The *Generalized Nehari manifold*
$\mathcal{\mathcal{N}}$ of the functional *J* consists of all nonzero
$u\in {H}^{1}$
such that

$({J}^{\prime}(u),u)=0\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}({J}^{\prime}(u),v)=0\mathit{\hspace{1em}}\text{for all}v\in {E}^{-}.$

Equivalently, these equations can be written as ${(\nabla J(u),u)}_{A}=0$ and ${P}_{-}\nabla J(u)=0$, respectively.

For any $w\notin {E}^{-}$, we set

${E}_{w}=\{sw+v:s>0,v\in {E}^{-}\}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\overline{E}}_{w}=\{sw+v:s\in \mathbb{R},v\in {E}^{-}\}.$

By the definition of $\mathcal{\mathcal{N}}$, if *u* is a critical point of ${J|}_{{E}_{w}}$, then $u\in \mathcal{\mathcal{N}}$.
As consequence, $\mathcal{\mathcal{N}}$ contains all nontrivial critical points of *J*.

The proof of the following lemma is essentially the same as the proof of [13, Lemma 5.1].

#### Lemma 4.1.

*For every $w\mathrm{\notin}{E}^{\mathrm{-}}$, the functional ${J\mathrm{|}}_{{E}_{w}}$ attains its positive global maximum.
As a consequence, ${E}_{w}\mathrm{\cap}\mathcal{N}\mathrm{\ne}\mathrm{\varnothing}$ for all $w\mathrm{\notin}{E}^{\mathrm{-}}$.*

It is convenient to introduce the functional

$I(u)=J(u)-\frac{1}{2}({J}^{\prime}(u),u).$

Obviously, $J(u)=I(u)$ for all $u\in \mathcal{\mathcal{N}}$.
Now we prove the following technical result.

#### Lemma 4.2.

*There exists a constant $C\mathrm{>}\mathrm{0}$ independent of $u\mathrm{\in}{H}^{\mathrm{1}}$ such that*

${\parallel u\parallel}^{2}\le C\left(|({J}^{\prime}(u),u)|+|({J}^{\prime}(u),{u}^{-})|+\mu ({\parallel u\parallel}_{\mathrm{\infty}}){\parallel u\parallel}_{\mathrm{\infty}}{\parallel u\parallel}^{2}\right)$(4.1)

*and*

${\parallel u\parallel}^{2}\le C\left(|({J}^{\prime}(u),u)|+|({J}^{\prime}(u),{u}^{-})|+({I}^{1/2}(u)+I(u))\parallel u\parallel \right)$(4.2)

*for all $u\mathrm{\in}{H}^{\mathrm{1}}$.*

#### Proof.

The identity

$({J}^{\prime}(u),{u}^{-})={l}_{A}({u}^{-})-{\int}_{\mathbb{R}}{F}_{u}(t,u)\cdot {u}^{-}\mathit{d}t$

and Proposition 2.5 imply

$\kappa {\parallel {u}^{-}\parallel}^{2}\le -({J}^{\prime}(u),{u}^{-})-{\int}_{\mathbb{R}}{F}_{u}(t,u)\cdot {u}^{-}\mathit{d}t.$(4.3)

Similarly, the identity

$({J}^{\prime}(u),u)={l}_{A}({u}^{+})-{\int}_{\mathbb{R}}{F}_{u}(t,u)\cdot {u}^{+}\mathit{d}t+({J}^{\prime}(u),{u}^{-})$

implies

$\kappa {\parallel {u}^{+}\parallel}^{2}\le ({J}^{\prime}(u),u)-({J}^{\prime}(u),{u}^{-})+{\int}_{\mathbb{R}}{F}_{u}(t,u)\cdot {u}^{+}\mathit{d}t.$(4.4)

Adding inequalities (4.3) and (4.4), we obtain immediately that

$\parallel u{\parallel}^{2}\le C(|({J}^{\prime}(u),u)|+|({J}^{\prime}(u),{u}^{-})|+{\int}_{\mathbb{R}}|{F}_{u}(t,u)||{u}^{+}|dt+{\int}_{\mathbb{R}}|{F}_{u}(t,u)||{u}^{-}|dt).$(4.5)

Now let $R={\parallel u\parallel}_{\mathrm{\infty}}$.
Then, by inequality (3.4),

${\int}_{\mathbb{R}}|{F}_{u}(t,u)||{u}^{\pm}|dt\le \mu (R)\parallel {u}^{\pm}{\parallel}_{\mathrm{\infty}}\parallel u{\parallel}_{2}^{2}.$

Hence,

${\parallel u\parallel}^{2}\le C\left(|({J}^{\prime}(u),u)|+|({J}^{\prime}(u),{u}^{-})|+\mu (R){\parallel u\parallel}_{2}^{2}({\parallel {u}^{+}\parallel}_{\mathrm{\infty}}+{\parallel {u}^{-}\parallel}_{\mathrm{\infty}})\right)$$\le C\left(|({J}^{\prime}(u),u)|+|({J}^{\prime}(u),{u}^{-})|+\mu (R){\parallel u\parallel}_{\mathrm{\infty}}{\parallel u\parallel}^{2}\right),$

which proves (4.1).

Now we prove inequality (4.2).
Given $u\in {H}^{1}$, let

${S}_{1}=\{t\in \mathbb{R}:|u(t)|\le 1\}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{S}_{2}=\mathbb{R}\setminus {S}_{1}.$

We introduce the following integrals:

${I}_{1}={\int}_{{S}_{1}}|{F}_{u}(t,u){|}^{2}dt\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{I}_{2}={\int}_{{S}_{2}}|{F}_{u}(t,u)|dt.$

By inequality (3.3) and assumption (v),

$I(u)\ge \nu ({I}_{1}+{I}_{2}),$(4.6)

where ν depends on *q*, ${\nu}_{1}$ and ${\nu}_{2}$.
Recall that we may take ${r}_{1}={r}_{2}=1$ in assumption (v).
Since

${\int}_{\mathbb{R}}}|{F}_{u}(t,u)||{u}^{\pm}|dt\le ({\displaystyle {\int}_{{S}_{1}}}{|{F}_{u}(u,t){|}^{2}dt)}^{1/2}({\displaystyle {\int}_{{S}_{1}}}{|{u}^{\pm}{|}^{2}dt)}^{1/2}+\parallel {u}^{\pm}{\parallel}_{\mathrm{\infty}}{\displaystyle {\int}_{{S}_{2}}}|{F}_{u}(u,t)|dt$$\le ({I}_{1}(u)+{I}_{2}^{1/2}(u))\parallel {u}^{\pm}\parallel ,$

equations (4.5) and (4.6) yield (4.2).
∎

As an immediate consequence of Lemma 4.2, we obtain the following corollary.

#### Corollary 4.3.

*There exists a constant ${\epsilon}_{\mathrm{0}}\mathrm{>}\mathrm{0}$ such that $\mathrm{\parallel}u\mathrm{\parallel}\mathrm{\ge}{\mathrm{\parallel}u\mathrm{\parallel}}_{\mathrm{\infty}}\mathrm{\ge}{\epsilon}_{\mathrm{0}}$,
$J\mathit{}\mathrm{(}u\mathrm{)}\mathrm{\ge}{\epsilon}_{\mathrm{0}}$ and*

${\int}_{\mathbb{R}}{F}_{u}(t,u)\cdot u\mathit{d}t\ge 2{\epsilon}_{0}$

*for all $u\mathrm{\in}\mathcal{N}$.*

Let ${\overline{E}}^{-}=\mathbb{R}\oplus {E}^{-}$.
Elements of this space are denoted by $[{h}_{0},h]$, where ${h}_{0}\in \mathbb{R}$ and $h\in {E}^{-}$.
We define the operator $G:{H}^{1}\to {\overline{E}}^{-}$ by

$G(u)=[{(\nabla J(u),u)}_{A},{P}^{-}\nabla J(u)],u\in {H}^{1}.$

It is not difficult to see that the operator *G* is a ${C}^{1,1}$ map, and its derivative is given by the formula

${G}^{\prime}(u)v=[{({\nabla}^{2}J(u)v,u)}_{A}+{(\nabla J(u),v)}_{A},{P}^{-}{\nabla}^{2}J(u)v]$

for all $u,v\in {H}^{1}$.

#### Lemma 4.5.

*Let ${u}_{\mathrm{0}}\mathrm{\in}{H}^{\mathrm{1}}$,*

${g}_{0}={(\nabla J({u}_{0}),{u}_{0})}_{A}\mathit{\hspace{1em}}\text{\mathit{a}\mathit{n}\mathit{d}}\mathit{\hspace{1em}}g={P}^{-}\nabla J({u}_{0})\in {E}^{-}.$

*Then, for all ${h}_{\mathrm{0}}\mathrm{\in}\mathrm{R}$ and $h\mathrm{\in}{E}^{\mathrm{-}}$,*

${({G}^{\prime}({u}_{0})({h}_{0}{u}_{0}+h),[{h}_{0},h])}_{A}\le 2|{g}_{0}|{h}_{0}^{2}-\kappa {\parallel h\parallel}^{2}+\frac{3}{2}{h}_{0}^{2}\parallel g\parallel +\frac{3}{2}\parallel g\parallel {\parallel h\parallel}^{2}-{h}_{0}^{2}(1-\theta ){\int}_{\mathbb{R}}{F}_{u}(t,{u}_{0})\cdot {u}_{0}\mathit{d}t,$

*where $\kappa \mathrm{>}\mathrm{0}$ and $\theta \mathrm{\in}\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{)}$ are the constants from Proposition 2.2 and assumption *(iv)*, respectively.*

#### Proof.

Since, by the assumptions,

$({L}_{A}{u}_{0},{u}_{0})={g}_{0}+{\int}_{\mathbb{R}}{F}_{u}(t,{u}_{0})\cdot {u}_{0}\mathit{d}t$

and

$({L}_{A}{u}_{0},h)={(g,h)}_{A}+{\int}_{\mathbb{R}}{F}_{u}(t,{u}_{0})\cdot h\mathit{d}t,$

a straightforward but a little bit tedious calculation yields the identity

${({G}^{\prime}({u}_{0})({h}_{0}{u}_{0}+h),[{h}_{0},h])}_{A}=2{h}_{0}^{2}{g}_{0}+({L}_{A}h,h)+3{h}_{0}{(g,h)}_{A}-{\int}_{\mathbb{R}}\left(H(t){h}_{0}^{2}+2{h}_{0}K(t)\cdot h+(M(t)h)\cdot h\right)\mathit{d}t,$

where

$H(t)=({F}_{uu}(t,{u}_{0}){u}_{0})\cdot {u}_{0}-{F}_{u}(t,{u}_{0})\cdot {u}_{0},$$K(t)={F}_{uu}(t,{u}_{0}){u}_{0}-{F}_{u}(t,{u}_{0}),$$M(t)={F}_{uu}(t,{u}_{0}).$

Since, obviously,

$|{h}_{0}{(g,h)}_{A}|\le \frac{1}{2}\parallel g\parallel ({h}_{0}^{2}+{\parallel h\parallel}^{2})$

and, by Proposition 2.2,

$({L}_{A}h,h)\le -\kappa {\parallel h\parallel}^{2},$

it is enough to prove that

$H(t){h}_{0}^{2}+2{h}_{0}K(t)h(t)+(M(t)h(t))\cdot h(t)\ge {h}_{0}^{2}(1-\theta ){F}_{u}(t,{u}_{0}(t))\cdot {u}_{0}(t).$

This is trivial for all $t\in \mathbb{R}$ such that ${u}_{0}(t)=0$.
Hence, we may assume that ${u}_{0}(t)\ne 0$.
Then the matrix $M(t)$ is positive definite. Making use of the trivial identities

$({M}^{1/2}h)\cdot ({M}^{1/2}h)=(Mh)\cdot h\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}({M}^{-1/2}K)\cdot ({M}^{-1/2}K)=({M}^{-1}K)\cdot K,$

we obtain that

$H{h}_{0}^{2}+2{h}_{0}Kh+(Mh)\cdot h=(H-({M}^{-1}K)\cdot K))h{}_{0}{}^{2}+({M}^{1/2}h+{h}_{0}{M}^{-1/2}K)\cdot ({M}^{1/2}h+{h}_{0}{M}^{-1/2}K)$$\ge (H-({M}^{-1}K)\cdot K))h{}_{0}{}^{2}.$

Making use of the following easily verified inequality:

$H-({M}^{-1}K)\cdot K={F}_{u}(t,{u}_{0})\cdot {u}_{0}-{F}_{uu}{(t,{u}_{0})}^{-1}{F}_{u}(t,{u}_{0})\cdot {F}_{u}(t,{u}_{0})$

and inequality (3.2), we obtain the required result.
∎

The key Lemmas 4.1, 4.2 and 4.5 permit us to transfer some of auxiliary results of [13] to the present setting just by inspection of the proofs given there.
In particular, the set $\mathcal{\mathcal{N}}$ is a non-empty closed ${C}^{1,1}$-submanifold of ${H}^{1}$.
The tangent space to $\mathcal{\mathcal{N}}$ at a point ${u}_{0}\in \mathcal{\mathcal{N}}$ is ${T}_{{u}_{0}}=\mathrm{ker}{G}^{\prime}({u}_{0})$, while ${\overline{E}}_{{u}_{0}}$ is a complementary subspace to ${T}_{{u}_{0}}$.
Furthermore, given $R>0$, there exist $\rho >0$ and $C>0$ such that for every ${u}_{0}\in \mathcal{\mathcal{N}}$, with $\parallel {u}_{0}\parallel \le R$, there exists a ${C}^{1,1}$-diffeomorphism from the ρ-neighborhood of 0 in ${T}_{{u}_{0}}$ onto a neighborhood of ${u}_{0}$ in $\mathcal{\mathcal{N}}$ such that the Lipschitz constant of its derivative is not greater than *C*.

Now we introduce the following quantities:

$m=inf\{J(u):u\in \mathcal{\mathcal{N}}\},$${m}_{h}=inf\{{J}_{h}(u):u\in {\mathcal{\mathcal{N}}}_{h}\},h\in \mathcal{\mathscr{H}},$

where ${\mathcal{\mathcal{N}}}_{h}$ stands for the generalized Nehari manifold of the functional ${J}_{h}$.
These numbers are strictly positive by Corollary 4.3 and Remark 4.4.

#### Proposition 4.6.

*We have ${m}_{h}\mathrm{=}m$ for all $h\mathrm{\in}\mathcal{H}$.
*

#### Proof.

See the proof of [13, Proposition 5.4].
∎

Let us turn to Palais–Smale sequences. Recall that a sequence ${u}_{n}\in {H}^{1}$ is a Palais–Smale sequence for the functional *J* at a level *c* if $J({u}_{n})\to c$ and ${J}^{\prime}({u}_{n})\to 0$ in ${H}^{-1}$ (equivalently, $\nabla J({u}_{n})\to 0$ in ${H}^{1}$).
A Palais–Smale sequence for ${J|}_{\mathcal{\mathcal{N}}}$ at a level *c* is a sequence ${u}_{n}\in \mathcal{\mathcal{N}}$ such that $J({u}_{n})\to c$ and ${\nabla}_{\tau}J({u}_{n})\to 0$ in ${H}^{1}$, where ${\nabla}_{\tau}$ stands for the tangent component of the gradient.

As an immediate consequence of Lemma 4.2, we obtain the following corollary.

#### Corollary 4.7.

*Let ${u}_{n}\mathrm{\in}{H}^{\mathrm{1}}$ be a Palais–Smale sequence for the functional **J* at a level *c*. Then $c\mathrm{\ge}\mathrm{0}$ and ${u}_{n}$ is bounded in ${H}^{\mathrm{1}}$. Furthermore, ${u}_{n}\mathrm{\to}\mathrm{0}$ in ${H}^{\mathrm{1}}$ if and only if $c\mathrm{=}\mathrm{0}$.

As in [13, Propositions 6.2 and 6.3], we obtain the following result.

#### Proposition 4.8.

*Every Palais–Smale sequence for ${J\mathrm{|}}_{\mathcal{N}}$ is a Palais–Smale sequence for **J*.
Conversely, if ${u}_{n}\mathrm{\in}{H}^{\mathrm{1}}$ is a Palais–Smale sequence for *J* at a level $c\mathrm{>}\mathrm{0}$, then there exists a Palais–Smale sequence ${\stackrel{\mathrm{~}}{u}}_{n}\mathrm{\in}\mathcal{N}$ for ${J\mathrm{|}}_{\mathcal{N}}$ such that $\mathrm{\parallel}{u}_{n}\mathrm{-}{\stackrel{\mathrm{~}}{u}}_{n}\mathrm{\parallel}\mathrm{\to}\mathrm{0}$ and, hence, $c\mathrm{\ge}m$.

#### Corollary 4.9.

*If ${u}_{n}\mathrm{\in}{H}^{\mathrm{1}}$ is a Palais–Smale sequence for **J* at a positive level, then $\mathrm{lim\; inf}\mathrm{\parallel}{u}_{n}{\mathrm{\parallel}}_{\mathrm{\infty}}\mathrm{\ge}{\epsilon}_{\mathrm{0}}$, where ${\epsilon}_{\mathrm{0}}\mathrm{>}\mathrm{0}$ is the constant from Corollary 4.3.

Inspecting the proof of [13, Proposition 6.4], we obtain the following statement on the existence of Palais–Smale sequences.
Notice that the main ingredient of the proof is the negative gradient flow of the functional ${J|}_{\mathcal{\mathcal{N}}}$.

#### Proposition 4.10.

*For any $\epsilon \mathrm{>}\mathrm{0}$, there exists a Palais–Smale sequence ${u}_{n}$ for ${J\mathrm{|}}_{\mathcal{N}}$, hence for **J*, at a level $c\mathrm{\in}\mathrm{[}m\mathrm{,}m\mathrm{+}\epsilon \mathrm{]}$ such that

$lim\parallel {u}_{n+1}-{u}_{n}\parallel =0.$

Also we need the following special version of concentration-compactness lemma (cf.
[13, Lemma 6.1]).

#### Lemma 4.11.

*Let ${u}_{n}\mathrm{\in}{H}^{\mathrm{1}}$ be a Palais–Smale sequence for **J* at a level $c\mathrm{\in}\mathrm{[}m\mathrm{,}\mathrm{2}\mathit{}m\mathrm{)}$. Suppose that
${u}_{n}\mathrm{\to}{u}_{\mathrm{0}}$ weakly in ${H}^{\mathrm{1}}$.

(a)

*If *
${u}_{0}\ne 0$
*, then *
${u}_{n}\to {u}_{0}$
* strongly in *
${H}^{1}$
*, and *
${u}_{0}$
* is a critical point of the
functional *
*J*
* with critical value *
*c*.

(b)

*If *
${u}_{0}=0$
*, then there exists a sequence *
${t}_{n}\in \mathbb{R}$
* such that
*
$|{t}_{n}|\to \mathrm{\infty}$
* and a nontrivial critical point *
${v}_{h}$
* of *
${J}_{h}$
* for some *
$h\in \mathcal{\mathscr{H}}$
* at the level *
*c*
* with the property that along a subsequence, *
${T}_{{t}_{n}}{u}_{n}\to {v}_{h}$
* and *
${T}_{-{t}_{n}}{v}_{h}-{u}_{n}\to 0$
* strongly in *
${H}^{1}$.

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