X(t)(I-\mathcal {P}^-) X(s)^{-1} \Vert \le K\, \text {e}^{\bar{\lambda }_1 (t-s)} &{} \text { for any }t<s<\tau \,,\\ \Vert X(t)\mathcal {P}^- X(s)^{-1} \Vert \le K\, \text {e}^{\bar{\lambda }_2 (t-s)} &{} \text { for any }s<t<\tau \,, \end{array} \end{aligned}$$
(6.2)

see, e.g. [12, Section 4]. Moreover the optimal choice for \(\bar{\lambda }_i\) is \(\bar{\lambda }_i=\lambda _i\) for \(i=1,2\), see [8, Appendix]. Let us denote by \(\mathcal {P}^-(\tau ):=X(\tau )\mathcal {P}^- X(\tau )^{-1}\), and by \(\ell ^u(\tau )\) the 1-dimensional kernel of \(\mathcal {P}^-(\tau )\); then \(\ell ^u(\tau )\) is the unstable space for (6.1). I.e let \(\mathbf {\xi } \in {\mathbb {R}}^2\), and denote by \(\mathbf {\xi }(t)\) the solution of (6.1) such that \(\mathbf {\xi }(\tau )=\mathbf {\xi }\); then \(\mathbf {\xi }(t)\) is bounded for \(t \le 0\) iff \(\mathbf {\xi } \in \ell ^u(\tau )\), cf. [12, Section 4]. Since \(\ell ^u(\tau )\) is 1-dimensional we see that there is \(c=c(\mathbf {\xi })\) such that \(\Vert \mathbf {\xi }(t)\Vert \text {e}^{-\lambda _1 t} \rightarrow c\) as \(t \rightarrow -\infty \). Also note that by construction \(\ell ^u(\tau )\) is a line, and \(\mathbf {\xi } \in \ell ^u(\tau )\) iff \(\mathbf {\xi }(t) \in \ell ^u(t)\).

Now assume \(\varvec{gu}\) and consider (S) where \(l=l_u\): we consider this problem as a nonlinear perturbation of (6.1). Thus, setting \(Q(\delta )=\{(x,y) \mid |x|\le \delta , \; |y| \le \delta \}\), we get the following, see [27, Theorem 2.25].

Lemma 6.1

Assume \(\varvec{gu}\); then for any \(N \in {\mathbb {R}}\) we can find \(\delta =\delta (N)\) such that the set

$$\begin{aligned} \begin{array}{l} W^u_{l_u,loc}(\tau ):= \Big \{ \varvec{Q}\in Q(\delta ) \mid \displaystyle \varvec{x_{l_u}}(t,\tau ;\varvec{Q}) \in Q(\delta ) \; \text {for any } t \le \tau , \\ \qquad \text { and } \displaystyle \; {\lim _{t \rightarrow -\infty }}\varvec{x_{l_u}}(t,\tau ;\varvec{Q}) = (0,0) \Big \} \end{array} \end{aligned}$$
(6.3)

is a graph on \(\ell ^u(\tau ) \cap Q(\delta )\) for any \(\tau \le N\). Moreover \(\ell ^u(\tau )\) is the tangent space to \(W^u_{l_u,loc}(\tau )\) in the origin.

We sketch the proof for completeness. Assume \(\varvec{gu}\) and suppose first that,

$$\begin{aligned} |g_{l_u}(x_2,t)-g_{l_u}(x_1,t)| \le c(\tau ) |x_2-x_1| \, , \qquad \text {for any }t \le \tau \end{aligned}$$

for some \(c(\tau )>0\) and for any \(x_1,x_2 \in {\mathbb {R}}\). Then, using a variation of constants formula, see e.g. [11, Section 3.3] or [27, Theorem 2.25], we prove that the set, cf. (2.7),

$$\begin{aligned} \tilde{W}^u_{l_u}(\tau ):= \Big \{ \varvec{Q}\mid {\lim _{t \rightarrow -\infty }}\varvec{x_{l_u}}(t,\tau ;\varvec{Q}) = (0,0) \Big \} \end{aligned}$$
(6.4)

is a graph on \(\ell ^u(\tau )\) (globally), for any \(\tau \in {\mathbb {R}}\). Then the proof follows from a truncation argument.

Using the flow of (S) we get the following.

Lemma 6.2

Assume \(\varvec{gu}\) and C. Then the set \(\tilde{W}^u_{l_u}(\tau )\) characterized as in (6.4) is a 1-dimensional immersed submanifold having \(\ell ^u(\tau )\) as tangent space in the origin.

Proof

Let us denote by \(\varPhi _{T,\tau }\) the diffeomorphism induced by the flow of (S): i.e. \(\varPhi _{T,\tau }(\varvec{Q})=\varvec{x_{l_u}}(T,\tau ;\varvec{Q})\). Then \(\varPhi _{T,\tau }(W^u_{l_u,loc}(\tau ))\) is a 1-dimensional submanifold for any \(\tau ,T \in {\mathbb {R}}\) and \(\varPhi _{T,\tau _1}(W^u_{l_u,loc}(\tau _1)) \supset \varPhi _{T,\tau _2}(W^u_{l_u,loc}(\tau _2))\) if \(\tau _1< \tau _2\).

Hence we may set \(\tilde{W}^u_{l_u}(T):=\bigcup _{\tau \in {\mathbb {R}}} \varPhi _{T,\tau }(W^u_{l_u,loc}(\tau ))\) and we see that \(\tilde{W}^u_{l_u}(T)\) is a 1-dimensional immersed manifold, and by construction it is characterized as in (6.4). \(\square \)

Remark 6.1

We stress that \(\tilde{W}^u_{l_u}(\tau )\) (and \(\tilde{W}^s_{l_s}(\tau )\) constructed below) is just an immersed 1-dimensional manifold, i.e. a \(C^1\) regular curve, and may be not an embedded submanifold: e.g. it may be 8 shaped in the origin as in the critical autonomous case, see e.g. Fig. 1. However it always contains \(W^u_{l_u,loc}(\tau )\) (respectively \({W}^s_{l_s, loc}(\tau )\)) which is tangent to \(\ell ^u(\tau )\) (resp. \(\ell ^s(\tau )\)).

Now we drop the assumption C and we prove Lemma 2.2.

Proof of Lemma 2.2

Fix \(\tau \in {\mathbb {R}}\); for every \(\varvec{Q}\in {\mathbb {R}}^2\) we can introduce

$$\begin{aligned} \mathfrak {T}(\varvec{Q},\tau )=\sup \big \{t \mid \varvec{x_{l_u}}(\cdot ,\tau ,\varvec{Q}) \text { is defined in }[\tau , t) \big \} \,. \end{aligned}$$

Then \(\lim _{t\rightarrow \mathfrak {T}(\varvec{Q},\tau )} |\varvec{x_{l_u}}(t,\tau ,\varvec{Q})|=+\infty \) if \(\mathfrak {T}(\varvec{Q},\tau )<+\infty \). It is easy to verify that \(\mathfrak {T}(\cdot ,\tau )\) is lower semicontinuous, i.e. the sets \( \{ \varvec{Q}\in {\mathbb {R}}^2 \mid \mathfrak {T}(\varvec{Q},\tau ) \le \mathfrak {t} \}\) are closed. In fact for every \(\varvec{Q}_0\in {\mathbb {R}}^2\) and every \(\varepsilon >0\) we can find neighbourhoods \(\mathcal U\) of \(\varvec{Q}_0\) and \(\mathcal V\) of \(\varvec{x_{l_u}}(\mathfrak {T}(\varvec{Q}_0,\tau ) -\varepsilon ,\tau ,\varvec{Q}_0)\) such that for every \(\varvec{Q}\in \mathcal U\) we have \(\varvec{x_{l_u}}(\mathfrak {T}(\varvec{Q}_0,\tau ) -\varepsilon ,\tau ,\varvec{Q})\in \mathcal V\), thus giving \(\mathfrak {T}(\varvec{Q},\tau ) > \mathfrak {T}(\varvec{Q}_0,\tau )-\epsilon \) for every \(\varvec{Q}\in \mathcal U\). Therefore if \({\varvec{Q_n}} \rightarrow \varvec{Q}_0\), then \(\liminf _{n \rightarrow \infty } \mathfrak {T}(\varvec{Q}_n,\tau ) \ge \mathfrak {T}(\varvec{Q}_0,\tau )\).

Then, we consider

$$\begin{aligned} \mathfrak {T}^u(\tau ):= \inf \big \{ \mathfrak {T}(\varvec{Q},\tau ) \mid \varvec{Q}\in W^{u,+}_{l_u,loc}(\tau ) \big \}\,. \end{aligned}$$
(6.5)

Notice that \(\mathfrak {T} ((0,0),\tau )=+\infty \) and that \(\mathfrak {T}^u(\tau )\) is increasing by construction. The lower semicontinuity gives us that either one has \(\mathfrak {T}^u(\tau )=+\infty \) or the infimum is in fact a minimum, being \(W^{u,+}_{l_u,loc}(\tau )\) bounded and \(\mathfrak {T}(\varvec{Q},\tau ) > \tau \). Moreover the subset containing the points \(\varvec{Q}\) which explode to infinity before a fixed time \(\mathfrak {t}\), i.e.

$$\begin{aligned} \mathcal X(\mathfrak {t},\tau ) = \big \{ \varvec{Q}\in W^{u,+}_{l_u,loc}(\tau ) \mid \mathfrak {T}(\varvec{Q},\tau ) \le \mathfrak {t} \big \}\,, \end{aligned}$$

is a closed subset. Conversely \(\mathcal {W}(\mathfrak {t},\tau ) = W^{u,+}_{l_u,loc}(\tau ) \setminus \mathcal X(\mathfrak {t},\tau )\) is a relatively open subset of \(W^{u,+}_{l_u,loc}(\tau )\).

If \(\mathfrak {t}<\mathfrak {T}^u(\tau )\), then \(\mathcal X(\mathfrak {t},\tau )=\emptyset \), so that \(\tilde{W}^{u,+}_{l_u}(\mathfrak {t}):=\varPhi _{\mathfrak {t},\tau }W^{u,+}_{l_u,loc}(\tau )\) is diffeomorph to \(W^{u,+}_{l_u,loc}(\tau )\), and it is a 1-dimensional manifold with boundary. In fact it is easy to check that the map \(\varPhi _{\tau ,{\mathfrak {t}}}= \varPhi _{{\mathfrak {t}},\tau }^{-1}\) is well defined in an open neighborhood of \(\varPhi _{{\mathfrak {t}},\tau }W^{u,+}_{l_u,loc}(\tau )\).

Fig. 3
figure3

Assume that \(\mathfrak {T}(\varvec{Q},\tau )\) is continuous and strictly decreasing in \(Q_x\). At the time \(\tau \) (on the left), the endpoint \(\varvec{Q}_\tau =(\delta ,Q_y)\) minimizes \(\mathfrak {T}(\cdot ,\tau )\) along \({W}^{u,+}_{l_u,loc}(\tau )\), thus having \(\mathcal {W}(\mathfrak {T}^u(\tau ),\tau )={W}^{u,+}_{l_u,loc}(\tau )\setminus \{\varvec{Q}_\tau \}\). If we consider \(\tau _0 < \tau \) (at the center), then \(\mathfrak {T}^u(\tau _0)<\mathfrak {T}^u(\tau )\) and the set \(\mathcal {W}(\mathfrak {T}^u(\tau ),\tau _0)\) consists of the points to the left with respect to \(\varvec{Q}_{\tau _0}=\varvec{x_{l_u}}(\tau _0,\tau ,\varvec{Q}_\tau )\). The images \(\varPhi _{\mathfrak {T}^u(\tau ),\tau } \mathcal {W}(\mathfrak {T}^u(\tau ),\tau )\) and \(\varPhi _{\mathfrak {T}^u(\tau ),\tau _0}\mathcal {W}(\mathfrak {T}^u(\tau ),\tau _0)\) gives us the unbounded 1-dimensional manifold \(W^{u,+}_{l_u}(\mathfrak {T}^u(\tau ))\) (on the right)

We assume first for illustrative purpose that, for any \(\varvec{Q}=(Q_x,Q_y)\in W^{u,+}_{l_u,loc}(\tau )\) the function \(\mathfrak {T}(\varvec{Q},\tau )\) is strictly decreasing in \(Q_x\): this is the case, e.g. if (S) is autonomous. This assumption will be removed later on.

If we set \(\mathfrak {t} = \mathfrak {T}^u(\tau )\) we have \(\mathcal {X}(\mathfrak {t},\tau )=\{\varvec{Q}_{\tau }\}\) where \(\varvec{Q}_{\tau }=(\delta ,Q_y)\) is the endpoint of \({W}^{u,+}_{l_u,loc}(\tau )\), while if \(\mathfrak {t} > \mathfrak {T}^u(\tau )\) the set \(\mathcal X(\mathfrak {t},\tau )\), which contains \(\varvec{Q}_\tau \) and \(\mathcal {W}(\mathfrak {t},\tau )\), is connected. In both the cases, the map \(\varPhi _{\mathfrak {t},\tau }\) is well defined on \(\mathcal {W}(\mathfrak {t},\tau )\), and the set \(\tilde{W}^{u,+}_{l_u}(\mathfrak {t}):=\varPhi _{\mathfrak {t},\tau }\mathcal {W}(\mathfrak {t},\tau )\) is diffeomorph to \(\mathcal {W}(\mathfrak {t},\tau )\). In fact there is an open neighborhood of \(\tilde{W}^{u,+}_{l_u}(\mathfrak {t})\) which is mapped by the inverse diffeomorphism \(\varPhi _{\tau ,\mathfrak {t}}\) into an open neighborhood of \(\mathcal {W}(\mathfrak {t},\tau )\). So \(\tilde{W}^{u,+}_{l_u,loc}( \mathfrak {t})\setminus \{(0,0) \}\) is a 1-dimensional manifold without boundary, see Fig. 3.

Now let us repeat the discussion replacing \(\tau \) by \(\tau _0<\tau \). It is easy to check that \(\varPhi _{\mathfrak {t},\tau _0}\mathcal {W}(\mathfrak {t},\tau _0) \supseteq \varPhi _{\mathfrak {t},\tau }\mathcal {W}(\mathfrak {t},\tau )\), and if \(\mathfrak {t} \ge \mathfrak {T}^u(\tau )\), then \(\varPhi _{\mathfrak {t},\tau _0}\mathcal {W}(\mathfrak {t},\tau _0) = \varPhi _{\mathfrak {t},\tau }\mathcal {W}(\mathfrak {t},\tau )=\tilde{W}^{u,+}_{l_u}(\mathfrak {t})\), and it is unbounded. Furthermore by construction

$$\begin{aligned} \tilde{W}^{u,+}_{l_u}(\mathfrak {t}) = \big \{ \varvec{Q}\mid {\lim _{t \rightarrow -\infty }}\varvec{x_{l_u}}(t,\mathfrak {t},\varvec{Q})=(0,0) \, , \,\, \dot{x}_{l_u}(t,\mathfrak {t},\varvec{Q})>0 \text { for }t \ll 0 \big \}\,, \end{aligned}$$
(6.6)

holds if \(\mathfrak {t} \ge \mathfrak {T}^u(\tau )\). Therefore the set

$$\begin{aligned} W^{u,+}_{l_u}(T)= \bigcup _{ \tau \le T} \varPhi _{T,\tau }\mathcal {W}(T,\tau ) \end{aligned}$$

is characterized by the property defined in (6.6).

If we remove the simplifying assumption that \(\mathfrak {T}({\varvec{Q}} ,\tau )\) is decreasing with respect to \(Q_x\), we can repeat the previous discussion with the following changes.

Fig. 4
figure4

If \(\mathfrak {T}(\varvec{Q},\tau )\) is not decreasing in \(Q_x\), at the time \(\tau \) (on the left), the minimum \(\mathfrak {T}^u(\tau )\) can be attained in a point \(\varvec{Q}_\tau =(Q_x^\tau ,Q_y^\tau )\) with \( Q_x^\tau <\delta \), while we denote by \(\varvec{R}_\tau \) the endpoint of \(W^{u,+}_{l_u,loc}(\tau )\). We consider also \(\tau _0 < \tau \) (in the center), where we can find the point \(\varvec{Q}_{\tau _0}=\varvec{x_{l_u}}(\tau _0,\tau ,\varvec{Q}_\tau )\) and we denote by \(\varvec{R}_{\tau _0}\) the endpoint of \(W^{u,+}_{l_u,loc}(\tau _0)\). In both the situations the sets \(\mathcal {W}(\mathfrak {T}^u(\tau ),\tau )\) and \(\mathcal {W}(\mathfrak {T}^u(\tau ),\tau _0)\) are disconnected respectively at the point \(\varvec{Q}_\tau \) and \(\varvec{Q}_{\tau _0}\). The images \(\varPhi _{\mathfrak {T}^u(\tau ),\tau } \mathcal {W}(\mathfrak {T}^u(\tau ),\tau )\subset \varPhi _{\mathfrak {T}^u(\tau ),\tau _0}\mathcal {W}(\mathfrak {T}^u(\tau ),\tau _0)\) give us two unbounded disconnected sets contained in \(\tilde{W}^{u,+}_{l_u}(\mathfrak {T}^u(\tau ))\) (on the right). The second components have endpoints respectively \(\varvec{R}_\tau ^e = \varPhi _{\mathfrak {T}^u(\tau ),\tau }(\varvec{R}_\tau )\) and \( \varvec{R}_{\tau _0}^e = \varPhi _{\mathfrak {T}^u(\tau ),\tau _0}(\varvec{R}_{\tau _0})\); while the point \(\varvec{Q}_\tau \) is, roughly speaking, sent to infinity by the flux \(\varPhi _{\mathfrak {T}^u(\tau ),\tau }\)

Fig. 5
figure5

In general \(\mathcal {W}(\mathfrak {t},\tau )\subset W^{u,+}_{l_u,loc}(\tau )\) may be disconnected when \(\mathfrak {t}> \mathfrak {T}^u(\tau )\). The picture sketches an example. Consider \(\mathfrak {t}_2> \mathfrak {t}_1 >\mathfrak {T}^u(\tau )\), the set \(\mathcal {W}(\mathfrak {t}_1,\tau )\) has two connected components, while \(\mathcal {W}(\mathfrak {t}_2,\tau )\) has three components. On the right we have drawn the corresponding images \(\varPhi _{\mathfrak {t}_1,\tau }\mathcal {W}(\mathfrak {t}_1,\tau )\subset W^{u,+}_{l_u}(\mathfrak {t}_1)\) and \(\varPhi _{\mathfrak {t}_2,\tau }\mathcal {W}(\mathfrak {t}_2,\tau )\subset W^{u,+}_{l_u}(\mathfrak {t}_2)\). We show how some points \(\varvec{Q}_1,\ldots ,\varvec{Q}_8\), are mapped by the fluxes \(\varPhi _{\mathfrak {t}_1,\tau }\) and \(\varPhi _{\mathfrak {t}_2,\tau }\), denoting the images again with \(\varvec{Q}_1,\ldots ,\varvec{Q}_8\) for simplicity. In particular, at the time \(\mathfrak {t}_2\) the solution \(\varvec{x_{l_u}}(\cdot ,\tau ,\varvec{Q}_j)\) is not defined for \(j=2,5,6\)

The open set \(\mathcal {W}(\mathfrak {T}^u(\tau ),\tau )\) can be disconnected, so its image \(\varPhi _{\mathfrak {T}^u(\tau ),\tau }\mathcal {W}(\mathfrak {T}^u(\tau ),\tau )\) may be disconnected too, see Fig. 4. However \(\mathcal {W}(\mathfrak {T}^u(\tau ),\tau )\) has a connected component containing the origin, say \(\mathcal {W}_1(\mathfrak {T}^u(\tau ),\tau )\), whose image is the connected component of the set \(\varPhi _{\mathfrak {T}^u(\tau ),\tau }\mathcal {W}(\mathfrak {T}^u(\tau ),\tau )\) containing the origin. Observe that \(\mathcal {W}_1(\mathfrak {T}^u(\tau ),\tau )\) is a connected one dimensional manifold, so this property is inherited by \(\varPhi _{\mathfrak {T}^u(\tau ),\tau }{\mathcal {W}_1}(\mathfrak {T}^u(\tau ),\tau )\) too.

When \(\mathfrak {t} \ge \mathfrak {T}^u(\tau )\), the map \(\varPhi _{\mathfrak {t},\tau }\) is well defined in \(\mathcal {W}(\mathfrak {t},\tau )\) and the set \(\mathcal X(\mathfrak {t},\tau )\) may disconnect \(\varPhi _{\mathfrak {t},\tau }\mathcal {W}(\mathfrak {t},\tau )\), see Fig. 5. Repeating the previous arguments we see that the image \(\varPhi _{\mathfrak {t},\tau }\mathcal {W}(\mathfrak {t},\tau )\) is unbounded and may have many components. However, \(\mathcal {W}(\mathfrak {t},\tau )\) has a connected component containing the origin, say \(\mathcal {W}_1(\mathfrak {t},\tau )\) and we can consider the image \(W^{u,+}_{l_u}(\mathfrak {t}):=\varPhi _{\mathfrak {t},\tau }\mathcal {W}_1(\mathfrak {t},\tau )\) which is a 1-dimensional connected manifold and it is unbounded.

Again, cf. Fig. 4, if we switch from \(\tau \) to \(\tau _0<\tau \) we see that \(\varPhi _{\mathfrak {t},\tau _0}\mathcal {W}_1(\mathfrak {t},\tau _0) \supseteq \varPhi _{\mathfrak {t},\tau }\mathcal {W}_1(\mathfrak {t},\tau )\), and if \(T = \mathfrak {T}^u(\tau )\), then

$$\begin{aligned} W^{u,+}_{l_u}(T)=\varPhi _{T,\tau }\mathcal {W}_1(T,\tau )=\varPhi _{T,\tau _0}\mathcal {W}_1(T,\tau _0) \supseteq \varPhi _{T,\tau _1} \mathcal {W}_1(T,\tau _1) \end{aligned}$$

for any \(\tau _0<\tau <\tau _1\). Hence, we can define for every \(T\in {\mathbb {R}}\) the set

$$\begin{aligned} W^{u,+}_{l_u}(T)= \bigcup _{\tau \le T} \varPhi _{T,\tau }\mathcal {W}_1(T,\tau ) \end{aligned}$$

which is a 1-dimensional connected manifold containing the origin in its boundary, and it is unbounded. Reasoning in the same way we see that if \(\tau _0 < \tau _1\) then

$$\begin{aligned} \varPhi _{T,\tau _0}\mathcal {W}(T,\tau _0) \supseteq \varPhi _{T,\tau _1} \mathcal {W}(T,\tau _1) \, , \end{aligned}$$

therefore we can define

$$\begin{aligned} \tilde{W}^{u,+}_{l_u}(T):= \bigcup _{\tau \le T} \varPhi _{T,\tau }\mathcal {W}(T,\tau ) \end{aligned}$$

Notice that \(\tilde{W}^{u,+}_{l_u}(T)\) may not be a manifold (it may break infinitely many times), since \(\mathcal {W}(T,\tau ) \) may be disconnected. However by construction \(\tilde{W}^{u,+}_{l_u}(T)\) may be still characterized as in (6.6); moreover \(W^{u,+}_{l_u}(T)\) is the connected component of \( \tilde{W}^{u,+}_{l_u}(T)\) containing the origin in its boundary, and it is, as shown above, a 1-dimensional connected manifold.

The construction of \(W^{u,-}_{l_u}(\tau )\) and of \(W^{u}_{l_u}(\tau )=W^{u,-}_{l_u}(\tau ) \cup W^{u,+}_{l_u}(\tau )\) is completely analogous and it is omitted. \(\square \)

The construction of the stable leaves is very similar and we just sketch it.

With a specular argument we assume \(\varvec{gs}\), so that \(\mathcal {A}_{l_s}(+\infty )\) has \(\nu _2<0<\nu _1\) as eigenvalues, where \(\nu _1:=\alpha _{l_s}-\kappa (\beta )\) and \(\nu _2:=\alpha _{l_s}+2-n+\kappa (\beta )\). So, let Y(t) be the fundamental matrix of (6.1), where \(\mathcal {A}_{l_u}(t)\) is replaced by \(\mathcal {A}_{l_s}(t)\). Then, for any \(\tau \in {\mathbb {R}}\) there is a constant \(K=K(\tau )>1\), and a projection \(\mathcal {P}^+\) such that

$$\begin{aligned} \begin{array}{cc} \Vert Y(t)(I-\mathcal {P}^+) Y(s)^{-1} \Vert \le K \text {e}^{\nu _1 (t-s)} &{} \text { for any }s>t>\tau \,,\\ \Vert Y(t)\mathcal {P}^+ Y(s)^{-1} \Vert \le K \text {e}^{\nu _2 (t-s)} &{} \text { for any }t>s>\tau \,, \end{array} \end{aligned}$$
(6.7)

see again [12, Section 4], and [8, Appendix]. Denote by \(\mathcal {P}^+(\tau ):=Y(\tau )\mathcal {P}^+ Y(\tau )^{-1}\), and by \(\ell ^s(\tau )\) the 1-dimensional range of \(\mathcal {P}^+(\tau )\). Then the solution \(\mathbf {\xi }(t)\) of (6.1), with \(l_s\) replacing \(l_u\), is bounded for \(t \ge {\tau }\) iff \(\mathbf {\xi }({\tau }) \in \ell ^s(\tau )\). Moreover \(\Vert \mathbf {\xi }(t)\Vert \text {e}^{-\nu _2 t} \rightarrow c\) as \(t \rightarrow +\infty \) for a suitable \(c>0\). This way we are able to construct a local manifold \(W^s_{l_s,loc}(\tau )\) and to reprove a result analogous to Lemma 6.1. Then, assuming temporarily \(\mathbf C \) and reasoning as in Lemma 6.2, we see that \(\varPhi _{T,\tau }(W^s_{l_s,loc}(\tau ))\) is a 1-dimensional submanifold for any \(\tau ,T \in {\mathbb {R}}\); moreover \(\varPhi _{T,\tau _2}(W^s_{l_s,loc}(\tau _2)) \supset \varPhi _{T,\tau _1}(W^s_{l_s,loc}(\tau _1))\) if \(\tau _1< \tau _2\). Hence, assuming \(\mathbf C \) and \(\varvec{gs}\), we obtain that the set

$$\begin{aligned} \tilde{W}^s_{l_s}(\tau ):=\bigcup _{\tau _0\ge \tau } \varPhi _{\tau ,\tau _0}(W^s_{l_s,loc}(\tau _0)) = \big \{ \varvec{Q}\mid {\lim _{t \rightarrow +\infty }}\varvec{x_{l_s}}(t,\tau ,\varvec{Q})=(0,0) \big \} \end{aligned}$$
(6.8)

is a 1-dimensional immersed manifold having \(\ell ^s(\tau )\) as tangent space in the origin.

Then we remove assumption C and, arguing as above, we see that \(\tilde{W}^s_{l_s}(\tau )\) may be disconnected, but its connected component containing the origin, denoted by \(W^s_{l_s}(\tau )\), is again a 1-dimensional manifold. Then, repeating the previous discussion, we conclude the proof of Lemma 2.2. The part of the proof concerning Lemmas 2.1 and 2.3 is given below.

Now we proceed with the proof of Lemma 2.3, which includes Lemma 2.1 as a particular case. The proof is adapted from [13, Lemma 2.10] where it is developed assuming C and \(h(r) \equiv 0\).

Proof of Lemma 2.3

Assume \(\varvec{gu}\) and \(\varvec{gs}\); recalling that \(\varvec{x_{l_u}}(t)=(u(\text {e}^t) \text {e}^{\alpha _{l_u} t},u'(\text {e}^t) \text {e}^{(1+\alpha _{l_u}) t})\), we find \(\varvec{x_{l_s}}(t) = \varvec{x_{l_u}}(t) \text {e}^{(\alpha _{l_s}-\alpha _{l_u})t}\). Therefore in particular \(\varvec{R}=\varvec{Q}\text {exp}[ -(\alpha _{l_u}-\alpha _{l_s}) \tau ]\).

Assume first C for simplicity. From roughness of exponential dichotomy, cf. [12, Chapter 4] and [27, Theorem 2.16], we see that, if \(\varvec{Q}\in W^u_{l_u}(\tau )\), then there is \(d=d(\varvec{Q}) \in {\mathbb {R}}\) such that \(\lim _{t\rightarrow -\infty } \varvec{x_{l_u}}(t,\tau ,\varvec{Q}) \text {e}^{-[\alpha _{l_u}-\kappa (\eta )]t} =d(1,-\kappa (\eta ))\). Assume \(d>0\) for definiteness; then for the corresponding solution u(r) of (Hr) we get

$$\begin{aligned} {u(r)r^{\kappa (\eta )}}={x_{l_u}}(\ln (r),\tau ,\varvec{Q}) r^{-\alpha _{l_u}+\kappa (\eta )} \rightarrow d \,, \quad \text {as }r \rightarrow 0 \,. \end{aligned}$$
(6.9)

Assume now \(\varvec{Q}\not \in W^u_{l_u}(\tau )\). Then, if \(l_u \ne 2^*\), we find that there are \(c>0\) and \(N \gg 1\) such that \(|\varvec{x_{l_u}}(t,\tau ,\varvec{Q})|>c\) for \(t<-N\), and if \(l_u = 2^*\) there is a sequence \(t_n \rightarrow -\infty \) such that \(|\varvec{x_{l_u}}(t_n,\tau ,\varvec{Q})|>c>0\): in both the cases the corresponding solution u(r) of (Hr) is not a \(\mathcal {R}\)-solution since \(u(\text {e}^{t_n})\text {e}^{\alpha _{l_u} t_n} \not \rightarrow 0\) as \(t_n \rightarrow -\infty \), hence \(u(r)r^{\kappa (\eta )}\) becomes unbounded as \(r \rightarrow 0\), see (6.9). Further we easily see that \(\varvec{Q}\in W^u_{l_u}(\tau ) \iff \varvec{R}\in W^u_{l_s}(\tau ) \).

Arguing similarly, if \(\varvec{Q}\in W^s_{l_s}(\tau )\) then there exists \(L=L(\varvec{Q})>0\) such that we have \(\lim _{t\rightarrow +\infty } \varvec{x_{l_s}}(t,\tau ,\varvec{Q}) \text {e}^{-[\gamma _{l_s}+\kappa (\beta )]t} = L(1,-(n-2)+\kappa (\beta ))\): hence the corresponding solution u(r) of (Lr) satisfies

$$\begin{aligned} \lim _{t\rightarrow +\infty } u(\text {e}^t)\text {e}^{(\alpha _{l_s}-\gamma _{l_s}-\kappa (\beta ))t}= \lim _{r\rightarrow +\infty } u(r)r^{n-2-\kappa (\beta )} = L\,. \end{aligned}$$
(6.10)

So we can easily conclude as above.

Hence, if we assume either \(\varvec{Gu}\) or \(\varvec{gu}\), we can construct the unstable manifold \(W^u_{l_u}(\tau )\) for any \(\tau \in {\mathbb {R}}\); similarly if either \(\varvec{Gs}\) or \(\varvec{gs}\) hold, we can construct the stable manifold \(W^s_{l_s}(\tau )\) for any \(\tau \in {\mathbb {R}}\). Moreover Remark 2.2 still holds and we can construct \(W^s_{l_u}(\tau )\) and \(W^u_{l_s}(\tau )\) via (2.8) too.

Now we drop assumption C. In this case, due to the presence of non-continuable trajectories, we need to distinguish between \(W^u_{l_u}(\tau )\) and \(\tilde{W}^u_{l_u}(\tau )\), and similarly for the other manifolds. In fact \({\varvec{x_{l_u}}}(t,\tau ,\varvec{Q}) \rightarrow (0,0)\) iff \(\varvec{Q}\in \tilde{W}^u_{l_u}(\tau )\). Further for any \(\varvec{Q}\in \tilde{W}^u_{l_u}(\tau )\) we can find \(N \gg 1\) such that \({\varvec{x_{l_u}}}(T,\tau ,\varvec{Q})\in W^u_{l_u}(T)\) for any \(T \le -N\). So we can repeat the previous argument and we see that the corresponding solution u(r) of (Hr) is a \(\mathcal {R}\)-solution. A similar argument holds for the stable manifold, too.

Summing up, for any \(\tau \) we find that \(\varvec{Q}\in W^u_{l_u}(\tau )\) iff \(\varvec{R}\in W^u_{l_s}(\tau )\) iff \(u(r)=u(r,d)\) is a \(\mathcal {R}\)-solution with \(0<d<d^+_\tau \), see (2.10). Similarly \(\varvec{Q}\in W^s_{l_u}(\tau )\) iff \(\varvec{R}\in W^s_{l_s}(\tau )\) if \(u(r)=u(r,L)\) is a \(\mathsf{fd}\)-solution with \(0<L<L^+_\tau \). \(\square \)

Proof of Lemmas 4.2 and 4.3

We prove now Lemma 4.2: such a result has been obtained in presence of continuability of the solutions and for \(h(r) \equiv 0\) in [13, Lemma 2.10].

Proof of Lemma 4.2

We will prove only the first part of the statement, the second follows similarly. Consider the parametrization \(\varSigma ^{u,+}_{l_u}(\cdot ,T)\).

Assume first C. Observe that, starting from \(\varSigma ^{u,+}_{l_u}(\cdot ,T)\), we can construct a parametrization of \(W^{u}_{l_u}(\tau )\) for any \(\tau \in {\mathbb {R}}\), by setting \(\varSigma ^{u,+}_{l_u}(\omega ,\tau ):= \varvec{x_{l_u}}(\tau ;T,\varSigma ^{u,+}_{l_u}(\omega ,T)) \). In fact, the function \(\varSigma ^{u,+}_{l_u}: [0,+\infty ) \times {\mathbb {R}}\rightarrow {\mathbb {R}}^2\) is continuous in both the variables, and the map \((\omega ,\tau ) \mapsto (\varSigma ^{u,+}_{l_u}(\omega ,\tau ), z(\tau ))\) is injective in \(\varvec{W^{u}}\). According to this parametrization, \(\varvec{x_{l_u}}(t;\tau ,\varSigma ^{u,+}_{l_u}(\omega ,\tau ))\) coincides with \(\varvec{x_{l_u}}(t;T,\varSigma ^{u,+}_{l_u}(\omega ,T))\) and corresponds to the given solution \(u(r,d(\omega ))\) for any \(\tau \in {\mathbb {R}}\).

Fix \(N\in {\mathbb {R}}\) and let \(\delta :=\delta (N)\) be the constant defined in Lemma 6.1; we can find \(\bar{\omega }>0\) and \(N(\bar{\omega })<N\) such that \(\varSigma ^{u,+}_{l_u}(\omega ,\tau ) \in W^{u,+}_{l_u, \text {loc}}(\tau )\), whenever \(0\le \omega \le \bar{\omega }\) and \(\tau \le N(\bar{\omega })\).

We now show that \(d(\omega )\) is strictly increasing. Once proved this claim for this particular parametrization we have it for any parametrization \( \varpi \rightarrow \varSigma ^{u,+}_{l_u}(\varpi ,\tau )\) of \(W^{u,+}_{l_u}(\tau )\) as in the assumption of Lemma 4.2, due to the monotonicity of the change of variables \(\varpi (\omega )\). Using Lemma 6.1 we see that we can choose \(\omega _1<\omega _2\), so that \(\varSigma ^{u,+}_{l_u}(\omega _i,\tau ) \in W^{u,+}_{l_u, \text {loc}}(\tau )\) for any \(\tau \le N(\omega _2)\) and for \(i=1,2\). Hence \(\varSigma ^{u,+}_{l_u}([0,\omega _2]\times \{\tau \})\subset W^{u,+}_{l_u}(\tau )\) is a graph on \(\ell ^u(\tau )\), for any \(\tau \le N(\omega _2)\), see Lemma 6.1. In particular \(x_{l_u}(t; T,\varSigma ^{u,+}_{l_u}(\omega _1,T))-x_{l_u}(t; T,\varSigma ^{u,+}_{l_u}(\omega _2,T))<0\) for \(t=N(\omega _2)\). We claim that \(W^{u,+}_{l_u,loc}(\tilde{\tau })\) is a graph on a segment of the x axis, for any \(\tilde{\tau } \le N(\omega _2)\). In fact when \(\eta =0\) the claim is obvious since \(\ell ^u(\tau )\) is contained in the x axis. If \(\eta \ne 0\), since \(\ell ^u(\tau )\) is not orthogonal to the x axis, possibly choosing a smaller \(\delta \) we can again assume that \(W^{u,+}_{l_u,loc}(\tilde{\tau })\) is a graph on the x axis too, so the claim is true.

Assume for contradiction that \(d(\omega _1)>d(\omega _2)\), then there is \(\tilde{\tau }<N(\omega _2)\) such that \(x_{l_u}(t; T,\varSigma ^{u,+}_{l_u}(\omega _1,T))-x_{l_u}(t; T,\varSigma ^{u,+}_{l_u}(\omega _2,T))\) is positive for any \(t<\tilde{\tau }\) and it is zero for \(t=\tilde{\tau }\). In particular \(W^{u,+}_{l_u,loc}(\tilde{\tau })\) is not a graph on the x axis, so we have found a contradiction. Hence \(d(\omega _1)<d(\omega _2)\), and the Lemma is concluded if C holds. Notice that we can redefine the parametrization and use directly d instead of \(\omega \) as parameter, so, with a little abuse of notation we find the parametrization \(\varSigma ^{u,+}_{l_u}(d,\tau )\) of \(W^{u,+}_{l_u}(\tau )\) which is continuous (and \(C^1\)) in both the variables for any \((d,\tau ) \in [0,+\infty ) \times {\mathbb {R}}\).

Now we drop C. Fix \(T \in {\mathbb {R}}\), and correspondingly \(d^+_T\) as in (2.10), so that, for any \(d\in (0,d^+_T\)), u(rd) is continuable for any \(0<r< \text {e}^{T}\). Using the previous discussion and a truncation argument, for any \(D\in (0,d^+_T)\), we can define the map \(\varSigma ^{u,+}_{l_u}(d,T)\) for \(d \in [0,D]\); so we get a parametrization of a connected branch of \(W^{u,+}_{l_u}(T)\), say \(\bar{W}^{u,+}_{l_u}(T)\). Since for any point \(\varvec{Q}\in \bar{W}^{u,+}_{l_u}(T)\) we have that \({\varvec{x_{l_u}}}(\tau ; T, \varvec{Q})\) exists for any \(\tau \le T\), arguing as above, we find that \(\varSigma ^{u,+}_{l_u}(d,\tau )= \varvec{x_{l_u}}(\tau ,T,\varSigma ^{u,+}_{l_u}(d,T)) \) is a parametrization of a connected branch of \(W^{u,+}_{l_u}(\tau )\), denoted again by \(\bar{W}^{u,+}_{l_u}(\tau )\), for any \(\tau \le T\).

Now let \(\tau >T\) and notice that the function \(d^+(\tau )=d^+_{\tau }\) defined in (2.10) is decreasing in \(\tau \). If \(D < d^+_{\tau }\), reasoning as above, we find that \(\varSigma ^{u,+}_{l_u}(d,\tau )= \varvec{x_{l_u}}(\tau ,T,\varSigma ^{u,+}_{l_u}(d,T)) \) gives again a parametrization of a connected branch of \(W^{u,+}_{l_u}(\tau )\), for \(0\le d \le D\). If \(D \ge d^+_{\tau }\) then \(\varSigma ^{u,+}_{l_u}(d,\tau )= \varvec{x_{l_u}}(\tau ,T,\varSigma ^{u,+}_{l_u}(d,T)) \) for \(0< d< d^+_{\tau }\) is unbounded and it is itself a parametrization of the whole manifold \(W^{u,+}_{l_u}(\tau )\). Summing up, by the previous reasoning, for every \(T\in {\mathbb {R}}\) and \(D < d^+_{T}\), we have defined a map \(\varSigma ^{u,+}_{l_u}(d,\tau ): \bar{E}\rightarrow {\mathbb {R}}^2\) in the domain

$$\begin{aligned} \bar{E}= \big ([0,D) \times (-\infty , T] \big ) \cup \big \{(d,\tau ) \mid \tau > T \,,\, 0<d<\min \{D,d^+_{\tau }\} \big \}, \end{aligned}$$

which is continuous in both the variables and such that \(d \rightarrow \varSigma ^{u,+}_{l_u}(d,\tau )\) is injective. For the arbitrariness of \(D<d^+_T\) we can let \(D \rightarrow d^+_T\). Then from the arbitrariness of \(T \in {\mathbb {R}}\) we define \(\varSigma ^{u,+}_{l_u}\) in the whole \(E =\{(d,\tau ) \mid 0<d< d^+_{\tau } \, , \; \tau \in {\mathbb {R}}\} \): it is continuous in both the variables and it gives a bijective parametrization of \(W^{u,+}_{l_u}(\tau )\) for any \(\tau \in {\mathbb {R}}\). \(\square \)

Now we prove Lemma 4.3: the argument is a modification of [13, Propositions 3.5, 3.8]. In fact, in this setting, we need to take into account the fact that \(\ell ^u(\tau )\) and \(\ell ^s(\tau )\) change with \(\tau \) (due to the presence of Hardy potentials), while in [13] there was not this difficulty. In particular we need to ask for \(l_s>2^*\) and to profit of Lemma 4.1.

Proof of Lemma 4.3

We introduce some definitions borrowed from [3, 24]. Following [3, 24, 30], given a curve \(\varvec{\gamma }:[a,b]\rightarrow {\mathbb {R}}^2 \setminus \{(0,0)\}\), we define its rotation number \(w(\varvec{\gamma })\) by setting

$$\begin{aligned} w(\varvec{\gamma }):=\text {Int}\left[ \frac{\theta _{\varvec{\gamma }}(b)-\theta _{\varvec{\gamma }}(a)}{2\pi }\right] , \end{aligned}$$
(6.11)

where \(\text {Int}[\cdot ]\) denotes the integer part, and \(\varvec{\gamma }(t)= (\rho _{\varvec{\gamma }}(t)\cos \theta _{\varvec{\gamma }}(t), \rho _{\varvec{\gamma }}(t)\sin \theta _{\varvec{\gamma }}(t)).\) As pointed out in [24], we can extend this definition to a curve \(\varvec{\gamma }\) defined in a semi-open interval [ab) if \(\lim _{t \rightarrow b^-} \theta _{\varvec{\gamma }}(t)\) exists (even if it is infinite).

Our argument will be rather sketchy since we just adapt [13, 24]. Let \(\varvec{\gamma ^i}(t):[a,b] \rightarrow {\mathbb {R}}^2\), for \(i=1,2\), be curves in \({\mathbb {R}}^2\) which do not intersect each other, and let \(\varphi (t)\) be a smooth monotone function such as \(\varphi (t)=z(t)=\text {e}^{\varpi t}\) as in (2.5), or \(\varphi (t)=\zeta (t)=\text {e}^{-\varpi t}\) as in (2.6) or \(\varphi (t)=t\) as in [3]. Then \(\varvec{\varGamma ^i}(t)=(\varvec{\gamma ^i}(t),\varphi (t))\) are curves in \({\mathbb {R}}^3\). Following [3], we call linking number of \(\varvec{\gamma ^1},\varvec{\gamma ^2}\) in [ab] the number \(w(\varvec{\gamma ^1}-\varvec{\gamma ^2})\), i.e. the number of complete rotations of a curve around the other. Such a quantity is invariant for homotopies in \({\mathbb {R}}^3\) which preserve the endpoints \(\varvec{\varGamma ^1}(a)=\varvec{\varGamma ^2}(a)\) and \(\varvec{\varGamma ^1}(b)=\varvec{\varGamma ^2}(b)\).

We want to establish an homotopy between two curves so that linking number and rotation number are equal. Let us fix \(\tau \in {\mathbb {R}}\) and \(\varvec{Q}\in W^{s,+}_{l_s}(\tau )\). Since \(\varvec{x_{l_s}}(t,\tau ,\varvec{Q})\) converges to the origin as \(t \rightarrow +\infty \) and \(\dot{x}_{l_s}(t,\tau ,\varvec{Q})<0\) for \(t \gg 1\), for every \(\delta >0\) we can find \(a=a(\delta ) \gg 1\) such that \(x_{l_s}(a,\tau ,\varvec{Q})= \delta \) and \( x_{l_s}(t,\tau ,\varvec{Q})> \delta \) for \(t \in [\tau , a)\).

Then we set \(\varvec{\gamma ^1}(t)=\varvec{x_{l_s}}(t,\tau ,\varvec{Q})\), where \(\varvec{Q}\in W^{s,+}_{l_s}(\tau )\), and we consider the trajectory \(\varvec{\varGamma ^1}(t)=(\varvec{x_{l_s}}(t,\tau ,\varvec{Q}),\zeta (t))\) of (2.5) for \(t\in [\tau ,a]\). We shrink further \(\delta = \delta (\tau )\) so that the sets \(W^s_{l_s, loc}(T)\) defined in Lemma 6.1 are graphs on \(\ell ^s(T)\) for any \(T \ge \tau \), and we denote by \(\varvec{\bar{C}}(T)\) the unique point in \(W^{s,+}_{l_s, loc}(T)\cap \{x=\delta \}\).

We recall that \(\varUpsilon ^{s,+}_{l_s}(\cdot ,T)\) is the parametrization of \(W^{s,+}_{l_s}(T)\) given by L, see Remark 4.2. We denote by \(L_b\) the positive value such that \(\varUpsilon ^{s,+}_{l_s}(L_b,\tau )= \varvec{Q}\) so that \(\varUpsilon ^{s,+}_{l_s}(L_b,T)=\varvec{x_{l_s}}(T,\tau ,\varvec{Q})\). Let \(0<L_a(T)<L_b\) be such that \(\varUpsilon ^{s,+}_{l_s}(L_a(T),T)=\varvec{\bar{C}}(T)\) and notice that \(L_a(a)=L_b\) and \(\varUpsilon ^{s,+}_{l_s}(L_a(a),a)=\varvec{x_{l_s}}(a,\tau ,\varvec{Q})\). We consider the curves \(\varvec{\varPsi _1}(T)=(\varvec{\bar{C}}(T),\zeta (T))\) for \(T \in [\tau ,a]\), the curve \(\varvec{\varPsi _2}({L},{T})=(\varUpsilon ^{s,+}_{l_s}({L},{T}),\zeta ({T}))\) for \({L} \in [L_a(T), {L_b}]\) and the curve \({\varvec{\varGamma ^2}}\) obtained following the graph of \(\varvec{\varPsi _1}\) and then the graph of \({\varvec{\varPsi _2}(\cdot ,\tau )}\). An homotopy between \(\varvec{\varGamma ^1}\) and \(\varvec{\varGamma ^2}\) is obtained by projecting \(\varvec{\varGamma ^1}\) on \(W^s_{l_s}(\tau )\) following the 2-dimensional manifold \(\varvec{W^s}\): we sketch the construction, see [24, Lemma 4.3] for more details. See also Fig. 6.

For any \(T \in [\tau ,a]\) we construct the function \(H(\cdot ,T)\) obtained following \(\varvec{\varPsi _1}(s)=(\varvec{\bar{C}}(s),\zeta (s))\) for \(s\in [T,a]\), then \(\varvec{\varPsi _2}(L)=(\varUpsilon ^{s,+}_{l_s}(L,T),\zeta (T))\) for \(L \in [L_a(T), L_b]\) and finally \(\varvec{\varGamma ^1}(t)=(\varvec{x_{l_s}}(t,\tau ,\varvec{Q}),\zeta (t))\) for \(t \in [\tau ,T]\) i.e.

$$\begin{aligned} H(S,T)=\left\{ \begin{array}{ll} \varvec{\varPsi _1}(a+S(T-a)) &{} \text {if } S \in [0,1] \\ \varvec{\varPsi _2}(L_a(T)+(S-1)(L_b-L_a(T))) &{} \text {if } S \in [1,2] \\ \varvec{\varGamma ^1}(T+(S-2)(\tau -T)) &{} \text {if } S \in [2,3] \end{array}\right. \end{aligned}$$
(6.12)
Fig. 6
figure6

In this figure we sketch the homotopy H introduced in (6.12). For simplicity, we focus our attention on the first two variables. The homotopy consists of curves linking the point \(\varvec{x_{l_s}}(a,\tau ,\varvec{Q})\) back to the point \(\varvec{Q}\). For every instant \(T\in [\tau ,a]\), in the interval [0, 1] we follow the line \(x=\delta \) (i.e. \({\varvec{\varPsi _1}}\)), then, in the interval [1, 2], we follow the set \(W^{s,+}_{l_s}(T)\) (i.e. \({\varvec{\varPsi _2}}\)) toward the point \(\varvec{x_{l_s}}(T,\tau ,\varvec{Q})\), finally in [2, 3] we follow the flux back to the point \(\varvec{Q}\) (along a subset of \({\varvec{\varGamma _1}}\)). In the case \(T=a\), we simply follow \({\varvec{\varGamma _1}}\) (the homotopy is constant in [0, 2]) and, in the case \(T=\tau \), following \({\varvec{\varPsi _1}}\) and \({\varvec{\varPsi _2}}\), we reach \(\varvec{Q}\) at \(S=2\) (the homotopy is constant in [2, 3])

Note that all the curves \(S \mapsto H(S,T)\) have the same endpoints and are homotopic; moreover \(H(\cdot ,a)\) is \(\varvec{\varGamma ^1}\) and \(H(\cdot ,\tau )\) is \(\varvec{\varGamma ^2}\). So the linking number of \(\varvec{\varGamma ^1}\) and \(\varvec{\varGamma ^2}\) is 0.

Hence, we have shown that

$$\begin{aligned} \theta _{\varUpsilon }(\tau ,a)+ \theta _{{\varvec{\bar{C}}}}(\tau ,a)= \theta _{\varvec{x}}(\tau ,a) \, , \end{aligned}$$
(6.13)

where \(\theta _{\varUpsilon }(\tau ,a)\), \(\theta _{{\varvec{\bar{C}}}}(\tau ,a)\), \(\theta _{\varvec{x}}(\tau ,a)\) denotes respectively the angles performed by \(\varUpsilon ^{s,+}_{l_s}(L,\tau )\) for \(L \in [L_a(T), L_b]\), by \(\varvec{\bar{C}}(T)\) for \(T \in [\tau ,a]\), and by \(\varvec{x_{l_s}}(t,\tau ,\varvec{Q})\) for \(t \in [\tau ,a]\).

So Lemma 4.3 follows from (6.13), being (4.9) equivalent. \(\square \)

On the Wazewski’s Principle

We conclude the appendix with a result, inspired by Wazewski’s principle, which allows to locate the unstable and the stable manifolds. Consider

$$\begin{aligned} \dot{x}=F(x,t) \,, \end{aligned}$$
(6.14)

where \(x \in {\mathbb {R}}^2\), \(t \in {\mathbb {R}}\), F is continuous, and assume that the origin \(\varvec{O}=(0,0)\) is a critical point for (6.14).

Let \(\mathcal {T}(\tau )\) be a closed set diffeomorphic to a full triangle. We call the vertices \(\varvec{O}\), \(\varvec{A}(\tau )\) and \(\varvec{B}(\tau )\), and \(o(\tau )\), \(a(\tau )\), \(b(\tau )\) the edges (without endpoints) which are opposite to the respective vertex. Let \(\mathcal {\hat{T}}(\tau )\) denote a further set diffeomorphic to a full triangle having \(\varvec{O}\) as vertex and with edges \(\hat{a}(\tau )\supset a(\tau )\) and \(\hat{b}(\tau )\supset b(\tau )\); it follows that \(\mathcal {\hat{T}}(\tau )\supset \mathcal {T}(\tau )\). We begin from a result requiring very weak regularity properties.

Lemma 6.3

Assume that local uniqueness for the solutions of (6.14) is ensured for any trajectory starting from \(\mathcal {T}(\tau ) \setminus \{\varvec{O} \}\).

Suppose that the flow on \(a(\tau )\cup b(\tau )\) points towards the interior of \(\mathcal {T}(\tau )\), and on \(o(\tau )\) points towards the exterior of \(\mathcal {T}(\tau )\) for any \(t \le \tau \). Assume further that the flow on \(\{\varvec{A}(\tau ), \varvec{B}(\tau ) \}\) points towards the interior of \(\mathcal {\hat{T}}(\tau )\) for any \(t \le \tau \). Finally suppose that if a solution \(\varvec{x}(t)\) of (6.14) satisfies \(\varvec{x}(t) \in T(\tau )\) for any \(t \le \tau \), then \({\lim _{t \rightarrow -\infty }}\varvec{x}(t)=\varvec{O}\).

Then there is a compact connected set \({\mathcal {W}^u}(\tau )\subset \mathcal {T}(\tau )\) such that \(\varvec{O} \in {\mathcal {W}^u}(\tau )\), \({\mathcal {W}^u}(\tau ) \cap o(\tau ) \ne \emptyset \), with the following property:

$$\begin{aligned} {\mathcal {W}^u}(\tau ) \subset \{ \varvec{Q}\mid {\lim _{t \rightarrow -\infty }}\varvec{x}(t,\tau ;\varvec{Q}) = \varvec{O} \, , \; \varvec{x}(t,\tau ;\varvec{Q}) \in \mathcal {T}(\tau ) \; \text { for any }t \le \tau \} \,. \end{aligned}$$

Proof

This Lemma is proved in [21, § 3], see also [22, Lemma 3.5]: the reasoning relies on a connection argument and a topological idea developed in [36, Lemma 4]. \(\square \)

Obviously the same idea can be applied to construct stable sets.

Lemma 6.4

Assume that local uniqueness for the solutions of (6.14) is ensured for any trajectory starting from \(\mathcal {T}(\tau ) \setminus \{\varvec{O} \}\).

Suppose that the flow on \(a(\tau )\cup b(\tau )\) points towards the exterior of \(\mathcal {T}(\tau )\), and on \(o(\tau )\) points towards the interior of \(\mathcal {T}(\tau )\) for any \(t \ge \tau \). Assume further that the flow on \(\{\varvec{A}(\tau ), \varvec{B}(\tau ) \}\) points towards the exterior of \(\mathcal {\hat{T}}(\tau )\) for any \(t \ge \tau \). Finally suppose that if a solution \(\varvec{x}(t)\) of (6.14) satisfies \(\varvec{x}(t) \in T(\tau )\) for any \(t \ge \tau \), then \({\lim _{t \rightarrow +\infty }}\varvec{x}(t)=\varvec{O}\).

Then there is a compact connected set \({\mathcal {W}^s}(\tau )\subset \mathcal {T}(\tau )\) such that \(\varvec{O} \in {\mathcal {W}^s}(\tau )\), \({\mathcal {W}^s}(\tau ) \cap o(\tau ) \ne \emptyset \), with the following property:

$$\begin{aligned} {\mathcal {W}^s}(\tau ) \subset \{ \varvec{Q}\mid {\lim _{t \rightarrow +\infty }}\varvec{x}(t,\tau ;\varvec{Q}) = \varvec{O} \, , \; \varvec{x}(t,\tau ;\varvec{Q}) \in \mathcal {T}(\tau ) \; \text { for any }t \ge \tau \} \,. \end{aligned}$$

If we are in the position to apply invariant manifold theory for non-autonomous systems, clearly we find that these sets are manifolds. So we get the following.

Lemma 6.5

Assume that we are in the hypotheses of Lemma 6.3, respectively of Lemma 6.4. Assume further that F is \(C^1\) and it is continuous in x uniformly with respect to \(t \in {\mathbb {R}}\). Suppose that the linearized system admits exponential dichotomy, i.e. there are projections \(\mathcal {P}^+\) and \(\mathcal {P}^-\) of rank 1 such that (6.2) and (6.7) hold, so that \(\varvec{O}\) admits unstable and stable manifolds \(W^u(\tau )\) and \(W^s(\tau )\) for any \(\tau \in {\mathbb {R}}\). Then the set \({\mathcal {W}^u}(\tau )\subset (\mathcal {T}(\tau )\cap W^u(\tau ))\) constructed in Lemma 6.3, resp. the set \({\mathcal {W}^s}(\tau )\subset (\mathcal {T}(\tau )\cap W^s(\tau ))\) constructed in Lemma 6.4, is a connected 1-dimensional manifold.

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Franca, M., Sfecci, A. Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials. J Dyn Diff Equat 30, 1081–1118 (2018). https://doi.org/10.1007/s10884-017-9589-z

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Keywords

  • Supercritical equations
  • Hardy potentials
  • Radial solutions
  • Regular/singular ground states
  • Fowler transformation
  • Invariant manifold
  • Continuability

Mathematics Subject Classification

  • 35J75
  • 35J91
  • 37D10
  • 34C37