Bifurcation from infinity for elliptic problems on ${\mathbb {R}}^N$

Abstract

In the paper the asymptotic bifurcation of solutions to a parameterized stationary semilinear Schrödinger equation involving a potential of the Kato-Rellich type is studied. It is shown that the bifurcation from infinity occurs if the parameter is an eigenvalue of the hamiltonian lying below the asymptotic bottom of the bounded part of the potential. Thus the bifurcating solution are related to bound states of the corresponding Schrödinger equation. The argument relies on the use of the (generalized) Conley index due to Rybakowski and resonance assumptions of the Landesman-Lazer or sign-condition type.

1 Introduction

We study a parameterized elliptic problem

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u(x) + V(x)u(x)=\lambda u(x)+f(x,u(x)),\, x\in {\mathbb {R}}^N, \lambda \in {\mathbb {R}},\\ u\in H^1({\mathbb {R}}^N), \end{array}\right. \end{aligned}$$

(1.1)

related to a nonlinear Schrödinger equation (1.9) and its bound states of the form (1.10). Solutions to (1.1) may also be interpreted as stationary states of the corresponding reaction-diffusion equation (1.8).

We are interested in a characterization of asymptotic bifurcation for (1.1).

Definition 1.1

A parameter $\lambda _0\in {\mathbb {R}}$ is a point of bifurcation from infinity or asymptotic bifurcation of solutions to (1.1) if there exists a sequence $(\lambda _n, u_n)_{n=1}^\infty$ such that $\lambda _n\rightarrow \lambda _0$, $u_n\in H^1({\mathbb {R}}^N)$ is a weak solution of (1.1) with $\lambda =\lambda _n$ for each $n\ge 1$, and $\Vert u_n\Vert _{H^1} \rightarrow +\infty$.

The study of asymptotic bifurcation, apparently started by M. Krasnoselskii [22], who introduced the notion of an asymptotically linear operator, and P. Rabinowitz [33], as well as the study of bifurcation from zero (i.e. from the zero solution), have been conducted by numerous authors from both the abstract and application viewpoints (e.g. by Toland, Dancer, Mawhin, Schmitt, Ward and many others; see e.g. [12, 25, 39, 44, 45]). These problems are related since it is often possible to adapt ideas and techniques coming from the study of bifurcation from zero to asymptotic bifurcation; this was effectively employed by Toland in [44] and in [33, 43] via the so-called Toland inversion. Most of applications to PDEs were concerned with bifurcation and multiplicity of solutions to elliptic problems of the form $-\Delta u=\lambda u+f(x,u)$ on a bounded domain $\Omega \subset {\mathbb {R}}^N$ together with various boundary conditions (see e.g. [3, 16, 24]). A careful analysis of interactions (i.e. crossing) of $\lambda$ with the (purely discrete) spectrum of $-\Delta$ subject to the boundary condition along with appropriate behavior of f such as, for instance, the so-called ‘sign condition’, leads to the existence and multiplicity of solution. In [25] (see also [8, 9, 39]) it was pointed out that a condition of the Landesman-Lazer type could substitute the sign condition. The topological tools used depend on the parity of the crossed eigenvalue of $-\Delta$: roughly speaking topological degree techniques are exploited if $\lambda$ crosses an eigenvalue of odd multiplicity while variational methods are used in the case of even multiplicity.

The problem of bifurcation of solutions to elliptic problems on ${\mathbb {R}}^N$ is not that well-recognized. A detailed study of bifurcation from zero is given e.g. in [14, 32, 42], while questions of asymptotic bifurcation were dealt with in [17, 43] (see also the references therein) and [23]. An important issue of the spectral theory of elliptic equations on ${\mathbb {R}}^N$, as opposed to its counterpart on bounded domains, is that the spectrum of $-\Delta +V(x)$ is not discrete in general and, depending on the potential, may be quite complicated. Results from [17, 23, 43] show that the existence of asymptotic bifurcation at an eigenvalue $\lambda _0$ relies on the appropriate relationship between $\lambda _0$, f and the essential spectrum of $-\Delta +V(x)$ inasmuch as bound states of the Schrödinger equation correspond to energies below the bottom of the essential spectrum.

Let us now present the standing assumptions. As concerns the potential generating the hamiltonian

$$\begin{aligned} {{\mathbf {A}}}:=-\Delta +V(x) \end{aligned}$$

we assume that

$$\begin{aligned}&V\in L^\infty ({\mathbb {R}}^N)+L^p({\mathbb {R}}^N),\; \text {i.e.}\;\; V=V_\infty +V_0,\; \text {where} \end{aligned}$$

(1.2)

$$\begin{aligned}&V_\infty \in L^\infty ({\mathbb {R}}^N)\;\; \text {and}\;\; V_0\in L^p({\mathbb {R}}^N),\; p\ge 2\;\; \text {if}\;\; N=1,\; p>2\;\; \text {if}\;\;\nonumber \\&N=2\;\; \text {and}\;\; p\ge N\;\; \text {for}\;\; N\ge 3, \end{aligned}$$

(1.3)

and, as concerns the nonlinear interaction term, we assume that $f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a Carathéodory function such that

$$\begin{aligned}&|f(x ,u)|\le m(x) \text{ for } \text{ all } u\in {\mathbb {R}} \text{ and } \text{ a.e. } x\in {\mathbb {R}}^N, \end{aligned}$$

(1.4)

$$\begin{aligned}&|f(x,u)-f(x,v)|\le l(x)|u-v| \text{ for } \text{ all } u,v\in {\mathbb {R}} \text{ and } \text{ for } \text{ a.e. } x\in {\mathbb {R}}^N, \end{aligned}$$

(1.5)

where $m\in L^2({\mathbb {R}}^N)$, $l=l_0+l_\infty$ with $l_0$ satisfying (1.3) (with $l_0$ instead of $V_0$) and $l_\infty \in L^\infty ({\mathbb {R}}^N)$.

Remark 1.2

Observe that V belongs the the so-called Kato class of potentials $K_N$ considered by Aizenman and Simon (see [37, A.2]) since $L^r({\mathbb {R}}^N)\subset K_N$ whenever $r\ge 2$ with $r>N/2$, $N\ge 2$, or a slightly more general class considered in [19]. If, for instance, V is the Coulomb type potential, i.e. $V(x):=c/|x-x_0|^{\alpha }$ for $x\ne x_0$, where $x_0\in {\mathbb {R}}^N$, $c\in {\mathbb {R}}$ and $\alpha \in [0,\nicefrac {1}{2})$ if $N=1$, $\alpha \in [0,1)$ for $N=2$ and $\alpha \in [0,1)$ for $N\ge 3$, then V satisfies conditions (1.2) and (1.3) since one may take $V_0=\chi V$ and $V_\infty =(1-\chi )V$, here $\chi$ is the characteristic function of the unit ball in ${\mathbb {R}}^N$ around $x_0$. $\square$

Since $\lim _{|s|\rightarrow +\infty }f(x,s)/s=0$ for $x\in {\mathbb {R}}^N$, one expects that, as in the classical situation (see e.g. [33]), if $\lambda$ approaches an eigenvalue of ${{\mathbf {A}}}$, then solutions to (1.1) bifurcate from infinity as the result of a produced resonance phenomenon. Indeed: as we shall see in Theorem 4.1, the necessary condition for $\lambda _0$ lying beyond the essential spectrum of the hamiltonian for inducing asymptotic bifurcation is that $\lambda _0\in \sigma _p({{\mathbf {A}}})$ the point spectrum of the hamiltonian. Conversely, if $\lambda _0$ is an isolated eigenvalue of odd multiplicity, then the asymptotic bifurcation occurs. In order to provide sufficient conditions for asymptotic bifurcation from an isolated eigenvalue of even multiplicity, one needs to impose additional assumptions concerning the behavior of f at infinity: the so-called Landesman-Lazer type or strong resonance conditions.

The Landesmann-Lazer type conditions state that either

or

where ${\hat{f}}_\pm (x):= \limsup _{s \rightarrow \pm \infty } f(x,s)$ and ${\check{f}}_\pm (x):= \liminf _{s \rightarrow \pm \infty } f(x,s)$ for a.e. $x\in {\mathbb {R}}^N$.

Remark 1.3

Conditions of this type has been considered by many authors; see e.g. [15] for a relatively up-to-date survey. Observe (see also the proof of Lemma 5.2) that $(LL)_+$ (resp. $(LL)_-$), together with the so-called unique continuation property, imply that

$$\begin{aligned} \int _{{\mathbb {R}}^N}({\check{f}}_+\varphi ^+-{\hat{f}}_-\varphi ^-)\,dx>0\;\; \left( \text {resp.}\;\; \int _{{\mathbb {R}}^N}({\hat{f}}_+\varphi ^+-{\check{f}}^-\varphi ^-)\,dx<0\right) \end{aligned}$$

(1.6)

for any eigenfunction $\varphi$ of the hamiltonian ${{\mathbf {A}}}$ and $\varphi ^\pm =\max \{0,\pm \varphi \}$. Clearly (1.6) is the classical Landesman-Lazer condition (see e.g. [15, eq. (LL)]); one can easily check by proof-inspection that each of the conditions stated in (1.6) is actually sufficient for our purposes.

$\square$

The so-called sign conditions or strong resonance conditions are fulfilled if $k_\pm (x):=\lim _{s\rightarrow \pm \infty } sf(x,s)$ exists for a.a. $x\in {\mathbb {R}}^N$, $k_\pm \in L^\infty ({\mathbb {R}}^N)$ and either

or

As we shall see (comp. Lemma 5.2) both assumption $(LL)_\pm$ and $(SR)_\pm$ lead to the geometric condition (5.2) concerning inward (or outward) behavior of the nonlinearity with respect to eigenspaces of ${{\mathbf {A}}}$. Such conditions were already studied on an abstract level in [25, Eq. (2.3) or (2.4)], [6] and [21]. A discussion of some other resonance conditions and their role is provided in [5].

Our main result is as follows. Let

$$\begin{aligned} \alpha _\infty :=\lim _{R\rightarrow \infty }\mathrm {essinf}\,_{|x|\ge R}V_\infty (x), \end{aligned}$$

(1.7)

be the asymptotic bottom of the potential $V_\infty$.

Theorem 1.4

Suppose that $\lambda _0\in \sigma ({\mathbf {A}})$. If either

(i) $\lambda _0$ is an isolated eigenvalue of odd multiplicity; or

(ii) $\lambda _0<\alpha _\infty$ (¹) and one of conditions $(LL)_\pm$ or $(SR)_\pm$ holds,

then $\lambda _0$ is a point of bifurcation from infinity for (1.1).

Remark 1.5

(1) It is clear that if $(\lambda _n,u_n)$ is a sequence bifurcating form infinity at $\lambda _0$, then $u_n\in H^2({\mathbb {R}}^N)$ and $\Vert u_n\Vert _{H^2}\rightarrow +\infty$. In Theorem 4.1 we show that under the assumptions of the above theorems also both sequences $(\Vert u_n\Vert _{L^2})$ and $(\Vert \nabla u_n\Vert _{L^2})$ tend to infinity; moreover these sequences have the same growth rate.

(2) Theorem 1.4 complements and generalizes results concerning the asymptotic bifurcation for equations of the form (1.1) from [43] and [23]. In [23] problem (1.1) was studied when $V\in L^\infty ({\mathbb {R}}^N)$ (i.e., $V_0\equiv 0$) and under hypotheses which, together with the ansatz $(f_4)$ (see [23, p. 415]), imply our standing assumptions with one important difference in comparison to (1.4): in the setting of [23], the bounding function $m\in L^\infty ({\mathbb {R}}^N)$. In [43] a similar problem is very thoroughly investigated with $f(x,u)=h(x)+{\tilde{f}}(u)$, where $h\in L^2({\mathbb {R}}^N))$, ${\tilde{f}}$ is Lipschitz and ${\tilde{f}}(u)/u\rightarrow 0 = {\tilde{f}}(0)$ as $|u|\rightarrow +\infty$ (see the assumption (G) in [43]). Hence, the results of this paper apply to a more general class of potentials but consider a different type of nonlinearities. It is also noteworthy that in both papers [43] and [23] the asymptotic bifurcation occurs at an eigenvalue $\lambda _0$ of ${{\mathbf {A}}}$ provided the distance $\mathrm {dist}(\lambda _0,\sigma _e({{\mathbf {A}}}))$ of $\lambda _0$ to $\sigma _e({{\mathbf {A}}})$, the essential spectrum of the hamiltonian, is larger than the Lipschitz constant of the nonlinearity g (in [43] a bit more restrictive bound is necessary). Such a condition was also implicitly contained in [12, Assumption D]. If the multiplicity of $\lambda _0$ is odd, then the proofs from [23, 43] use the degree theory (via the Toland inversion in [43]), while for an eigenvalue of even multiplicity the existence of asymptotic bifurcation in [23] relies on a variational approach based on the Morse theory. In [17] the principal eigenvalue (being simple) of the linearization at infinity is shown to be a point of asymptotic bifurcation and the result is obtained by the Toland inversion.

In our approach the physically relevant unbounded part $V_0$ of the potential is not trivial, but, at least in case the multiplicity of $\lambda _0$ is even, we need that $\lambda _0<\alpha _\infty$ which, as we shall see, implies that $\lambda _0$ lies below the bottom of $\sigma _e({{\mathbf {A}}})$ since $\sigma _e({{\mathbf {A}}})\subset [\alpha _\infty ,\infty )$. We do not require any relations of the distance $\mathrm {dist}(\lambda _0,\sigma _e({{\mathbf {A}}}))$ with the Lipschitz constant, but instead we make use of the estimate (1.4). If $V_0\ne 0$ (making V look like a potential well) is sufficiently deep and steep, then $\sigma ({{\mathbf {A}}})\cap (-\infty ,\alpha _\infty )\ne \emptyset$ (this holds for instance if V is the Coulomb type potential from Remark 1.2; see also eg. [34, Theorem XIII.6] and [40]).

(3) Our attitude to the first part of Theorem 1.4 is based on the Leray-Schauder degree theory; in this context condition (1.5) is not necessary since the continuity of the Nemytskii operator generated by f is sufficient. In the second part we shall rely on the Conley index theory applied to the semiflow generated by the parabolic equation

$$\begin{aligned} u_t=\Delta u-V(x)u+\lambda u+f(x,u),\;\; x\in {\mathbb {R}}^N,\; u\in {\mathbb {R}},\; t>0, \end{aligned}$$

(1.8)

related to (1.1). We shall show that our assumptions imply that this semiflow is well-defined and its Conley indices ‘at infinity’ change when the parameter $\lambda$ crosses $\lambda _0$. To meet the quite demanding requirements concerning compactness issues (i.e. the so-called admissibility of the semiflow with respect to bounded sets) we adopt some ideas of Prizzi [30, 31]. The use of the (generalized) Conley type index of Rybakowski [36] in the context of bifurcation has been started by Ward [45, 46] and applied for elliptic problems on bounded domains. Quite recently this approach has been thoroughly complemented and expanded in [24] (see also the rich bibliography therein) and applied to bifurcation problems on bounded domains. To the best of our knowledge the present paper is the first one to employ Conley index to the asymptotic bifurcation for elliptic problems in ${\mathbb {R}}^N$. $\square$

Let us now discuss the physical context of the studied problem. We consider the externally driven nonlinear Schrödinger equation of the form

$$\begin{aligned} i\psi _t=-\Delta \psi +V(x)\psi -W'(x,\psi ), \end{aligned}$$

(1.9)

and its bound states, i.e. wave-functions $\psi :[0,+\infty )\times {\mathbb {R}}^N\rightarrow {\mathbb {C}}$ that vanish at infinity; here V satisfies assumptions (1.2) and (1.3), $W:{\mathbb {R}}^N\times {\mathbb {C}}\rightarrow {\mathbb {R}}$ and $W'(x,z):=\frac{\partial }{\partial z_1}W(x,z)+i\frac{\partial }{\partial z_2}W(x,z)$, $x\in {\mathbb {R}}^N$, $z=z_1+iz_2$. One usually assumes that W depends on $x\in {\mathbb {R}}^N$ and |z| only, i.e. $W(x,z)=H(x,|z|)$ where $H:{\mathbb {R}}^N\times [0,+\infty )\rightarrow {\mathbb {R}}$ has the form

$$\begin{aligned} H(x,s)=\int _0^sh(x,\xi )\,d\xi ,\;\;x\in {\mathbb {R}}^N,\;s\ge 0, \end{aligned}$$

and $h:{\mathbb {R}}^N\times [0,+\infty )\rightarrow {\mathbb {R}}$ is a Carathéodory function satisfying conditions analogous to (1.4) and (1.5). Therefore for all $x\in {\mathbb {R}}^N$

$$\begin{aligned} W'(x,z)=h(x,|z|)\frac{z}{|z|}\;\;\text {for}\;\;z\in {\mathbb {C}}{\setminus }\{0\},\;\; W'(x,0)=0. \end{aligned}$$

Problems concerning (1.9) play an important role in different physical contexts, especially in the description of macroscopic quantum systems like, for instance, plasma physics, nonlinear optics and others – see e.g. [28, 41]. For appropriate choice of h the equation (1.9) has standing wave solutions, i.e. satisfying the ansatz

$$\begin{aligned} \psi (t,x)=e^{-i\lambda t}u(x),\;\;t\ge 0,\;x\in {\mathbb {R}}^N,\end{aligned}$$

(1.10)

with the time-independent profile $u\in H^1({\mathbb {R}}^N)$ and $\lambda \in {\mathbb {R}}$. Substituting (1.10) into (1.9) and putting for $x\in {\mathbb {R}}^N$ and $u\in {\mathbb {R}}$

$$\begin{aligned} f(x,u):= h(x,|u|)\frac{u}{|u|}\;\;\text {if}\; u\ne 0,\;\;f(x,0)=0, \end{aligned}$$

(1.11)

we get (1.1) along with our standing assumptions; clearly any solution $(\lambda ,u)\in {\mathbb {R}}\times H^1({\mathbb {R}}^N)$ gives via (1.10) a bound state $\psi$ for (1.9).

The energy (see [4]) of a wave-function $\psi$ satisfying (1.9), given by

$$\begin{aligned} E(\psi ):=\frac{1}{2}\int _{{\mathbb {R}}^N}(|\nabla \psi |^2+V(x)|\psi |^2)\,dx-\int _{{\mathbb {R}}^N} W(x,\psi )\,dx \end{aligned}$$

is time invariant and, in case (1.10),

$$\begin{aligned} E(\psi )=\frac{1}{2}\int _{{\mathbb {R}}^N}(|\nabla u|^2+V(x)u^2)\,dx-\int _{{\mathbb {R}}^N}H(x,|u|)\,dx. \end{aligned}$$

Theorem 1.6

Suppose that $\lambda _0<\alpha _\infty$, where $\alpha _\infty$ is given by (1.7), $\lambda _0\in \sigma (-\Delta +V)$ and one of the following conditions is satisfied:

$(i)_+$ for a.a. $x\in {\mathbb {R}}^N$, ${\check{h}}(x):=\liminf _{\xi \rightarrow +\infty }h(x,\xi )\ge 0$ and ${\check{h}}$ is positive on a set of positive measure;

$(i)_-$ for a.a. $x\in {\mathbb {R}}^N$, ${\hat{h}}(x):=\limsup _{\xi \rightarrow +\infty }h(x,\xi )\le 0$ and ${\hat{h}}$ is negative on a set of positive measure;

$(ii)_+$ for a.a. $x\in {\mathbb {R}}^N$ and all $\xi \ge 0$, $h(x,\xi )\ge 0$ and $\lim _{\xi \rightarrow +\infty }\xi \, h(x,\xi )$ is positive on a set of positive measure;

$(ii)_-$ for a.a. $x\in {\mathbb {R}}^N$ and all $\xi \ge 0$, $h(x,\xi )\le 0$ and $\lim _{\xi \rightarrow +\infty }\xi h(x,\xi )$ is negative on a set of positive measure.

Then there is a sequence $(\psi _n)$ of bound states of (1.9) of the form $\psi _n(t,x)=e^{-i\lambda _n t}u_n(x)$ for $t\ge 0$, $x\in {\mathbb {R}}^N$, where $\lambda _n\in {\mathbb {R}}$, $u_n\in H^1({\mathbb {R}}^N)$ for all $n\ge 1$, $\lambda _n\rightarrow \lambda _0$ and $\Vert u_n\Vert _{H^1}\rightarrow +\infty$. If $\lambda _0\ne 0$, then $|E(\psi _n)|\rightarrow +\infty$.

Proof: It is easy to see that if f is given by (1.11), then condition $(i)_\pm$ (resp. $(ii)_\pm$) implies $(LL)_\pm$ (resp. $(SR)_\pm$); hence, in view of Theorem 1.4, there is a sequence $(\lambda _n,u_n)$ of solutions to (1.1), yielding via (1.10) the existence of the required sequence of bound states $\psi _n$. Observe that

$$\begin{aligned} E(\psi _n)= & {} \frac{1}{2}\left( \lambda _n\Vert u_n\Vert ^2_{L^2}+\int _{R^N} (h(x,|u_n|)|u_n|-2H(x,|u_n|))\,dx\right) \\\ge & {} \frac{1}{2}\lambda _n\Vert u_n\Vert ^2_{L^2}-2\Vert m\Vert _{L^2}\Vert u_n\Vert _{L^2}\rightarrow +\infty \end{aligned}$$

when $\lambda _0>0$ and $E(\psi _n)\rightarrow -\infty$ if $\lambda _0<0$.$\square$

The paper is organized as follows. Section 2 is devoted to basic notation and a brief exposition of the Conley index theory. In Section 3 we construct the semiflow related to the considered problem, study its basic properties such as continuity and admissibility; we also recall a linearizaton method to compute the Conley index of the set of bounded trajectories. Section 4 deals with necessary conditions as well as further properties of bifurcating sequences. Finally, Section 5 is devoted to the proof of the main results.

2 Preliminaries

By $L^p(\Omega )$, $1\le p\le \infty$, and $H^{k}(\Omega )$, $k\in {\mathbb {N}}$, we denote the standard Lebesgue and Sobolev spaces on an open domain $\Omega \subset {\mathbb {R}}^N$, $N\ge 1$, with their standard norms and inner products. For brevity, in the sequel we will write $L^p$ or $H^k$ instead of $L^p({\mathbb {R}}^N)$ and $H^k({\mathbb {R}}^N)$.

If (X, A) is a topological pair with a closed and nonempty $A\subset X$, then X / A denotes the quotient space, obtained by collapsing the subset A to a point [A]. Pointed spaces $(X,x_0)$ and $(Y,y_0)$ are homotopy equivalent or have the same homotopy type if there are pointed maps $f:(X,x_0)\rightarrow (Y,y_0)$ and $g:(Y,y_0)\rightarrow (X,x_0)$ such that $f\circ g$ (resp. $g\circ f$) is homotopic to the identity on $(Y,y_0)$ (resp. on $(X,x_0)$). The homotopy class represented by a space $(X,x_0)$ is denoted by $[(X,x_0)]$.

2.1 Conley index due to Rybakowski

We shall briefly recall a version of the Conley index due to Rybakowski (see [35] or [36]). Let $\Phi :[0,+\infty )\times X\rightarrow X$ be a semiflow on a complete metric space X. We write $\Phi _t(x):=\Phi (t,x)$ and $\Phi _{[0,t]}(x):=\{\Phi _s(x)\mid 0\le s\le t\}$ for $t\ge 0$, $x\in X$. A continuous $u:J\rightarrow X$, where $J\subset {\mathbb {R}}$ is an interval, is a solution of $\Phi$ if $u(t+s) = \Phi _t(u(s))$ for all $t\ge 0$ and $s\in J$ such that $t+s\in J$. If, in addition $0\in J$ and $u(0)=x$, then u is a solution through x.

If $a\in {\mathbb {R}}$ and $u:[a,+\infty )\rightarrow X$ is a solution of $\Phi$, then the $\omega$-limit set of u is defined by

$$\begin{aligned} \omega (u):=\{x=\lim _{n\rightarrow \infty } u(t_n)\mid t_n\ge a,\; t_n\rightarrow +\infty \}; \end{aligned}$$

if $u:(-\infty , a]\rightarrow X$ is a solution of $\Phi$, then the $\alpha$-limit set of u is defined by

$$\begin{aligned} \alpha (u):=\{x=\lim _{n\rightarrow \infty } u(t_n)\mid t_n\le a,\;t_n\rightarrow -\infty \}. \end{aligned}$$

Note that both sets $\omega (u)$ and $\alpha (u)$ are closed.

Let $N\subset X$. We define the invariant part $\mathrm {Inv}_\Phi (N)$ of N by

$$\begin{aligned} x\in \mathrm {Inv}_\Phi (N)\,\Longleftrightarrow \,\text {there is a solution}\;u:{\mathbb {R}}\rightarrow N\;\text {through}\;x. \end{aligned}$$

A set $K\subset X$ is a $\Phi$-invariant or invariant (w.r.t. $\Phi$) if $\mathrm {Inv}_\Phi (K)=K$. A set K is an isolated invariant if there exists an isolating neighborhood of K, i.e. $N\subset X$ such that $K=\mathrm {Inv}_\Phi (N) \subset \mathrm {int}\,N$.

A set $N\subset X$ is $\Phi$-admissible or admissible (w.r.t. $\Phi$) if, for any sequences $(t_n)$ in $[0,+\infty )$, $(x_n)$ in X such that $t_n\rightarrow +\infty$ and $\Phi _{[0,t_n]} (x_n) \subset N$, the sequence of end-points $\left( \Phi _{t_n}(x_n) \right)$ has a convergent subsequence. It is easy to see that if $N\subset X$ is $\Phi$-admissible, then the invariant part $\mathrm {Inv}_\Phi (N)$ is compact.

Suppose that $\{\Phi ^\lambda \}_{\lambda \in \Lambda }$, where $\Lambda$ is a metric space, is a family of semiflows on X. This family is continuous if the map $[0,+\infty )\times X\times \Lambda \ni (t,x,\lambda )\mapsto \Phi ^\lambda _t(x)$ is continuous. A set $N\subset X$ is admissible w.r.t. $\{\Phi ^\lambda \}$ if, for any sequences $(t_n)$ in $[0,+\infty )$, $(x_n)$ in X and $(\lambda _n)$ such that $t_n\rightarrow +\infty$, $\lambda _n\rightarrow \lambda _0$ in $\Lambda$ and $\Phi ^{\lambda _n}_{[0,t_n]}(x_n)\subset N$, the sequence $(\Phi ^{\lambda _n}_{t_n}(x_n))$ has a convergent subsequence.

Let ${{\mathcal {I}}}(X)$ be the family of all pairs $(\Phi , K)$, where $\Phi$ is a semiflow on X and a set $K\subset X$ is isolated invariant w.r.t. $\Phi$ having a $\Phi$-admissible isolating neighborhood. If $(\Phi ,K)\in {{\mathcal {I}}}(X)$, then the Conley homotopy index $h(\Phi ,K)$ of K relative to $\Phi$ is defined by

$$\begin{aligned} h(\Phi , K):=[(B/B^-, [B^-])], \end{aligned}$$

where B is an isolating block of K (relative to $\Phi$; see [35] for the details) with the exit set $B^- \ne \emptyset$; if $B^-=\emptyset$ we put $h(\Phi , K):=[(B\cup \{ a\}, a)]$ where a is an arbitrary point out of B. In particular, $h(\Phi , \emptyset )={\overline{0}}$ where ${\overline{0}}:=[(\{a\},a)]$.

Let us enumerate several important properties of homotopy index:

• (H1) for any $(\Phi , K)\in {{\mathcal {I}}}(X)$, if $h(\Phi , K)\ne \overline{0}$, then $K\ne \emptyset$;

• (H2) if $(\Phi , K_1), (\Phi , K_2)\in {{\mathcal {I}}}(X)$ and $K_1\cap K_2=\emptyset$, then $(\Phi , K_1\cup K_2)\in {{\mathcal {I}}}(X)$ and $h(\Phi , K_1\cup K_2) = h(\Phi , K_1)\vee h(\Phi , K_2)$;

• (H3) for any $(\Phi _1, K_1)\in {{\mathcal {I}}}(X_1)$ and $(\Phi _2, K_2)\in {{\mathcal {I}}}(X_2)$, $(\Phi _1\times \Phi _2, K_1\times K_2)\in {{\mathcal {I}}} (X_1\times X_2)$ and $h(\Phi _1\times \Phi _2, K_1\times K_2) = h(\Phi _1, K_1) \wedge h(\Phi _2, K_2)$;

• (H4) if the family of semiflows $\{\Phi ^\lambda \}_{\lambda \in [0,1]}$ is continuous and there exists an admissible (with respect to this family) N such that $K_\lambda = \mathrm {Inv}_{\Phi ^\lambda } (N) \subset \mathrm {int}\ N$, $\lambda \in [0,1]$, then

  $$\begin{aligned} h(\Phi ^0, K_0) = h(\Phi ^1, K_1). \end{aligned}$$

In a linear case the following formula for computation of the Conley index is used.

Theorem 2.1

(See [35, Ch. I, Th. 11.1]) Assume that a $C_0$ semigroup $\{T(t)\}_{t\ge 0}$ of bounded linear operators on a Banach space X is hyperbolic (see e.g. [13, Def. V.1.14]). If the dimension $\dim X_u=k$ of the unstable subspace $X_u$ (²) is finite, then $\Phi :[0,+\infty ) \times X\rightarrow X$, given by $\Phi (t,x):=T(t)x$ for $x\in X$ and $t\ge 0$, is a semiflow on X, $\{ 0\}$ is the maximal bounded invariant set with respect to $\Phi$, $(\Phi , \{ 0 \})\in {{\mathcal {I}}}(X)$ and $h(\Phi , \{ 0\})=\Sigma ^k$ where $\Sigma ^k=[(S^k, {\overline{s}})]$ is the homotopy type of the pointed k-dimensional sphere.$\square$

3 Admissibility and compactness properties of semiflow

Let us consider problems (1.1) in its abstract form

$$\begin{aligned} ({\mathbf {A}}-\lambda {\mathbf {I}})u={\mathbf {F}}(u),\;\;u\in H^2,\;\lambda \in {\mathbb {R}}, \end{aligned}$$

(3.1)

where ${\mathbf {I}}$ is the identity on $L^2$, with the linear operator ${{\mathbf {A}}}:D({{\mathbf {A}}})\subset L^2 \rightarrow L^2$ given by

$$\begin{aligned}&D({\mathbf {A}}):=H^2,\;\; {\mathbf {A}}:={\mathbf {A}}_0+{\mathbf {V}}_0+{\mathbf {V}}_\infty ,\;\; \text {where:} \end{aligned}$$

(3.2)

$$\begin{aligned}&{\mathbf {A}}_0u:=-\Delta u,\;\;\text {i.e.,}\;\; {\mathbf {A}}_0u:=-\sum _{j=1}^N\frac{\partial ^2u}{\partial x_j^2}\;\; \text {for}\;\; u\in D({\mathbf {A}}_0)=D({\mathbf {A}}), \end{aligned}$$

(3.3)

$$\begin{aligned}&{\mathbf {V}}_\infty u:=V_\infty \cdot u\;\; \text {for}\;\; u\in D({\mathbf {V}}_\infty ):=L^2\;\; \text {and} \end{aligned}$$

(3.4)

$$\begin{aligned}&{{\mathbf {V}}}_0u=V_0\cdot u\;\; \text {for}\;\; u\in D({{\mathbf {V}}}_0):=L^q,\;\; \text {where}\;\;q\;\;\text {is given by } 3.8\, below; \end{aligned}$$

(3.5)

and ${{\mathbf {F}}}:H^1\rightarrow L^2$ is the superposition operator generated by f, i.e.:

$$\begin{aligned} {{\mathbf {F}}}(u):= f(\cdot ,u(\cdot )),\text {for}u\in L^2.\end{aligned}$$

(3.6)

Let us discuss the above abstract setting.

Remark 3.1

(1) By [27, Th. 7.3.5], ${{\mathbf {A}}}_0$ is self-adjoint and sectorial. Clearly ${{\mathbf {V}}}_\infty$ is a bounded linear operator. By [13, Proposition III.1.12] ${{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty$, defined on $D({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )=D({{\mathbf {A}}})$, is sectorial, too. By the Kato-Rellich theorem (see [40, Theorem 8.5]) it is self-adjoint. It is also clear that

$$\begin{aligned} s_\infty :=\inf \sigma ({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )=\inf _{u\in H^1,\,\Vert u\Vert _{L^2}=1}\int _{{\mathbb {R}}^N}(|\nabla u|^2+V_\infty (x)u^2)\,dx, \end{aligned}$$

i.e. $\sigma ({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )\subset [s_\infty ,+\infty )$. In view of the Persson theorem [29, Theorem 2.1] we have that

$$\begin{aligned} s_\infty ^*:=&\inf \sigma _e({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )\\ =&\lim _{R\rightarrow \infty } \inf \left\{ \int _{{\mathbb {R}}^N}(|\nabla u|^2+V_\infty (x)u^2)\,dx\mid u\in C^\infty _0(\{|x|\ge R\}),\,\Vert u\Vert _{L^2}=1\right\} . \end{aligned}$$

It is immediate to see that $\alpha _\infty \le s^*_\infty$ (recall (1.7)). Therefore

$$\begin{aligned} \sigma _e({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )\subset [\alpha _\infty ,+\infty ). \end{aligned}$$

(3.7)

At most instances $\alpha _\infty <s^*_\infty$ (see [29]); if, however, $\lim _{R\rightarrow \infty }\mathrm {esssup}\,_{|x|\ge R}|V_\infty (x)-\alpha _\infty |=0$, then $\sigma _e({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )=[\alpha _\infty ,+\infty )$.

(2) Let p be as in (1.3) and let

$$\begin{aligned} q:=\frac{2p}{p-2}\;\; \text {if}\;\; p>2,\;\; q:=\infty \;\; \text {for}\;\; p=2. \end{aligned}$$

(3.8)

Observe that, in view of the Sobolev embeddings (see [1, Theorem 4.12]), our assumptions imply that for any $N\ge 1$, $H^1\hookrightarrow L^q$ (continuous embeddings) and, in view of the Rellich-Kondrachov theorem (see [1, Theorem 6.3]), $H^2(\Omega )$ is compactly embedded in $L^q(\Omega )$ provided $\Omega \subset {\mathbb {R}}^N$ is a smooth bounded domain.

(3) By the above, $H^1\hookrightarrow D({{\mathbf {V}}}_0)=L^q$. In view of the Hölder inequality ${{\mathbf {V}}}_0$ is well-defined and, as the operator $L^q\rightarrow L^2$, continuous. It is symmetric, hence, closable. In view of Lemma 3.2 below, ${{\mathbf {V}}}_0$ is relatively $({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )$-compact. Therefore, by [13, Corollary III.2.17 (ii)], ${{\mathbf {A}}}$ is sectorial and, in view of [40, Proposition 8.14 (ii), Theorem 8.5], ${{\mathbf {A}}}$ is self-adjoint; see also [34, Corollary XIII.4.2]. Hence $\sigma ({{\mathbf {A}}})\subset {\mathbb {R}}$.

(4) The relative compactness of ${{\mathbf {V}}}_0$ w.r.t. ${{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty$ (see Lemma 3.2) implies, in view of the Weyl theorem (see e.g. [38, Theorem 1.4.6] or [40, Theorem 8.15]) and (1.7), that

$$\begin{aligned} \sigma _e({{\mathbf {A}}})=\sigma _e({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )\subset [\alpha _\infty ,+\infty ). \end{aligned}$$

(3.9)

Therefore $\sigma ({{\mathbf {A}}})\cap (-\infty ,\alpha _\infty )$ is contained in the discrete part of the spectrum $\sigma _d({\mathbf {A}})$; hence it consists of at most countable number of isolated eigenvalues with finite multiplicity.

(5) Observe that in view of (1.4) $\mathbf F$ is well-defined and continuous as an operator $L^2\rightarrow L^2$ since

$$\begin{aligned} \Vert {{\mathbf {F}}}(u)\Vert _{L^2}\le \Vert m\Vert _{L^2},\;\; u\in L^2, \end{aligned}$$

(3.10)

and, by (1.5),

$$\begin{aligned}&\Vert {{\mathbf {F}}}(u)-{{\mathbf {F}}}(v)\Vert _{L^2}\le \Vert (l_0+l_\infty )|u-v|\Vert _{L^2}\nonumber \\&\quad \le \Vert l_0\Vert _{L^p}\Vert u-v\Vert _{L^q}+\Vert l_\infty \Vert _{L^\infty }\Vert u-v\Vert _{L^2} \nonumber \\&\quad \le L \Vert u-v\Vert _{H^1}, \end{aligned}$$

(3.11)

for $u,v\in H^1$, with an appropriately chosen Lipschitz constant L. Clearly, if $u\in H^1$, then $u\in L^2\cap L^q$ and $\max \{\Vert u\Vert _{L^2},\Vert u\Vert _{L^q}\}\le \mathrm {const.}\Vert u\Vert _{H^1}$ (³). Hence ${\mathbf {F}}$ is Lipschitz continuous as a map $H^1\rightarrow L^2$.

(6) By [20, Theorem 3.3.3] (comp. [7, Chapter 3]), the sectoriality of ${{\mathbf {A}}}$, conditions (3.10) and (3.11) imply that for each ${\bar{u}}\in H^1$ and $\lambda \in {\mathbb {R}}$ there is a unique global solution u of

$$\begin{aligned}&\dot{u}= - {{\mathbf {A}}}u+\lambda u + {{\mathbf {F}}}(u),\;\; t>0,\;\lambda \in {\mathbb {R}},\;u\in H^1, \end{aligned}$$

(3.12)

i.e. a continuous function $u=u(\cdot ;{\bar{u}},\lambda ):[0,+\infty )\rightarrow H^1$ such that $u\in C((0,+\infty ),H^2)\cap C^1((0,+\infty ),L^2)$, $u(0)={\bar{u}}$ and (3.12) holds for all $t>0$. $\square$

Lemma 3.2

The operator ${{\mathbf {V}}}_0$ is relatively $({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )$-compact, i.e. $D({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )\subset D({{\mathbf {V}}}_0)$ and ${{\mathbf {V}}}_0$ is compact as a map on $D({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )$ endowed with the graph-norm.

Proof

In view of Remark 3.1 (2), $D({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )=H^2({\mathbb {R}}^N)\subset L^q=D({{\mathbf {V}}}_0)$. Assume that a sequence $(u_n)_{n=1}^\infty$ is bounded in the $H^2$ sense, i.e. $\sup \Vert u_n\Vert _{H^2}\le R$ for some $R>0$. Clearly $\sup \Vert u_n\Vert _{L^q}\le \mathrm {const.}R$. Let $v_n:={{\mathbf {V}}}_0u_n$, $n\ge 1$; we will show that the set $\{v_n\}_{n=1}^\infty$ is precompact in $L^2$. Take an arbitrary $\varepsilon >0$. For any $n, k\ge 1$,

$$\begin{aligned}&\int _{\{|x|\ge k\}}v_n^2\,dx\le \left( \int _{\{|x|\ge k\}}|V_0|^p\,dx\right) ^{2/p}\left( \int _{\{|x|\ge k\}}|u_n|^q\,dx\right) ^{2/q}\nonumber \\&\quad \le \mathrm {const.}R^q\left( \int _{\{|x|\ge k\}}|V_0|^p\,dx\right) ^{2/p}<\varepsilon ^2 \end{aligned}$$

(3.13)

provided k is large enough. Take such k, let $B:=\{x\in {\mathbb {R}}^N\mid |x|<k\}$ and $u_n'=u_n|_B$, $n\ge 1$. Then $u_n'\in H^2(B)$, $(u_n')$ is bounded in $H^2(B)$ and, in view of the compactness of the embedding $H^2(B)\subset L^q(B)$, without loss of generality we may assume that $u_n'\rightarrow u_0'$ in $L^q(B)$ as $n\rightarrow \infty$. For $n\ge 0$ let

$$\begin{aligned} w_n={\left\{ \begin{array}{ll}V_0u_n\;\;&{}\text {on}\;\;B,\\ 0\;\;&{}\text {on}\;\;{\mathbb {R}}^N\setminus B. \end{array}\right. } \end{aligned}$$

Then $w_n\rightarrow w_0$ in $L^2$ and, by (3.13), $\Vert v_n-w_n\Vert _{L^2}<\varepsilon$. It follows that $\{v_n\}_{n=1}^\infty$ is precompact.

$\square$

Remark 3.3

(1) The above argument shows actually that ${{\mathbf {V}}}_0$ is relatively $({{\mathbf {A}}}_0+{{\mathbf {V}}}_\infty )$-compact if $p\ge 2$ for $N\le 3$ and $p>N/2$ for $N\ge 3$; comp. [40, Theorem 8.19]. The restrictions put on p in (1.3) are necessary to ensure that $H^1\subset L^q$.

(2) An argument similar to the one used in the above proof shows that a bounded subset $M\subset H^1$ is relatively compact in $L^2$ provided for any $\varepsilon >0$ there is $R>0$ such that

$$\begin{aligned} \forall \,u\in M\;\;\;\;\; \int _{\{|x|\ge R\}}|u(x)|^2 \,dx < \varepsilon . \end{aligned}$$

$\square$

In view of Remark 3.1 (6), for any $\lambda \in {\mathbb {R}}$, we are in a position to define $\Phi ^\lambda :[0,\infty )\times H^1\rightarrow H^1$ by putting

$$\begin{aligned} \Phi ^\lambda _t({\bar{u}}):=u(t;{\bar{u}},\lambda ),\;\;{\bar{u}}\in H^1,\;t\ge 0. \end{aligned}$$

(3.14)

It is immediate to see that $\Phi ^\lambda$ is a semiflow on $H^1$. By invoking [30, Prop. 2.3] (comp. [7, Theorem 3.2.1], [10, Prop. 4.3]) we get the following continuity result.

Proposition 3.4

Given sequences $({\bar{u}}_n)$ in $H^1$ and $\lambda _n\rightarrow \lambda$ in ${\mathbb {R}}$,

(i) if  ${\bar{u}}_n\rightarrow {\bar{u}}$ in $H^1$, then $\Phi ^{\lambda _n}_t({\bar{u}}_n)\rightarrow \Phi ^\lambda _t({\bar{u}})$ uniformly with respect to t in compact subsets of ${\mathbb {R}}$; as a consequence the family $\{\Phi ^\lambda \}_{\lambda \in {\mathbb {R}}}$ is continuous;

(ii) if  $T>0$, $R>0$, $\Vert \Phi ^{\lambda _n}_t({\bar{u}}_n)\Vert _{H^1}\le R$ for all $t\in [0,T]$ and ${\bar{u}}_n\rightarrow {\bar{u}}$ in $L^2$, then $\Phi ^{\lambda _n}_t({\bar{u}}_n)\rightarrow \Phi ^\lambda _t({\bar{u}})$ uniformly with respect to t in compact subsets of (0, T]. $\square$

Recall the standing assumptions and, as in Theorem 1.4 (i), suppose that

$$\begin{aligned}&\lambda _0\;\;\text {is an isolated eigenvalue of}\;\;{\mathbf {A}}\;\;\text {of finite multiplicity and let }\nonumber \\&\quad \;\;0<\delta <\mathrm {dist}(\lambda _0,\sigma ({\mathbf {A}}){\setminus }\{\lambda _0\}). \end{aligned}$$

(3.15)

Let $X_0:=\mathrm {Ker}\,({{\mathbf {A}}} -\lambda _0 {{\mathbf {I}}})$, $X_\pm$ be the closed subspaces of $L^2$ corresponding to $\sigma ({\mathbf {A}})\cap (-\infty ,\lambda _0)$, $\sigma ({\mathbf {A}})\cap (\lambda _0,+\infty )$, respectively; let $X:=X_-\oplus X_+$ ($\oplus$ stands for the orthogonal sum). It is clear that $X_0$, $X_\pm$ are ${\mathbf {A}}$-invariant, $L^2=X_0\oplus X$, $\dim X_0,\dim X_-<\infty$ and $X_0, X_-\subset H^2$ since these spaces are spanned by a finite number of eigenfunctions. Let ${\mathbf {Q}}_\pm :L^2\rightarrow L^2$ be the orthogonal projections onto $X_\pm$, ${\mathbf {Q}}:={\mathbf {Q}}_-+{\mathbf {Q}}_+$ and ${\mathbf {P}}:={\mathbf {I}}-{\mathbf {Q}}$. Observe that ${\mathbf {P}},{\mathbf {Q}}_-\in {{\mathcal {L}}}(L^2,H^2)$, ${\mathbf {Q}}_+(H^2)\subset H^2\cap X_+$ and ${\mathbf {Q}}_+|_{H^1}\in {{\mathcal {L}}}(H^1,H^1)$, i.e.

$$\begin{aligned} \Vert {{\mathbf {Q}}}|_{H^1}\Vert _{{{\mathcal {L}}}(H^1,H^1)}<\infty . \end{aligned}$$

(3.16)

If $|\lambda -\lambda _0|\le \delta$, then $\lambda \not \in \sigma ({\mathbf {A}}|_X)$. Hence $({\mathbf {A}}-\lambda {\mathbf {I}})_{|X}$ is inveritble and the map

$$\begin{aligned}{}[\lambda _0-\delta ,\lambda _0+\delta ]\times X\ni (\lambda ,w)\mapsto [({\mathbf {A}}-\lambda {\mathbf {I}})_{|X}]^{-1}w\in X\cap H^2 \end{aligned}$$

(3.17)

is continuous and $\Vert [({\mathbf {A}}-\lambda {\mathbf {I}})_{|X}]^{-1}w\Vert _{H^2}\le \mathrm {const.}\Vert w\Vert _{L^2}.$

Lemma 3.5

The map

$$\begin{aligned}{}[\lambda _0-\delta ,\lambda _0+\delta ]\times L^2\ni (\lambda ,u)\mapsto {\mathbf {G}}(\lambda ,u):={\mathbf {F}}({\mathbf {P}}u+[({\mathbf {A}}-\lambda {\mathbf {I}})_{|X}]^{-1}{\mathbf {Q}}u)\in L^2 \end{aligned}$$

is completely continuous.

Proof

The continuity of ${\mathbf {G}}$ is evident. Let sequence $(u_n)$ in $L^2$ and $(\lambda _n)$ in $[\lambda _0-\delta ,\lambda _0+\delta ]$ be bounded. Let $v_n={\mathbf {P}}u_n$, $w_n:={\mathbf {Q}}u_n$, ${\tilde{w}}_n:=[({\mathbf {A}}-\lambda _n{\mathbf {I}})|_X]^{-1}w_n$ and $z_n:={\mathbf {G}}(\lambda _n, u_n)$, $n\ge 1$. Without loss of generality we may assume that $v_n\rightarrow v_0\in X_0$. Take an arbitrary $\varepsilon >0$. In view of (1.4) there is $R>0$ such that for all $n\ge 1$

$$\begin{aligned} \int _{\{|x|\ge R\}}z_n^2\,dx\le \int _{\{|x|\ge R\}}m^2\,dx<\varepsilon ^2. \end{aligned}$$

(3.18)

Let $B=\{x\in {\mathbb {R}}^N\mid |x|<R\}$, $v_n':={v_n}_{|B}$, ${\tilde{w}}_n':=\tilde{w_n}_{|B}$, $n\ge 1$. Then $v_n'\rightarrow v_0':={v_0}_{|B}$; the sequence $({\tilde{w}}_n')$ is bounded in $H^2(B)$ and, thus, we may assume that ${\tilde{w}}_n'\rightarrow {\tilde{w}}_0'\in L^2(B)$ as $n\rightarrow \infty$. For $n\ge 0$ let

$$\begin{aligned} z_n'={\left\{ \begin{array}{ll}f(x,v_n'(x)+{\tilde{w}}_n'(x))\;\;&{}\text {on}\;\;B,\\ 0\;\;&{}\text {on}\;\;{\mathbb {R}}^N{\setminus } B.\end{array}\right. } \end{aligned}$$

Then $z_n'\rightarrow z_0'$ in $L^2$ and, in view of (3.18), $\Vert z_n-z_n'\Vert _{L^2}<\varepsilon$. This implies that $\{z_n\}$ is precompact. $\square$

Now, in the context of Theorem 1.4 (ii) we suppose that

$$\begin{aligned} \lambda _0\in \sigma ({\mathbf {A}}) \;\; \text { and }\;\; \lambda _0<\alpha _\infty . \end{aligned}$$

(3.19)

In view of Remark 3.1 (4), $\lambda _0$ is an isolated eigenvalue of finite multiplicity. Take $\delta >0$ such that

$$\begin{aligned}&0<\delta <\min \{\alpha _\infty -\lambda _0,\mathrm {dist}(\lambda _0, \sigma ({{\mathbf {A}}}){\setminus }\{\lambda _0\}\}. \end{aligned}$$

(3.20)

Lemma 3.6

(comp. [31, Proposition 2.2], [10]) Let $R>0$ and let $\delta >0$ be given as in (3.20). There is $\alpha >0$ and a sequence $(\alpha _n)$ with $\alpha _n\searrow 0$ such that if $u:[t_0,t_1] \rightarrow H^1$ is a solution of the semiflow $\Phi ^\lambda$ corresponding to (3.12) for some $\lambda \in [\lambda _0-\delta , \lambda _0 +\delta ]$ such that $\Vert {\mathbf {Q}}u(t)\Vert _{H^1}\le R$ for all $t\in [t_0,t_1]$, then there is $n_0\ge 1$ such that

$$\begin{aligned} \forall \,n\ge n_0\quad \quad \int _{{\mathbb {R}}^N {\setminus } B(0,n)} |{\mathbf {Q}}u(t_1)|^2\, dx \le e^{-2\alpha (t_1-t_0)} \Vert \mathbf {Q} u(t_0) \Vert _{L^2}^{2} + \alpha _n. \end{aligned}$$

(3.21)

Proof

Since u is a solution of $\Phi ^\lambda$, we have $u(t+t_0)=\Phi ^\lambda _t(u(t_0))$ for $t\in [0,t_1-t_0]$, i.e., in the case of (3.12),

$$\begin{aligned} \dot{u} (t)= -{{\mathbf {A}}}u(t) + \lambda u(t) + {{\mathbf {F}}}(u(t))\;\; \text {for}\;\; t\in (t_0,t_1]. \end{aligned}$$

For $w:={\mathbf {Q}}u$ and $t\in (t_0, t_1]$ we have

$$\begin{aligned} \dot{w}(t)=-{{\mathbf {A}}}w(t)+\lambda w(t)+{\mathbf {Q}}{{\mathbf {F}}}(u(t)). \end{aligned}$$

Let $\phi :[0,+\infty )\rightarrow [0,1]$ be a smooth function such that $\phi (s)=0$ if $s\in [0,1/2]$ and $\phi (s)=1$ if $s\ge 1$. Putting

$$\begin{aligned} \phi _n(x):=\phi (|x|^2/n^2),\;\; x\in {\mathbb {R}}^N, \end{aligned}$$

we get, for $t\in (t_0, t_1]$ and $n\ge 1$,

$$\begin{aligned} \frac{1}{2}\frac{d}{d t}\langle w(t), \phi _n w(t)\rangle _{L^2}&= \langle \phi _n w(t),\dot{w}(t)\rangle _{L^2}\\&=\langle \phi _nw(t),-({\mathbf {A}}_0 +{\mathbf {V}}_0+{\mathbf {V}}_\infty -\lambda {{\mathbf {I}}})w(t)+{\mathbf {Q}}{\mathbf {F}}(u(t))\rangle _{L^2}\\&=I_1(t) + I_2(t) + I_3(t), \end{aligned}$$

where

$$\begin{aligned} I_1(t)&=\;\langle \phi _n w(t), - {{\mathbf {A}}}_0w(t)\rangle _{L^2}=-\langle \nabla (\phi _nw(t)),\nabla w(t)\rangle _{L^2}= -\int _{{\mathbb {R}}^N}\phi _n(x)|\nabla w(t)|^2\,dx+\\&\quad -\frac{2}{n^2}\int _{\{\frac{\sqrt{2}}{2}n\le |x|\le n\}} \phi '(|x|^2/n^2)\langle w(t)x,\nabla w(t)\rangle _{{\mathbb {R}}^N}\,dx\\&\quad \le \frac{2L_\phi }{n}\Vert w(t)\Vert _{L^2}\Vert \Vert w(t)\Vert _{H^1}\le \frac{2L_\phi R^2}{n}, \end{aligned}$$

with

$$\begin{aligned} L_\phi := \sup _{s\in [0,+\infty )}|\phi '(s)|; \end{aligned}$$

(3.22)

note that $L_\phi < \infty$.

In order to estimate the second term $I_2(t)$, take $0<\eta \le \frac{1}{2}(\alpha _\infty -\lambda _0-\delta )$. By definition of $\alpha _\infty$ (see (1.7)), there is a positive integer $n_0$ such that $V_\infty (x)>\alpha _\infty -\eta$ for a.a. $|x|\ge \sqrt{2}n_0/2$. For $n\ge n_0$ we have

$$\begin{aligned} I_2(t)&=\langle \phi _nw(t),-({{\mathbf {V}}}_0+{{\mathbf {V}}}_\infty -\lambda {{\mathbf {I}}})w(t)\rangle _{L^2}\\&=-\langle \phi _nw(t),({{\mathbf {V}}}_\infty -\lambda {{\mathbf {I}}})w(t)\rangle _{L^2}-\langle \phi _nw(t),V_0w(t)\rangle _{L^2}\\&=-\int _{\{\frac{\sqrt{2}}{2}n\le |x|\le n\}}\phi _n(x)(V_\infty (x)-\lambda )|w(t)|^2\,dx-\int _{{\mathbb {R}}^N} \phi _n(x)V_0(x)|w(t)|^2\,dx\\&\le -\alpha \langle \phi _nw(t),w(t)\rangle _{L^2}+\mathrm {const.}\Vert w(t)\Vert _{H^1}^2 \left( \int _{\{|x|\ge \frac{\sqrt{2}}{2}n\}}|V_0(x)|^p\,dx\right) ^{1/p}, \end{aligned}$$

where $\alpha :=\alpha _\infty -\lambda _0-\delta -\eta >0$; the last estimate follows in view of the Hölder inequality since $\Vert w(t)\Vert _{L^{2p/p-1}}\le \mathrm {const.}\Vert u(t)\Vert _{H^1}$. Finally for all $n\ge 1$

$$\begin{aligned} \begin{aligned} I_3(t)&=\langle \phi _nw(t),{{\mathbf {Q}}} {{\mathbf {F}}}(u(t))\rangle _{L^2} \Vert w(t)\Vert _{L^2} \left( \Vert \phi _n {{\mathbf {F}}}(u(t))\Vert _{L^2} + \Vert \phi _n {{\mathbf {P}}} {{\mathbf {F}}}(u(t))\Vert _{L^2}\right) \\&\le R\left( \left( \int _{\{|x|>\frac{\sqrt{2}}{2}n\}} |m(x)|^2 dx \right) ^{1/2} +\ \kappa _n \right) , \end{aligned} \end{aligned}$$

(3.23)

where $\kappa _n:= \sup \left\{ \left( \int _{\{|x|>\frac{\sqrt{2}}{2}n\}} |z(x)|^2 dx \right) ^{1/2} \left| \right. z\in {{\mathbf {P}}}\left( B\left( 0 , \Vert m\Vert _{L^2} \right) \right) \right\}$ for $n\ge 1$. Since ${{\mathbf {P}}}\left( B\left( 0,\Vert m\Vert _{L^2}\right) \right)$ is relatively compact (as a bounded subset of the finite dimensional space) with respect to the $L^2$ topology, in view of the Kolmogorov-Riesz compactness criterion (see e.g. [18, Theorem 5]), we see that $\kappa _n \rightarrow 0^+$ as $n\rightarrow \infty$.

Combining these estimates we get that for any $n\ge n_0$

$$\begin{aligned} \frac{d}{d t}\langle w(t), \phi _n w(t)\rangle _{L^2} \le -2\alpha \langle w(t), \phi _n w(t)\rangle _{L^2} + 2\tilde{\alpha }_n, \end{aligned}$$

where

$$\begin{aligned} \tilde{\alpha }_n:=&\frac{2 R^2 L_\phi }{n} + \mathrm {const.}R^2 \bigg (\int _{\big \{|x|\ge \frac{\sqrt{2}}{2}n\big \}} |V_0(x)|^p d x\bigg )^{1/p} \\&+ R \bigg (\!\!\int _{\big \{|x|\ge \frac{\sqrt{2}}{2}n\big \}}|m(x)|^2 d x\bigg )^{1/2}+R\kappa _n. \end{aligned}$$

Multiplying by $e^{2\alpha (t-t_0)}$ and integrating over $[t_0,t_1]$ one obtains

$$\begin{aligned} e^{2\alpha (t_1-t_0 )}\langle w(t_1), \phi _n w(t_1)\rangle _{L^2}- \langle w(t_0), \phi _n w(t_0)\rangle _{L^2} \le \frac{e^{2\alpha (t_1-t_0)}-1}{\alpha }\tilde{\alpha }_n, \end{aligned}$$

This clearly implies

$$\begin{aligned} \int _{{\mathbb {R}}^N{\setminus } B(0,n)} |w(t_1)|^2 d x \le \langle w(t_1), \phi _n w(t_1)\rangle _{L^2} \le e^{-2\alpha (t_1-t_0)}\Vert w(t_0)\Vert ^2_{L^2} + \alpha ^{-1} \tilde{\alpha }_n, \end{aligned}$$

which finally yields the assertion with $\alpha _n:=\frac{\tilde{\alpha }_n}{\alpha }$.$\square$

Proposition 3.7

Let $R>0$, $\delta$ be as in Lemma 3.6 and $M_R$ be the set of ${\bar{u}} \in H^1({\mathbb {R}}^N)$ such that there exists a solution $u:(-\infty , 0]\rightarrow H^1({\mathbb {R}}^N)$ of $\Phi ^\lambda$ for some $\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$ with $u(0)={\bar{u}}$ and $\Vert {\mathbf {Q}}u(t)\Vert _{H^1} \le R$ for all $t\le 0$. Then ${\mathbf {Q}}M_R$ is relatively compact in $L^2({\mathbb {R}}^N)$.

Proof

We will use Remark 3.3 (2). Take $\varepsilon >0$ and $t_0<0=t_1$. In view of Lemma 3.6 there is $\alpha >0$ and a sequence $\alpha _n \searrow 0^+$ (recall that $\alpha _n$ is independent of the choice of $t_0$) such that, for all ${\bar{u}}\in M_{R}$ and $n\ge n_0$,

$$\begin{aligned} \int _{{\mathbb {R}}^N {\setminus } B(0,n)} |{\mathbf {Q}} {\bar{u}}|^2 d x \le e^{2\alpha t_0} \Vert {\mathbf {Q}} u(t_0)\Vert _{L^2}^2 + \alpha _n \le e^{2\alpha t_0} R^2 + \alpha _n<\varepsilon , \end{aligned}$$

where $u:(-\infty ,0]\rightarrow H^1({\mathbb {R}}^N)$ is the solution of $\Phi ^\lambda$ such that $u(0)={\bar{u}}$, provided that $e^{2\alpha t_0}R^2<\varepsilon /2$ and $\alpha _n<\varepsilon /2$ for $n\ge n_0$.$\square$

Remark 3.8

Conclusions of Lemma 3.6 and Proposition 3.7 stay true if the projection ${{\mathbf {Q}}}$ is replaced by the identity on $L^2({\mathbb {R}}^N)$.$\square$

Corollary 3.9

(Comp. [30]) Any bounded set $M\subset H^1$ is admissible with respect to $\{\Phi ^\lambda \}_{\lambda \in [\lambda _0-\delta ,\lambda +\delta ]}$.

Proof

Take sequences $t_m\rightarrow \infty$, $(u_m)\in H^1$ and $\lambda _m\rightarrow \lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$ such that $\Phi ^{\lambda _m}_{[0,t_m]}(u_m)\subset M$ and $R>0$ such that $M \subset D_{H^1}(0,R):=\{u\in H^1\mid \Vert u\Vert _{H^1}\le R\}$. With no loss of generality we may assume that $t_m>t_0$ for all m. Then, for all m,

$$\begin{aligned} \Phi _{t_m}^{\lambda _m} (u_m) = \Phi _{t_0}^{\lambda _m} (z_m) \end{aligned}$$

where $z_m := \Phi _{t_m-t_0}^{\lambda _m}(u_m)$. It follows from Lemma 3.6 that, for all $m, n\in {\mathbb {N}}$,

$$\begin{aligned} \int _{{\mathbb {R}}^N{\setminus } B(0,n)} |z_m(x)|^2\, dx \le e^{-2\alpha (t_m-t_0)} \Vert u_m\Vert _{L^2}+\alpha _n \le R^2 e^{-2\alpha (t_m-t_0)}+\alpha _n \end{aligned}$$

where $\alpha _n\rightarrow 0^+$ as $n\rightarrow \infty$. This, in view of Remark 3.3 (2), means that the sequence $(z_m)$ is relatively compact in $L^2$. Now, by the weak relative compactness of bounded sets in $H^1$, there exists $z\in H^1$ such that (up to a subsequence), $z_m \rightharpoonup z$ (weakly) in $H^1$ and $z_m \rightarrow z$ in $L^2$. Thus, by Proposition 3.4, $\Phi ^{\lambda _m}_{t_m}(u_m)=\Phi ^{\lambda _m}_{t_0}(z_n)\rightarrow \Phi _{t_0}^{\lambda }(z)$. $\square$

Remark 3.10

(1) Observe that if $u:{\mathbb {R}}\rightarrow H^1$ is a full bounded solution of $\Phi ^\lambda$ for some $\lambda \in [\lambda _0-\delta , \lambda _0 + \delta ]$, then the set $u({\mathbb {R}})$ is relatively compact (in $H^1$). Indeed: for any $(t_n) \in {\mathbb {R}}$ one has $u(t_n)=\Phi _{n}^{\lambda } (z_n)$ with $z_n = u(t_n-n)$, $n\in {\mathbb {N}}$, that are contained in a bounded set; hence, by Corollary 3.9, $(u(t_n))$ contains a convergent subsequence.

(2) Let the functional $J_\lambda :H^1\rightarrow {\mathbb {R}}$, $\lambda \in [\lambda _0-\delta , \lambda _0+\delta ]$, be given by

$$\begin{aligned} J_\lambda (u):= \frac{1}{2} \int _{{\mathbb {R}}^N} (|\nabla u|^2 + V(x)|u|^2 - \lambda |u|^2)\, dx - \int _{{\mathbb {R}}^N} F (x,u)\, dx \end{aligned}$$

where $F(x,s):=\int _{0}^{s} f(x,\tau )\,d\tau$. Then, for any solution $u:(t_0,t_1)\rightarrow H^1$ of $\Phi ^\lambda$, one has

$$\begin{aligned} \frac{d}{dt}\left[ J_\lambda (u(t)) \right] = - \Vert \dot{u}(t)\Vert _{L^2}^2 \ \text{ for } \text{ each } \ t\in (t_0,t_1). \end{aligned}$$

This means that $J_\lambda$ is a Liapunov-function for $\Phi ^\lambda$, i.e. it decreases along solutions of $\Phi ^\lambda$. It is also clear that if a solution u is nonconstant, then so is $t\rightarrow J(u(t))$. Therefore, if $u:{\mathbb {R}}\rightarrow H^1$ is a full bounded solution of $\Phi ^\lambda$, then the limit sets $\alpha (u)$ and $\omega (u)$ consists only of equilibria of $\Phi ^\lambda$ (see [35, Prop. 5.3]).$\square$

The following Conley index formula, obtained by linearization and Theorem 2.1, will be used in the sequel.

Proposition 3.11

(comp. [30, Theorem 3.3]) Under assumptions (1.3), (1.4) and (1.5), suppose that $\lambda \not \in \sigma ({{\mathbf {A}}})$ and $\lambda <\alpha _\infty$. Denote by $K(\Phi ^\lambda )$ the set of all ${\bar{u}}\in H^1$ such that there exists a bounded solution $u:{\mathbb {R}}\rightarrow H^1$ of $\Phi ^\lambda$ such that $u(0)={\bar{u}}$. Then $K(\Phi ^\lambda )$ is bounded, isolated invariant with respect to $\Phi ^\lambda$, $(\Phi ^\lambda ,K(\Phi ^\lambda ))\in {{\mathcal {I}}}(H^1)$ and the Conley index

$$\begin{aligned} h(\Phi ^\lambda , K(\Phi ^\lambda )) = \Sigma ^{k(\lambda )} \end{aligned}$$

where $k(\lambda )$ is the total multiplicity of the negative eigenvalues of ${{\mathbf {A}}}-\lambda {{\mathbf {I}}}$, i.e. eigenvalues of $-\Delta +V$ less than $\lambda$.$\square$

4 Necessary conditions

Below we provide necessary conditions for bifurcation from infinity and study additional properties of bifurcation sequences.

Theorem 4.1

If a bifurcation from infinity for (1.1) occurs at $\lambda _0\not \in \sigma _e({{\mathbf {A}}})$, i.e., there is a sequence $(u_n,\lambda _n)$ solving (1.1) with $\lambda =\lambda _n$, $\Vert u_n\Vert _{H^1}\rightarrow \infty$, $\lambda _n\rightarrow \lambda _0$, then $\lambda _0$ lies in $\sigma _p({{\mathbf {A}}})$ the point spectrum of ${\mathbf {A}}$ and $\Vert {{\mathbf {P}}}u_n\Vert _{L^2},\Vert \nabla {{\mathbf {P}}}u_n\Vert _{L^2}\rightarrow \infty$ as $n\rightarrow \infty$. This implies that $\Vert u_n\Vert _{L^2}, \Vert \nabla u_n\Vert _{L^2}\rightarrow \infty$, too. Moreover the sequences $(\Vert {\mathbf {Q}}u_n\Vert _{L^2})$ and $(\Vert \nabla {\mathbf {Q}}u_n\Vert _{L^2})$ are bounded.

If, additionally $\lambda _0<\alpha _\infty$, then the sequences $(\Vert u_n\Vert _{L^2})$ and $(\Vert \nabla u_n\Vert _{L^2})$ have the same growth rates, i.e., there are constants $C_1, C_2>0$ such that, for all large n,

$$\begin{aligned} C_1 \Vert u_n\Vert _{L^2} \le \Vert \nabla u_n\Vert _{L^2} \le C_2 \Vert u_n\Vert _{L^2}; \end{aligned}$$

(4.1)

a similar estimate holds for $\Vert {{\mathbf {P}}}u_n\Vert _{L^2}$ and $\Vert \nabla {{\mathbf {P}}}u_n\Vert _{L^2}$ with large n.

Proof. Let $\rho _n:=\Vert u_n\Vert _{H^1}$; we may assume that $\rho _n>0$ for all n. Let $z_n := \rho _n^{-1} u_n$; then $\Vert z_n\Vert _{H^1}=1$ and $\Vert z_n\Vert _{L^2}\le \mathrm {const.}$ Suppose to the contrary that $\lambda _0\not \in \sigma _p({{\mathbf {A}}})$. Since $\lambda _0\not \in \sigma _e({{\mathbf {A}}})$, this implies that $\lambda _0\in \rho ({{\mathbf {A}}})$, the resolvent set of ${\mathbf {A}}$. We have

$$\begin{aligned} ({\mathbf {A}}-\lambda _0{{\mathbf {I}}}) z_n= (\lambda _n-\lambda _0) z_n + \rho _{n}^{-1} {\mathbf {F}}(\rho _n z_n). \end{aligned}$$

Clearly $v_n:=(\lambda _n-\lambda _0)z_n+\rho _n^{-1}{\mathbf {F}}(\rho _n z_n)\rightarrow 0$ as $n\rightarrow \infty$ (in $L^2$). Hence $z_n=({{\mathbf {A}}}-\lambda _0{{\mathbf {I}}})^{-1}v_n\rightarrow 0$ in $H^1$: a contradiction.

Since $\lambda _0$ is isolated in $\sigma ({{\mathbf {A}}})$, there is $c>0$ such that for large n we have $\langle ({{\mathbf {A}}}-\lambda _n{{\mathbf {I}}})v,v\rangle _{L^2}\ge c\Vert v\Vert _{L^2}^{2}$ for $v\in X_+$ and $\langle ({{\mathbf {A}}}-\lambda _n{{\mathbf {I}}})w,w\rangle _{L^2}\le -c\Vert w\Vert _{L^2}^{2}$ for $w\in X_-$. This implies that for large n

$$\begin{aligned} c\Vert {{\mathbf {Q}}}_\pm u_n\Vert ^2_{L^2}&\le \pm \langle ({{\mathbf {A}}}-\lambda _n{{\mathbf {I}}}){{\mathbf {Q}}}_\pm u_n,{{\mathbf {Q}}}_\pm u_n\rangle _{L^2}=\pm \langle ({{\mathbf {A}}}-\lambda _n{{\mathbf {I}}})u_n,{{\mathbf {Q}}}_\pm u_n\rangle _{L^2}=\\&=\pm \langle {{\mathbf {F}}}(u_n),{{\mathbf {Q}}}_\pm u_n\rangle _{L^2}\le \Vert m\Vert _{L^2}\Vert {{\mathbf {Q}}}_\pm u_n\Vert _{L^2}.\end{aligned}$$

Therefore for large n

$$\begin{aligned} \Vert {\mathbf {Q}}u_n\Vert _{L^2}\le 2c^{-1}\Vert m\Vert _{L^2}. \end{aligned}$$

(4.2)

On the other hand

$$\begin{aligned} \Vert \nabla {{\mathbf {Q}}}u_n\Vert _{L^2}^2+\langle ({{\mathbf {V}}} -\lambda _n{{\mathbf {I}}}){{\mathbf {Q}}}u_n,{{\mathbf {Q}}}u_n\rangle _{L^2}=\langle ({{\mathbf {A}}}- \lambda _n{{\mathbf {I}}})u_n,{{\mathbf {Q}}}u_n\rangle _{L^2} =\langle {{\mathbf {F}}}(u_n),{{\mathbf {Q}}}u_n\rangle _{L^2}. \end{aligned}$$

Hence

$$\begin{aligned} \Vert \nabla {{\mathbf {Q}}}u_n\Vert ^2_{L^2}\le \Vert V_\infty -\lambda _n\Vert _{L^\infty }\Vert {{\mathbf {Q}}}u_n\Vert _{L^2}^2+ \Vert V_0\Vert _{L^p}\Vert {{\mathbf {Q}}}u_n\Vert ^2_{L^s}+\Vert m\Vert _{L^2}\Vert {{\mathbf {Q}}}u_n\Vert _{L^2}, \end{aligned}$$

where $s:=2p/(p-1)$. Clearly, $s>2$ and, if $N\ge 3$, one has also $s<2_{N}^*=2N/(N-2)$. In view of the Gagliardo-Nirenberg inequality (see Remark 4.2)

$$\begin{aligned} \Vert \nabla {{\mathbf {Q}}}u_n\Vert ^2_{L^2}&\le \Vert V_\infty - \lambda _n\Vert _{L^\infty }\Vert {{\mathbf {Q}}}u_n\Vert _{L^2}^2 +C^2\Vert V_0\Vert _{L^p}\Vert \nabla {{\mathbf {Q}}}u_n\Vert _{L^2}^ {2\theta }\Vert {{\mathbf {Q}}}u_n\Vert _{L^2}^{2(1-\theta )}\nonumber \\&\quad + \Vert m\Vert _{L^2}\Vert {{\mathbf {Q}}}u_n\Vert _{L^2} \end{aligned}$$

(4.3)

for some $C>0$ and $\theta \in (0,1)$. This, together with (4.2), implies that the sequence $(\Vert \nabla {{\mathbf {Q}}}u_n\Vert _{L^2})$ is bounded.

The same argument (replacing ${{\mathbf {Q}}}$ in (4.3) by the identity ${\mathbf {I}}$) shows that $(\Vert \nabla u_n\Vert _{L^2})$ would be bounded if $(\Vert u_n\Vert _{L^2})$ were bounded. Since $\Vert u_n\Vert _{H^1}\rightarrow \infty$, we deduce therefore that $\Vert u_n\Vert _{L^2}\rightarrow \infty$. Now $\Vert {{\mathbf {P}}}u_n\Vert _{L^2}^2=\Vert u_n\Vert _{L^2}^2-\Vert {{\mathbf {Q}}}u_n\Vert ^2_{L^2}$, so $\Vert {{\mathbf {P}}}u_n\Vert _{L^2}\rightarrow \infty$ in view of (4.2). This implies that also $\Vert \nabla {{\mathbf {P}}}u_n\Vert _{L^2}\rightarrow \infty$ because $\dim X_0<\infty$. Since

$$\begin{aligned} \Vert \nabla u_n\Vert _{L^2}\ge |\Vert \nabla {{\mathbf {P}}}u_n\Vert _{L^2}-\Vert \nabla {{\mathbf {Q}}}u_n\Vert _{L^2}|, \end{aligned}$$

we finally infer that $\Vert \nabla u_n\Vert _{L^2}\rightarrow \infty$.

Now assume that $\lambda _0<\alpha _\infty$. Take $\eta >0$ such that $\lambda _0+3\eta <\alpha _\infty$ and $R>0$ such that $V_\infty (x)\ge \alpha _\infty -\eta$ for a. a. $x\in {\mathbb {R}}^N$ with $|x|>R$. Then for large $n\ge 1$, $V_\infty (x)-\lambda _n>\eta$ a.e. on $\{x\in {\mathbb {R}}^N\mid |x|>R\}$.

For large n we have

$$\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla u_n|^2\,dx+\eta \int _{{\mathbb {R}}^N}u^2_{n}\,dx&\le \int _{{\mathbb {R}}^N}|\nabla u_{n}|^2\,dx\\&\quad +\int _{\{|x|>R\}}(V_\infty (x)-\lambda _n)u_n^2\,dx+\eta \int _{\{|x|\le R\}}u_n^2\,dx\\&=\int _{{\mathbb {R}}^N}(|\nabla u|^2\\&\quad +(V_\infty (x)-\lambda _n)u_n^2)\,dx+\int _{\{|x|\le R\}}(\eta -V_\infty (x)+\lambda _n)u_n^2\,dx. \end{aligned}$$

Hence

$$\begin{aligned}&\Vert \nabla u_n\Vert _{L^2}^2+\eta \Vert u_n\Vert _{L^2}^2\le -\int _{{\mathbb {R}}^N}V_0(x)u_n^2\, dx+\int _{\{|x|\le R\}}(\eta -V_\infty (x)+\lambda _n)u_n^2\,dx+\int _{{\mathbb {R}}^N}f(x,u_n)u_n\,dx.\nonumber \\ \end{aligned}$$

(4.4)

Take $\xi >0$ such that $\xi \ge |\eta -V_\infty (x)-\lambda _n|$ for all large n and let $V_1(x)=\xi$ if $|x|\le R$ and $V_1(x)=0$ otherwise. Then $V_1\in L^p$ and, by (4.4) we have

$$\begin{aligned} \Vert \nabla u_n\Vert _{L^2}^2+\eta \Vert u_n\Vert _{L^2}^2\le \Vert V_0+V_1\Vert _{L^p}\Vert u_n\Vert ^2_{L^s}+\Vert m\Vert _{L^2}\Vert u_n\Vert _{L^2} \end{aligned}$$

and, again in virtue of the Gagliardo-Nirenberg inequlaity, we get that

$$\begin{aligned} \Vert \nabla u_n\Vert _{L^2}^{2}+\eta \Vert u_n\Vert _{L^2}^{2} \le C^2 \Vert V_0+V_1\Vert _{L^p} \Vert \nabla u_n\Vert _{L^2}^{2\theta }\Vert u_n\Vert _{L^2}^{2(1-\theta )} +\Vert m\Vert _{L^2} \Vert u_n\Vert _{L^2} \end{aligned}$$

(4.5)

with constants $C>0$ and $\theta \in (0,1)$. For large n,

$$\begin{aligned} \left( \frac{\Vert \nabla u_n\Vert _{L^2}}{\Vert u_n\Vert _{L^2} }\right) ^2 + \eta \le C^2\Vert V_0+V_1\Vert _{L^p} \left( \frac{\Vert \nabla u_n\Vert _{L^2}}{\Vert u_n\Vert _{L^2} }\right) ^{2\theta } + 1 \end{aligned}$$

and

$$\begin{aligned} 1+ \eta \left( \frac{\Vert u_n\Vert _{L^2} }{\Vert \nabla u_n\Vert _{L^2}}\right) ^2 \le C^2\Vert V_0+V_1\Vert _{L^p} \left( \frac{\Vert u_n\Vert _{L^2}}{\Vert \nabla u_n\Vert _{L^2}}\right) ^{2(1-\theta )} + \frac{\Vert u_n\Vert _{L^2} }{\Vert \nabla u_n\Vert _{L^2}}, \end{aligned}$$

which gives the existence of $C_1, C_2>0$ satisfying (4.1). A similar argument shows that growth rates of $(\Vert {{\mathbf {P}}}u_n\Vert _{L^2})$ and $\Vert \nabla {{\mathbf {P}}}u_n\Vert _{L^2})$ are the same. $\square$

Remark 4.2

The Gagliardo-Nirenberg inequality (see [26] and [2]) states that given $1<r<s$ (with $s<2_{N}^*=\frac{2N}{N-2}$ if $N\ge 3$) there are $C>0$ and $\theta \in (0,1)$ such that for any $u\in H^1$

$$\begin{aligned} \Vert u\Vert _{L^s}\le C\Vert \nabla u\Vert _{L^2}^{\theta } \Vert u\Vert _{L^r}^{1-\theta } \ \ \text{ for } \text{ all } \ \ u\in H^1. \end{aligned}$$

$\square$

Theorem 4.1 corresponds to [43, Theorem 5.2(i)] where a different situation is considered: $V\in L^\infty ({\mathbb {R}}^N)$, f is of the form given in Remark 1.5 (2) and $\mathrm {dist}(\lambda _0, \sigma _e ({{\mathbf {A}}})) > 2\cdot \sup _{u\ne 0} |{\tilde{f}}(u)/u| +l$ where l is the Lipschitz constant of ${\tilde{f}}$. Theorem 4.1 also shows that bifurcating sequences $(u_n)$ are localized around the eigenspace $\mathrm {Ker}\,({{\mathbf {A}}}-\lambda _0{{\mathbf {I}}})$ having mass $\Vert u_n\Vert _{L^2}$ and energy of the same growth rate as in [43, Theorem 5.2 (iii)] where the case of simple eigenvalue (with the mentioned assumptions on V and f) was considered.

5 Sufficient conditions - proof of Theorem 1.4

Recall the notation introduced in front of Lemma 3.5. We start with the proof of Theorem 1.4 (i): assume (3.15), let $\dim X_0$ be odd and suppose that there is no asymptotic bifurcation at $\lambda _0$. Taking smaller $\delta >0$ if necessary there is $r>0$ such that for all $\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$ if $w\in H^2$ and $({\mathbf {A}}-\lambda {\mathbf {I}})w={\mathbf {F}}(w)$, then $\Vert w\Vert _{H^1}\le r$.

Observe that $w\in H^2$, solves (3.1) with some $\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$, i.e. $({\mathbf {A}}-\lambda {\mathbf {I}})w={\mathbf {F}}(w)$, if and only if $u:={\mathbf {P}}w+({\mathbf {A}}-\lambda {\mathbf {I}}){\mathbf {Q}}w\in L^2$ solves

$$\begin{aligned} u={\mathbf {K}}(\lambda ,u):=&(1+\lambda -\lambda _0){\mathbf {P}}u+{\mathbf {F}}({\mathbf {P}}u+[({\mathbf {A}}-\lambda {\mathbf {I}})_{|X}]^{-1}{\mathbf {Q}}u)\nonumber \\ =&(1+\lambda -\lambda _0){\mathbf {P}}u+{\mathbf {G}}(\lambda ,u); \end{aligned}$$

(5.1)

see Lemma 3.5. Here the nonlinearity ${\mathbf {K}}:[\lambda _0-\delta ,\lambda _0+\delta ]\times L^2\rightarrow L^2$ is continuous and, in view of Lemma 3.5, completely continuous. Moreover (5.1) has no solutions if $|\lambda -\lambda _0|\le \delta$ and $\Vert u\Vert _{L^2}$ is sufficiently large. Indeed if $u\in L^2$ solves (5.1), where $|\lambda -\lambda _0|\le \delta$, then $w:={\mathbf {P}}u+[({\mathbf {A}}-\lambda {\mathbf {I}})|_X]^{-1}{\mathbf {Q}}u$ solves (3.1), i.e., $\Vert {\mathbf {P}}u\Vert _{L^2}=\Vert {\mathbf {P}}w\Vert _{L^2}\le \Vert w\Vert _{H^1}\le r$. Hence $\Vert u\Vert _{L^2}\le (1+\delta )r+\Vert m\Vert _{L^2}:= R_0$. Therefore the Leray-Schauder fixed-point index $\mathrm {ind}_{LS}({\mathbf {K}}(\lambda ,\cdot ),B)$, where B is the ball around 0 of radius $R>\max \{R_0,\delta ^{-1}\Vert m\Vert _{L^2}\}$ in $L^2$, is well-defined and independent of $\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$. It is immediate to see that if $\lambda =\lambda _0\pm \delta$, then $u\ne (1+\lambda -\lambda _0){\mathbf {P}}u+t{\mathbf {G}}(\lambda ,u)$ for $u\not \in B$ and $t\in [0,1]$. Hence, in view of the homotopy invariance and the restriction property of the index, for $\lambda =\lambda _0\pm \delta$

$$\begin{aligned} \mathrm {ind}_{LS}({\mathbf {K}}(\lambda ,\cdot ),B)=\mathrm {ind}_{LS}((1\pm \delta ){\mathbf {P}},B)=\mathrm {ind}_{LS}((1\pm \delta ){\mathbf {I}},B\cap X_0). \end{aligned}$$

However

$$\begin{aligned} \mathrm {ind}_{LS}((1-\delta ){\mathbf {I}},B\cap X_0)=1, \;\mathrm {ind}_{LS}((1+\delta ){\mathbf {I}},B\cap X_0)=(-1)^{\dim X_0}=-1. \end{aligned}$$

This is a contradiction.$\square$

Remark 5.1

The standard use of the Kuratowski-Whyburn lemma makes it easy to get a slightly better result in the context of Theorem 1.4 (i). Namely it appears that there exists a closed connected set $\Gamma \subset H^2\times {\mathbb {R}}$ of solutions to (1.1) which contains a sequence $(u_n,\lambda _n)$ such that $\Vert u_n\Vert _{H^2}\rightarrow \infty$, $\lambda _n\rightarrow \lambda _0$.

Now we shall pass to the proof of Theorem 1.4 (ii). We start with the geometric interpretation of the resonance assumptions in spirit of [6] and [21].

Lemma 5.2

Assume that $M\subset X$. If either

(i) condition $(LL)_\pm$ holds and M is bounded in $L^2$, or

(ii) condition $(SR)_\pm$ holds and M relatively compact in $L^2$, then there exist $R_0>0$ and $\alpha >0$ such that for all ${\bar{v}}\in X_0$ with $\Vert {\bar{v}}\Vert _{L^2}\ge R_0$ and ${\bar{w}}\in M$

$$\begin{aligned} \pm \langle {\bar{v}}, {\mathbf {F}}({\bar{v}}+{\bar{w}})\rangle _{L^2} >\alpha . \end{aligned}$$

(5.2)

Proof

We carry out the proof for $(LL)_+$ and $(SR)_+$; other cases may be treated analogously. Suppose to the contrary that for any $n\in {\mathbb {N}}$ there are ${\bar{v}}_n\in X_0$ and ${\bar{w}}_n\in M$ such that $\Vert {\bar{v}}_n\Vert _{L^2}\ge n$ and

$$\begin{aligned} \langle {\bar{v}}_n, \mathbf{F}({\bar{v}}_n+{\bar{w}}_n)\rangle _{L^2} \le n^{-1}.\end{aligned}$$

(5.3)

Let $\rho _n:= \Vert {\bar{v}}_n\Vert _{L^2}$ and ${\bar{z}}_n:= \rho _n^{-1}{\bar{v}}_n$, $n\in {\mathbb {N}}$. Since $\dim X_0<\infty$, we may assume that $\Vert {\bar{z}}_n-{\bar{z}}_0\Vert _{L^2}\rightarrow 0$ as $n\rightarrow \infty$, where ${\bar{z}}_0\in X_0$ and $\Vert {\bar{z}}_0\Vert _{L^2 ({\mathbb {R}}^N)}=1$. Therefore we may assume that ${\bar{z}}_n(x)\rightarrow {\bar{z}}_0(x)$ for a.a. $x\in {\mathbb {R}}^N$ and there is $\kappa \in L^2$ such that $|{\bar{z}}_n|\le \kappa$ a.e. In view of the so-called unique continuation property (see e.g. [11, Proposition 3, Remark 2]), ${\bar{z}}_0\ne 0$ a.e. Hence the set ${\mathbb {R}}^N{\setminus } (A_+\cup A_-)$, where $A_\pm :=\{x\in {\mathbb {R}}^N\mid \pm {\bar{z}}_0>0\}$, is of measure zero.

Dividing (5.3) by $\rho _n$ we get

$$\begin{aligned} n^{-2}\ge \rho _n^{-1}/n \ge \langle {\bar{z}}_n, {\mathbf {F}}( \rho _n {\bar{z}}_n+{\bar{w}}_n)\rangle _{L^2} = \int _{{\mathbb {R}}^N} {\bar{z}}_n(x) f(x, \rho _n {\bar{z}}_n(x)+{\bar{w}}_n(x))\, d x. \end{aligned}$$

Assume (i); then $\rho _n^{-1}{\bar{w}}_n\rightarrow 0$ in $L^2$ since M is bounded. We may assume without loss of generality that $\rho _n^{-1}{\bar{w}}_n(x)\rightarrow 0$ for a.a. $x\in {\mathbb {R}}^N$. Hence ${\bar{z}}_n+\rho _n^{-1}{\bar{w}}_n\rightarrow {\bar{z}}_0$ a.e. This implies that $\rho _n{\bar{z}}_n+{\bar{w}}_n\rightarrow \pm \infty$ for a.a. $x\in A_\pm$. Using (1.4) we are in a position to use the Fatou lemma to get

$$\begin{aligned} 0\ge \liminf _{n\rightarrow \infty } \int _{{\mathbb {R}}^N} {\bar{z}}_n f(x, \rho _n {\bar{z}}_n+{\bar{w}}_n)\,dx&\ge \int _{{\mathbb {R}}^N}\liminf _{n\rightarrow \infty }{\bar{z}}_n f(x,\rho _n{\bar{z}}_n+{\bar{w}}_n)\,dx\ge \\&\ge \int _{A_+} {\check{f}}_+{\bar{z}}_0\, d x+ \int _{A_-} {\hat{f}}_-{\bar{z}}_0\, d x >0, \end{aligned}$$

in view of Remark 1.3; this is a contradiction.

Assume (ii). Since now M is $L^2$-precompact, we may assume that ${\bar{w}}_n\rightarrow {\bar{w}}_0\in L^2({\mathbb {R}}^N)$, ${\bar{w}}_n(x)\rightarrow {\bar{w}}_0(x)$ for a.e. $x\in {\mathbb {R}}^N$ and there is $\gamma \in L^2({\mathbb {R}}^N)$ such that $|{\bar{w}}_n|\le \gamma$ a.e. on ${\mathbb {R}}^N$ for all $n\in {\mathbb {N}}$.

Clearly $\langle {\bar{v}}_n,{\mathbf {F}}({\bar{v}}_n+{\bar{w}}_n) \rangle _{L^2} = \langle {\bar{v}}_n + {\bar{w}}_n, {\mathbf {F}}({\bar{v}}_n+{\bar{w}}_n)\rangle _{L^2} - \langle {\bar{w}}_n, {\mathbf {F}}({\bar{v}}_n+{\bar{w}}_n)\rangle _{L^2}$. In view of $(SR)_+$, $\lim _{s\rightarrow \pm \infty }f(x,s)=0$ for a.a. $x\in {\mathbb {R}}^N$. Hence, again by (1.4) and the Lebesgue dominated convergence theorem we have

$$\begin{aligned} \langle {\bar{w}}_n, {\mathbf {F}}({\bar{v}}_n+{\bar{w}}_n)\rangle _{L^2}=\int _{{\mathbb {R}}^N} {\bar{w}}_n(x) f(x, \rho _n {\bar{z}}_n(x)+{\bar{w}}_n(x))\, d x \rightarrow 0, \text{ as } n\rightarrow +\infty , \end{aligned}$$

and, in view of (5.3), arguing as before

$$\begin{aligned}&0\ge \liminf _{n\rightarrow \infty }\langle {\bar{v}}_n + {\bar{w}}_n, {\mathbf {F}}({\bar{v}}_n+{\bar{w}}_n)\rangle _{L^2}\\&\quad =\liminf _{n\rightarrow \infty }\int _{{\mathbb {R}}^N} (\rho _n {\bar{z}}_n(x)+{\bar{w}}_n(x)) f(x, \rho _n z_n(x)+{\bar{w}}_n(x))\,dx>0 \\&\quad \ge \int _{A_+} {\check{k}}_+ (x)d x + \int _{A_-} {\check{k}}_- (x)\,d x>0, \end{aligned}$$

we reach a contradiction. $\square$

The set of stationary points of the semiflow $\Phi ^\lambda$ related to (3.12), where $|\lambda -\lambda _0|\le \delta$ and $\delta$ is given by (3.20) will be denoted by ${{\mathcal {E}}}_\lambda$ and let

$$\begin{aligned} {{\mathcal {E}}}:= \bigcup _{\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]} {{\mathcal {E}}}_\lambda . \end{aligned}$$

Lemma 5.3

Suppose that there is $r>0$ such that ${{\mathcal {E}}}\subset B_{H^1} (0,r)$ (⁴). Then there exists $R_\infty =R_\infty (r) > 0$ such that, for any bounded solution $u:{\mathbb {R}}\rightarrow H^1$ of $\Phi ^\lambda$ with $\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$, one has

$$\begin{aligned} \sup _{t\in {\mathbb {R}}} \Vert {\mathbf {Q}}\, u(t)\Vert _{H^1} < R_\infty . \end{aligned}$$

Proof

Since $\delta <\mathrm {dist}(\lambda _0,\sigma ({{\mathbf {A}}}){\setminus }\{\lambda _0\})$, there is $c>0$ such that $\sigma (({{\mathbf {A}}}-\lambda {{\mathbf {I}}})_{|X_-}) \subset (-\infty , -c)$ and $\sigma ({{\mathbf {A}}}-\lambda {{\mathbf {I}}})_{|X_+}) \subset (c,+\infty )$ whenever $|\lambda -\lambda _0|\le \delta$.

Fix $\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$ and let ${{\mathbf {B}}}_\pm :=({{\mathbf {A}}}-\lambda {{\mathbf {I}}})_{|X_\pm }$. Clearly ${{\mathbf {B}}}_+$ is sectorial and positive. By [7] (see Corollary 1.3.5 and comp. Corollary 1.3.4) the domain $D({{\mathbf {B}}}_+^{1/2})=D({{\mathbf {A}}}_0^{1/2})=H^1\cap X_+$; thus, in view of [7, Proposition 1.3.6], there is $K>0$ independent of $\lambda \in [\lambda _0-\delta , \lambda _0+\delta ]$ such that for all $\tau >0$

$$\begin{aligned} \forall \,v\in X_+\;\;\;\;\ \Vert e^{-\tau {{\mathbf {B}}_+}}v\Vert _{H^1}=\Vert {{\mathbf {B}}}_+^{1/2}e^{-\tau {{\mathbf {B}}_+}}v\Vert _{L^2}\le K\frac{e^{-c\tau }}{\tau ^{1/2}}\Vert v\Vert _{L^2},\;\; \tau >0, \end{aligned}$$

(5.4)

and

$$\begin{aligned} \forall \,v\in H^1\cap X_+\;\;\;\; \Vert e^{-\tau {{\mathbf {B}}_+}}v\Vert _{H^1}\le Ke^{-c\tau }\Vert v\Vert _{H^1}. \end{aligned}$$

(5.5)

where $\{e^{-\tau {{\mathbf {B}}}_+}\}_{\tau \ge 0}$ denotes the semigroup generated by $-{{\mathbf {B}}}_+$.

The semigroup $\{e^{-\tau {{\mathbf {B}}}_-}\}_{\tau \ge 0}$ generated by ${{\mathbf {B}}}_-$ is uniformly continuous, i.e. it extends to a strongly continuous group and there is $K'> 0$ independent of $\lambda \in [\lambda _0-\delta , \lambda _0+\delta ]$ such that

$$\begin{aligned} \forall \,v\in X_-\;\; \Vert e^{-\tau {{\mathbf {B}}}_-}v\Vert _{L^2}\ge \frac{1}{K'}e^{c\tau }\Vert v\Vert _{L^2},\;\; \tau \ge 0 \end{aligned}$$

(5.6)

since $\sigma ({{\mathbf {B}}}_-)<-c$.

Now take a solution $u:{\mathbb {R}}\rightarrow H^1$ of the semiflow $\Phi ^\lambda$ corresponding to (3.12). It is well-known that u is a mild solution (see [20]), i.e. the so-called Duhamel formula holds

$$\begin{aligned} u(t) = e^{-(t-s)( {{\mathbf {A}}}-\lambda {{\mathbf {I}}})} u(s)+\int _{s}^{t} e^{-(t-\tau )({{\mathbf {A}}}-\lambda {{\mathbf {I}}})} {\mathbf {F}}(u(\tau ))\, d\tau \text{ for } \text{ all } s,t\in {\mathbb {R}}, \, t>s, \end{aligned}$$

(5.7)

where $\{e^{-\tau ({{\mathbf {A}}}-\lambda {{\mathbf {I}}})}\}_{\tau \ge 0}$ denotes the analytic semigroup generated by $-({{\mathbf {A}}}-\lambda {{\mathbf {I}}})$.

Since, due to Remark 3.10 (2), $\alpha (u)\subset {{\mathcal {E}}}_{\lambda }$, there exists $t_u<0$ such that $\Vert u(\tau ) \Vert _{H^1}<2r$ for all $\tau \le t_u$. Thus, by (5.5), (3.16) and (5.4), for $t\ge t_u$

$$\begin{aligned} \Vert {{\mathbf {Q}}}_+u(t)\Vert _{H^1} \le \Vert e^{-(t-t_u) {{\mathbf {B}}}_+} {{\mathbf {Q}}}_+ u(t_u) \Vert _{H^1} + \int _{t_u}^{t} \Vert e^{-(t-\tau ){{\mathbf {B}}}_+} {{\mathbf {Q}}}_+{{\mathbf {F}}}(u(\tau ))\Vert _{H^1}\, d\tau \le \\ K\left( \Vert {{\mathbf {Q}}}_+\Vert _{{{\mathcal {L}}}(H^1,H^1)}e^{-c (t-t_u)}2r + \int _{t_u}^{t} (t-\tau )^{-1/2} e^{-c (t-\tau )}\Vert {{\mathbf {Q}}}_+{{\mathbf {F}}}(u(\tau ))\Vert _{L^2}\,d \tau \right) . \end{aligned}$$

In view of (1.4)

$$\begin{aligned} \Vert {{\mathbf {Q}}}_+{{\mathbf {F}}}(u(\tau ))\Vert _{L^2}\le \Vert m\Vert _{L^2},\;\; t_u\le \tau \le t; \end{aligned}$$

(5.8)

thus

$$\begin{aligned} \Vert {{\mathbf {Q}}}_+u(t)\Vert _{H^1}\le K\left( \Vert {{\mathbf {Q}}}_+\Vert _{{{\mathcal {L}}}(H^1,H^1)} 2r + \Vert m\Vert _{L^2} \int _{0}^{+\infty } s^{-1/2} e^{-c s}\,ds\right) =: R'_{1,\infty }. \end{aligned}$$

This means that $\Vert {{\mathbf {Q}}}_+ u(t)\Vert _{H^1} \le R_{1,\infty }=\max \{2r,R'_{1,\infty }\}$ for all $t\in {\mathbb {R}}$.

Since, due to Remark 3.10 (2), $\omega (u)\subset {\mathcal {E}}_\lambda$ we can take $s_u\in {\mathbb {R}}$ such that $\Vert u(\tau )\Vert _{H^1} \le 2r$, for all $\tau \ge s_u$, and observe that, in view of (5.7), we have for each $t< s_u$

$$\begin{aligned} {{\mathbf {Q}}}_- u(s_u) = e^{-(s_u-t){{\mathbf {B}}}_-}{{\mathbf {Q}}}_- u(t) + \int _{t}^{s_u} e^{-(s_u-\tau ){{\mathbf {B}}}_-} {{\mathbf {Q}}}_- {{\mathbf {F}}}(u(\tau ))\, d \tau . \end{aligned}$$

Hence, using (5.6), we get

$$\begin{aligned} \Vert {{\mathbf {Q}}}_- u(t)\Vert _{L^2}\le K'\left( e^{c (t-s_u)} \Vert {{\mathbf {Q}}}_- u(s_u)\Vert _{L^2}+\int _{t}^{s_u}e^{c(t-\tau )} \Vert {{\mathbf {Q}}}_- {{\mathbf {F}}} (u(\tau ))\Vert _{L^2}\,d \tau \right) . \end{aligned}$$

Again in view of (1.4)

$$\begin{aligned} \Vert {{\mathbf {Q}}}_-{{\mathbf {F}}}(u(\tau ))\Vert _{L^2}\le \Vert m\Vert _{L^2},\;\; t\le \tau \le s_u.\end{aligned}$$

(5.9)

Therefore

$$\begin{aligned} \Vert {{\mathbf {Q}}}_- u(t)\Vert _{L^2}\le K'(2r +\Vert m\Vert _{L^2} c^{-1})=:R'_{2,\infty } \end{aligned}$$

and thus $\Vert {{\mathbf {Q}}}_- u(t)\Vert _{L^2} \le {\tilde{R}}_{2,\infty }:=\max \{2r,R'_{2,\infty }\}$ for all $t\in {\mathbb {R}}$. Since $X_-$ is finite dimensional, there is a constant $R_{2,\infty }>0$ such $\Vert {{\mathbf {Q}}}_- u(t)\Vert _{H^1}\le R_{2,\infty }$ for all $t\in {\mathbb {R}}$.$\square$

Lemma 5.4

If $u:[t_0,t_1]\rightarrow H^{1}({\mathbb {R}}^N)$ is a solution of $\Phi ^\lambda$ for some $\lambda \in {\mathbb {R}}$, then

$$\begin{aligned} \frac{1}{2}\frac{d}{d t} \Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2} = (\lambda -\lambda _0) \Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2} + \langle {\mathbf {P}}u(t), {\mathbf {F}}(u(t)\rangle _{L^2}, t\in (t_0,t_1). \end{aligned}$$

when u solves (3.12).

Proof

The symmetry of ${{\mathbf {A}}}$ implies that $X_0$ is orthogonal to to the range $R({{\mathbf {A}}}-\lambda _0 {{\mathbf {I}}})$ in $L^2$. Hence

$$\begin{aligned} \frac{1}{2}\frac{d}{d t}\Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2}&= \langle {\mathbf {P}}u(t), \dot{u}(t)\rangle _{L^2} = \langle {\mathbf {P}}u(t), -({{\mathbf {A}}}-\lambda _0 {{\mathbf {I}}}) u(t) +(\lambda -\lambda _0) u(t)+{\mathbf {F}}(u(t))\rangle _{L^2}\\&=(\lambda -\lambda _0) \Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2} + \langle {\mathbf {P}}u(t), {\mathbf {F}}(u(t)\rangle _{L^2} \end{aligned}$$

for all $t\in (t_0,t_1)$. $\square$

Proof of Theorem 1.4 (ii)

Assume (3.19) and suppose to the contrary that $\lambda _0$ is not a point of bifurcation from infinity. Thus there are $r>0$ and $\delta >0$ satisfying condition (3.20) such that

$$\begin{aligned} {{\mathcal {E}}} \subset B_{H^1} (0,r). \end{aligned}$$

(5.10)

By Proposition 3.11, there is $R>0$ such that the $K(\Phi ^{\lambda _0\pm \delta })\subset B_{H^1}(0,R)$. By Lemma 5.3 one has $R_\infty =R_\infty (r) \ge R$ such that, for any bounded solution of $u:{\mathbb {R}}\rightarrow H^1$ of $\Phi ^\lambda$, $|\lambda -\lambda _0|\le \delta$, one has

$$\begin{aligned} \sup _{t\in {\mathbb {R}}} \Vert {\mathbf {Q}} u (t)\Vert _{H^1} < R_\infty . \end{aligned}$$

(5.11)

Let $M_{R_\infty }$ be the set of all ${\bar{u}}\in H^1$ such that there exists a solution $u:(-\infty , 0]\rightarrow H^1$ of $\Phi ^\lambda$, $|\lambda -\lambda _0|\le \delta$, with $u(0)={\bar{u}}$ and $\Vert {\mathbf {Q}} u(t)\Vert _{H^1} \le R_\infty$ for all $t\le 0$. In view of Proposition 3.7, the set $M:={\mathbf {Q}}M_{R_\infty }\subset X$ is relatively compact in $L^2$. By Lemma 5.2 there are $R_0 \ge R_\infty$ and $\alpha >0$ such that for all ${\bar{v}}\in X_0{\setminus } B_{L^2}(0,R_0)$ and ${\bar{w}}\in M$

$$\begin{aligned} \langle {\bar{v}}, {{\mathbf {F}}}({\bar{v}}+{\bar{w}})\rangle _{L^2} >\alpha \end{aligned}$$

(5.12)

if $(LL)_+$ or $(SR)_+$ is satisfied, or

$$\begin{aligned} \langle {\bar{v}}, {{\mathbf {F}}}({\bar{v}}+{\bar{w}})\rangle _{L^2}<-\alpha \end{aligned}$$

(5.13)

if $(LL)_-$ or $(SR)_-$ is satisfied. Put

$$\begin{aligned} B:=\{{\bar{u}}\in H^1 \mid \Vert {\mathbf {P}}{\bar{u}}\Vert _{L^2} \le R_0,\;\; \Vert {\mathbf {Q}}{\bar{u}}\Vert _{H^1}\le R_\infty \}. \end{aligned}$$

Taking $\delta$ smaller if necessary we may assume that

$$\begin{aligned} \delta R_{0}^{2}<\alpha . \end{aligned}$$

(5.14)

Then, for any $\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$, B is an isolating neighborhood for the semiflow $\Phi ^\lambda$. To see this, suppose to the contrary that there is ${\bar{u}}\in \mathrm {Inv}_{\Phi ^\lambda }(B)\cap \partial B$. Hence there is a solution $u:{\mathbb {R}}\rightarrow B$ of $\Phi ^\lambda$ through ${\bar{u}}$, i.e. ${\bar{u}}=u(0)$. Since u is bounded, we have $\Vert {{\mathbf {Q}}}{\bar{u}}\Vert _{H^1}<R_\infty$ in view of (5.11). Therefore $\Vert {\mathbf {P}}{\bar{u}}\Vert _{L^2}=R_0$. Let ${\bar{u}}={\bar{v}}+{\bar{w}}$, where ${\bar{v}}:={{\mathbf {P}}}{\bar{u}}$ and ${\bar{w}}:={{\mathbf {Q}}}{\bar{u}}$. Then ${\bar{v}}\in X_0{\setminus } B_{L^2}(0,R_0)$ and ${\bar{w}}\in M$. By Lemma 5.4,

$$\begin{aligned} \frac{1}{2}\left. \frac{d}{d t} \Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2} \right| _{t=0}=&(\lambda -\lambda _0)\Vert {{\mathbf {P}}}u(0)\Vert _{L^2}^2 + \langle {\mathbf {P}}u(0), {\mathbf {F}}(u(0))\rangle _{L^2}\\ =&(\lambda -\lambda _0)R_0^2+\langle {\bar{v}},{{\mathbf {F}}}({\bar{v}}+{\bar{w}})\rangle _{L^2}. \end{aligned}$$

Due to (5.14) and (5.12) (or (5.13))

$$\begin{aligned} \frac{1}{2}\left. \frac{d}{dt}\Vert {\mathbf {P}}u(t)\Vert _{L^2}^2 \right| _{t=0} >0\quad \left( \text {or}\;\; \frac{1}{2}\left. \frac{d}{d t}\Vert {\mathbf {P}}u(t)\Vert _{L^2}^2\right| _{t=0} <0\right) . \end{aligned}$$

This contradicts the assumption $u({\mathbb {R}})\subset B$ and proves that B is an isolating neighborhood for the semiflows $\Phi ^\lambda$, $\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$. Using the continuation property (H4) of the homotopy index, we obtain

$$\begin{aligned} h(\Phi ^{\lambda _0-\delta }, K_{\lambda _0-\delta }) = h (\Phi ^{\lambda _0+\delta },K_{\lambda _0+\delta }) \end{aligned}$$

(5.15)

where $K_\lambda := \mathrm {Inv}_{\Phi ^{\lambda }} (B)$ for $\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]$.

We also claim that, for $\lambda \in [\lambda _0-\delta , \lambda _0+\delta ]$, one has

$$\begin{aligned} K_\lambda = K(\Phi ^\lambda ). \end{aligned}$$

(5.16)

Indeed, the inclusion $K_\lambda \subset K(\Phi ^\lambda )$ is self-evident. Conversely, any bounded full solution $u:{\mathbb {R}}\rightarrow H^1({\mathbb {R}}^N)$ of $\Phi ^\lambda$ satisfies (5.11). Therefore if u leaves B, then for some $t\in {\mathbb {R}}$ we have $\Vert {\mathbf {P}}u(t)\Vert _{L^2}>R_0$. Put $t_-:=\inf \{t\in {\mathbb {R}}\mid \Vert {\mathbf {P}} u(t) \Vert _{L^2}>R_0\}$ and $t_+:= \sup \{t\in {\mathbb {R}}\mid \Vert {\mathbf {P}} u(t) \Vert _{L^2}>R_0 \}$. In view of (5.10) and the fact that $R_0\ge R_\infty >r$ we see that $-\infty<t_-<t_+<+\infty$. It is clear that $\Vert {\mathbf {P}}u(t_\pm )\Vert _{L^2} = R_0$ and

$$\begin{aligned} \Vert {\mathbf {P}} u(t) \Vert _{L^2} <R_0 \text{ for } \text{ all } t\in (-\infty ,t_-)\cup (t_+,+\infty ), \end{aligned}$$

which means that

$$\begin{aligned} \left. \frac{d}{d t} \Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2} \right| _{t=t_-}\!\!\!\! \ge 0 \ \text{ and } \ \left. \frac{d}{d t} \Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2} \right| _{t=t_+} \!\!\!\!\le 0. \end{aligned}$$

(5.17)

But on the other hand, as before,

$$\begin{aligned} \frac{1}{2}\left. \frac{d}{d t} \Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2} \right| _{t=t_\pm }=&(\lambda -\lambda _0)\Vert {{\mathbf {P}}}u(t_\pm )\Vert _{L^2}^2 + \langle {\mathbf {P}}u(t_\pm ), {\mathbf {F}}(u(t_\pm ))\rangle _{L^2}\\ =&(\lambda -\lambda _0)R_0^2+\langle {\bar{v}},{{\mathbf {F}}}({\bar{v}}+{\bar{w}})\rangle _{L^2} \end{aligned}$$

where $\bar{v} := \mathbf {P} u(t_\pm )$ and $\bar{w} := \mathbf {Q} u(t_\pm )$,

which together with (5.12) (or (5.13)) yields

$$\begin{aligned} \frac{1}{2}\left. \frac{d}{d t} \Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2} \right| _{t=t_\pm }>0 \ \ \ \left( \text{ or } \frac{1}{2}\left. \frac{d}{d t} \Vert {\mathbf {P}}u(t)\Vert _{L^2}^{2} \right| _{t=t_\pm } <0\right) . \end{aligned}$$

This contradicts one of the inequalities in (5.17) and shows (5.16). Therefore, by Proposition 3.11, one has

$$\begin{aligned} h(\Phi ^{\lambda _0\pm \delta },K_{\lambda _0\pm \delta })= h(\Phi ^{\lambda _0\pm \delta },K(\Phi ^{\lambda _0\pm \delta }))= \Sigma ^{k(\lambda _0\pm \delta )}. \end{aligned}$$

and this together with (5.15) leads to a contradiction, since $k(\lambda _0+\delta )-k(\lambda _0-\delta )=\dim X_0>0$. $\square$