In order to prove our main conclusions, we need the following lemmas. First, in (E_{0}^{alpha}), let (V=E^{-}oplus E^{0}) and (X=E^{+}), then the dimension of subspace V is finite and (E_{0}^{alpha}=V oplus X).

Lemma 3.1

Assuming that the conditions (I1), (G3), and (G4) are satisfied, we can find constants (rho , sigma , zeta ^{*}> 0) such that (Phi{|_{partial {B_{rho }} cap X }}geq sigma ), provided that (zeta in [0,zeta ^{*})).

Proof

Based on (G3) and (G4), for any (varepsilon >0), we can find a constant (c_{varepsilon}) such that for (tin [0,T]),

$$begin{aligned} G(t,u)leq varepsilon vert u vert ^{2}+c_{varepsilon} vert u vert ^{mu}, end{aligned}$$

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which shows that

$$begin{aligned} int _{0}^{T}G(t,u),dt leq & varepsilon int _{0}^{T} vert u vert ^{2} ,dt+c_{ varepsilon} int _{0}^{T} vert u vert ^{mu},dt \ leq &varepsilon T S^{2}_{infty} Vert u Vert ^{2}_{alpha}+c_{varepsilon} T S^{mu}_{infty} Vert u Vert ^{mu}_{alpha}. end{aligned}$$

Hence, for (uin E_{0}^{alpha}), by (I1), one has

$$begin{aligned} Phi (u) geq & kappa Vert u Vert _{alpha}^{2}+ sum _{j = 1}^{m} { int _{0}^{u(t_{j} )} { bigl(I_{1j} (t)+I_{2j} (t) bigr),dt } } + int _{0}^{T} biglvert {}_{0} D_{t}^{alpha}u(t) bigrvert ^{2} biglvert u(t) bigrvert ^{2},dt \ &{} – int _{0}^{T} {G bigl(t,u(t) bigr)},dt – frac{zeta}{nu} int _{0}^{T} h(t) biglvert u(t) bigrvert ^{ nu},dt \ geq & kappa Vert u Vert _{alpha}^{2} – int _{0}^{T} {G bigl(t,u(t) bigr)},dt – frac{zeta}{nu} int _{0}^{T} h(t) biglvert u(t) bigrvert ^{nu},dt \ geq & Vert u Vert ^{nu}_{alpha} biggl( bigl(kappa – varepsilon TS^{2}_{infty} bigr) Vert u Vert ^{2-nu}_{alpha}-c_{varepsilon} TS^{mu}_{infty} Vert u Vert ^{mu – nu}_{alpha}-frac{zeta}{nu} TS^{v}_{infty} Vert h Vert _{L^{1}} biggr). end{aligned}$$

Letting (varepsilon =frac{zeta}{2T{S}^{2}_{infty}}), leads to

$$ Phi (u)geq Vert u Vert ^{nu}_{alpha} biggl( frac{kappa}{2} Vert u Vert ^{2-nu}_{ alpha}-c_{varepsilon} TS^{mu}_{infty} Vert u Vert ^{mu -nu}_{alpha}- frac{zeta}{nu} TS^{v}_{infty} Vert h Vert _{L^{1}} biggr).$$

Set

$$ y(t)=frac{kappa}{2}t^{2-nu}-c_{varepsilon} TS^{mu}_{infty}t^{ mu -nu},quad tgeq 0.$$

Clearly, there exists a (rho = [ frac{kappa (2-nu )}{2c_{varepsilon} TS^{mu}_{infty}(mu -nu )} ]^{frac{1}{mu -2}}) such that

$$ y(rho )=max_{tgeq 0}y(t)=frac{kappa (mu -2)}{2(mu -nu )} biggl[ frac{kappa (2-nu )}{2c_{varepsilon} TS^{mu}_{infty}(mu -nu )} biggr]^{frac{2-nu}{mu -2}}>0.$$

Therefore, we can find

$$ zeta ^{*}= frac{nu kappa (mu -2)}{TS^{v}_{infty}(mu -nu ) Vert h Vert _{L^{1}}} biggl[ frac{kappa (2-nu )}{2c_{varepsilon} TS^{mu}_{infty}(mu -nu )} biggr]^{frac{2-nu}{mu -2}}.$$

If (zeta in [0,zeta ^{*})), there exists a constant (sigma >0) such that (Phi{|_{X cap partial {B_{rho }}}}geq sigma ). □

Lemma 3.2

If the conditions (I2) and (G1) are satisfied, there exists a constant (l>0) such that for each finitedimensional subspace (widetilde{X}subset E_{0}^{alpha}), (Phi (u) le 0), (forall u in {widetilde{X}}backslash {B_{l}}), provided that (zeta in [0,+infty )).

Proof

Actually, for (zeta in [0,+infty )), the key point is to prove that (Phi (u)) is anticoercive, i.e.,

$$begin{aligned} Phi (u)to -infty quad text{as } Vert u Vert _{alpha}to +infty text{ for } uin widetilde{X}. end{aligned}$$

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If not, let the sequence ({u_{n}}subset widetilde{X}) and τR such that

$$begin{aligned} Phi (u_{n})geq tau quad text{when } Vert u_{n} Vert _{alpha}to +infty text{ as } nto +infty . end{aligned}$$

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Setting (omega _{n}=frac{u_{n}}{|u_{n}|_{alpha}}), then (|omega _{n}|_{alpha}=1). Since dim (widetilde{X}<infty ), we can find a subsequence of ({omega _{n}}) (named again ({omega _{n} })) such that (omega _{n}rightarrow omega ) in (E_{0}^{alpha}), which implies (|omega |_{alpha}=1). From (omega neq 0), one has (|u_{n}(t)|to +infty ) as (nto +infty ). Define

$$ W(t,u)=G(t,u)+frac{zeta}{nu}h(t) vert u vert ^{nu}- frac{1}{2}b(t) vert u vert ^{2}.$$

In view of (G1), it follows that for any (tin [0,T]),

$$begin{aligned} lim_{ vert u vert to +infty} frac{W(t,u)}{ vert u vert ^{4}}=+ infty . end{aligned}$$

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Moreover, by a standard measure estimation on a finite-dimensional space (see [4]), it follows that there exists a positive constant (epsilon >0 ) such that

$$begin{aligned} text{meas} bigl{ tin [0,T]: biglvert u(t) bigrvert geq epsilon Vert u Vert _{alpha} bigr} geq epsilonquad text{for } uin widetilde{X} setminus {0}. end{aligned}$$

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Let (Pi ={tin [0,T]:|u(t)|geq epsilon |u|_{alpha}}). Based on (3.4), it means that for (frac{2S_{infty}^{2}}{epsilon ^{4}}>0), there exists (eta >0) such that

$$begin{aligned} W(t,u)geq frac{2S_{infty}^{2}}{epsilon ^{4}} vert u vert ^{4}quad text{for } vert u vert geq eta . end{aligned}$$

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Hence, for (uin widetilde{X}) with (|u|_{alpha}geq frac{eta}{epsilon}), we can obtain that

$$begin{aligned} W(t,u)geq 2S_{infty}^{2} Vert u Vert ^{4}_{alpha} quad text{for } tin Pi . end{aligned}$$

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Let (|u_{n}|_{alpha}geq frac{eta}{epsilon}) for n large enough. From (I2), one has

$$begin{aligned} Phi (u) =& frac{1}{2} int _{0}^{T} biglvert {}_{0} D_{t}^{alpha}u_{n}(t) bigrvert ^{2},dt + sum_{j = 1}^{m} { int _{0}^{u_{n}(t_{j} )} { bigl(I_{1j} (t)+I_{2j} (t) bigr),dt } } \ &{} + int _{0}^{T} biglvert {}_{0} D_{t}^{alpha}u_{n}(t) bigrvert ^{2} biglvert u_{n}(t) bigrvert ^{2},dt \ &{} – int _{0}^{T} {W bigl(t,u_{n}(t) bigr)},dt \ leq & frac{1}{2} Vert u_{n} Vert ^{2}_{alpha}+ sum_{j = 1}^{m} a_{1j}S_{infty} Vert u_{n} Vert _{alpha}+ sum _{j = 1}^{m} a_{2j}S_{infty} Vert u_{n} Vert _{alpha}+sum_{j = 1}^{m} d_{1j}S^{ gamma _{1j}+1}_{infty} Vert u_{n} Vert ^{gamma _{1j}+1}_{alpha} \ &{}+sum_{j = 1}^{m} d_{2j}S^{gamma _{2j}+1}_{infty} Vert u_{n} Vert ^{gamma _{2j}+1}_{alpha}+S^{2}_{infty} Vert u_{n} Vert ^{4}_{alpha}- int _{0}^{T} {W bigl(t,u(t) bigr)},dt \ =& Vert u_{n} Vert ^{4}_{alpha} Biggl( frac{1}{2 Vert u_{n} Vert ^{2}_{alpha}}+sum_{j = 1}^{m} a_{1j}S_{infty}frac{1}{ Vert u_{n} Vert ^{3}_{alpha}}+sum _{j = 1}^{m} a_{2j}S_{infty} frac{1}{ Vert u_{n} Vert ^{3}_{alpha}} \ &{} +sum_{j = 1}^{m} d_{1j}S_{infty}^{gamma _{1j}+1} frac{1}{ Vert u_{n} Vert ^{3-gamma _{1j}}_{alpha}} \ &{}+ {sum_{j = 1}^{m} d_{2j}S_{infty}^{gamma _{2j}+1} frac{1}{ Vert u_{n} Vert ^{3-gamma _{j}}_{alpha}}}+S_{infty}^{2}- int _{0}^{T} frac{W(t,u_{n})}{ Vert u_{n} Vert ^{4}_{alpha}},dt Biggr) \ leq & Vert u_{n} Vert ^{4}_{alpha} Biggl( frac{1}{2 Vert u_{n} Vert ^{2}_{alpha}}+sum_{j = 1}^{m} a_{1j}S_{infty}frac{1}{ Vert u_{n} Vert ^{3}_{alpha}}+sum _{j = 1}^{m} a_{2j}S_{infty} frac{1}{ Vert u_{n} Vert ^{3}_{alpha}} \ &{} +sum_{j = 1}^{m} d_{1j}S_{infty}^{gamma _{1j}+1} frac{1}{ Vert u_{n} Vert ^{3-gamma _{1j}}_{alpha}} \ &{}+ {sum_{j = 1}^{m} d_{2j}S_{infty}^{gamma _{2j}+1} frac{1}{ Vert u_{n} Vert ^{3-gamma _{j}}_{alpha}}}+S_{infty}^{2}- int _{ Pi} frac{W(t,u_{n})}{ Vert u_{n} Vert ^{4}_{alpha}},dt Biggr) \ to& -inftyquad text{if } Vert u_{n} Vert _{alpha}to + infty text{ as } n to +infty , end{aligned}$$

which is in contradiction to (3.3). Hence, (Phi (u)) is anticoercive. Therefore, there exists a constant (l>0) such that (Phi (u) le 0), (forall u in {widetilde{X}}backslash {B_{l}}) for (zeta in [0,+infty )). □

Lemma 3.3

If the assumptions (I2), (I3), (G1), (G2), and (G4) are satisfied, (Phi (u) ) meets the (PS)-condition, provided that (zeta in [0,+infty )).

Proof

Let ({ u_{n} } subset E_{0}^{alpha}) such that (Phi (u_{n})) is bounded and (Phi ‘(u_{n})to 0) as (nto +infty ), which implies that there exists a constant (beta >0) such that

$$ biglvert Phi (u_{n}) bigrvert leq beta ,qquad biglVert Phi ‘(u_{n}) bigrVert _{(E_{0}^{alpha})^{*}} leq beta .$$

We claim that the sequence ({ u_{n} }) is bounded. If not, let (| u_{n}|to +infty ) as (nto +infty ). Setting (omega _{n}=frac{u_{n}}{|u_{n}|_{alpha}} ), it follows that (omega _{n}) is bounded in (E_{0}^{alpha}). Noting that (E_{0}^{alpha}) is a reflexive Banach space, it implies that ({omega _{n}}) has a convergent subsequence (named again ({omega _{n} })) such that (omega _{n}rightharpoonup omega ) in (E_{0}^{alpha}) and (omega _{n}rightarrow omega ) uniformly in C.

In view of (I2), one has

$$begin{aligned} int _{0}^{T} W bigl(t,u_{n}(t) bigr) ,dt =&frac{1}{2} int _{0}^{T} biglvert {}_{0} D_{t}^{ alpha}u_{n}(t) bigrvert ^{2},dt + sum_{j = 1}^{m} { int _{0}^{u_{n}(t_{j} )} { bigl(I_{1j} (t)+I_{2j} (t) bigr),dt } } \ &{}+ int _{0}^{T} biglvert {}_{0} D_{t}^{alpha}u_{n}(t) bigrvert ^{2} biglvert u_{n}(t) bigrvert ^{2},dt- Phi (u_{n}) \ leq & frac{1}{2} Vert u_{n} Vert ^{2}_{alpha}+ sum_{j = 1}^{m} a_{1j}S_{infty} Vert u_{n} Vert _{alpha}+ sum _{j = 1}^{m} a_{2j}S_{infty} Vert u_{n} Vert _{alpha} \ &{}+sum_{j = 1}^{m} d_{1j}S^{ gamma _{1j}+1}_{infty} Vert u_{n} Vert ^{gamma _{1j}+1}_{alpha} \ &{}+sum_{j = 1}^{m} d_{2j}S^{gamma _{2j}+1}_{infty} Vert u_{n} Vert ^{gamma _{2j}+1}_{alpha} +S^{2}_{infty} Vert u_{n} Vert ^{4}_{alpha}+ beta , end{aligned}$$

which shows that for n large enough,

$$begin{aligned} int _{0}^{T} frac{W(t,u_{n})}{ Vert u_{n} Vert ^{4}_{alpha}},dtleq S^{2}_{ infty}+o(1). end{aligned}$$

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Based on the continuity of g, we can find a constant (vartheta _{1}>0) such that

$$begin{aligned} biglvert ug(t,u)-theta G(t,u) bigrvert le vartheta _{1} quad text{for } vert u vert leq L_{1}, tin [0,T], end{aligned}$$

which together with (G2) yields

ug(t,u)θG(t,u) M 1 |u | 2 ϑ 1 for |u|R,t[0,T].

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In view of (I3) and (3.9), we have

$$begin{aligned} theta beta +beta Vert u_{n} Vert _{alpha} geq & theta Phi (u_{n} ) – bigllangle Phi ‘(u_{n}),u_{n} bigrrangle \ =& biggl(frac{theta}{2}-1 biggr) Vert u_{n} Vert ^{2}_{alpha} +( theta -4) int _{0}^{T} biglvert {}_{0} D_{t}^{alpha}u_{n}(t) bigrvert ^{2} biglvert u_{n}(t) bigrvert ^{2},dt \ &{} + biggl( frac{theta}{2}-1 biggr) int _{0}^{T} b(t)u_{n}^{2}(t) ,dt \ &{}+theta sum_{j = 1}^{m} int _{0}^{{u_{n}}(t_{j} )} bigl(I_{1j} (t)+I_{2j} (t) bigr),dt -sum_{j = 1}^{m} { bigl(I_{1j} bigl({u_{n}} (t_{j} ) bigr)+I_{1j} bigl({u_{n}} (t_{j} ) bigr) bigr){u_{n}}(t_{j} )} \ &{}+ int _{0}^{T} { bigl(u_{n} (t) g bigl(t,u_{n} (t) bigr) – theta G bigl(t,u_{n} (t) bigr) bigr)},dt-zeta frac{theta -nu}{nu} int _{0}^{T} h(t) biglvert u_{n}(t) bigrvert ^{nu},dt \ geq & biggl(frac{theta}{2}-1 biggr) Vert u_{n} Vert ^{2}_{alpha} + int _{0}^{T} { bigl(u_{n} (t) g bigl(t,u_{n} (t) bigr) – theta G bigl(t,u_{n} (t) bigr) bigr)},dt \ &{}+ biggl(frac{theta}{2}-1 biggr) int _{0}^{T} b(t)u_{n}^{2}(t) ,dt-zeta frac{theta -nu}{nu} int _{0}^{T} h(t) biglvert u_{n}(t) bigrvert ^{nu},dt \ geq & biggl(frac{theta}{2}-1 biggr) Vert u_{n} Vert ^{2}_{alpha}- biggl( M_{1}T+ biggl( frac{theta}{2}-1 biggr) Vert b Vert _{L^{1}} biggr) Vert u_{n} Vert _{infty}^{2} \ &{}- zeta frac{theta -nu}{nu}S^{nu}_{infty} Vert h Vert _{L^{1}} Vert u_{n} Vert ^{nu}_{ alpha}- vartheta _{1}T, end{aligned}$$

which means that there exists a positive constant (vartheta _{2}) such that

$$ lim_{nto +infty} Vert omega _{n} Vert _{infty}= lim_{nto +infty} frac{ Vert u_{n} Vert _{infty}}{ Vert u_{n} Vert _{alpha}}geq vartheta _{2} >0.$$

Therefore, we can obtain (omega neq 0). Define

$$ Xi _{1}= bigl{ tin [0,T]:omega neq 0 bigr} ,qquad Xi _{2}=[0,T]setminus Xi _{1}.$$

In view of (G1), there exists a constant (vartheta _{3}>0) such that (G(t,u)geq 0), (text{for} tin [0,T]), (|u|geq vartheta _{3}), which together with (G4) yields that there exist constants (vartheta _{4}, vartheta _{5}>0) such that

G(t,u) ϑ 4 u 2 ϑ 5 for t[0,T],uR.

Based on Fatou’s lemma, it follows that

$$ mathop {lim inf} _{nto +infty} int _{Xi _{2}} frac{G(t,u_{n})}{ Vert u_{n} Vert ^{4}_{alpha}},dt>-infty .$$

By (G1), for (tin [0,T]), we can obtain that

$$begin{aligned}& mathop {lim inf} _{nto +infty} int _{0}^{T} frac{G(t,u_{n})}{ Vert u_{n} Vert ^{4}_{alpha}},dt \& quad =mathop {lim inf} _{nto +infty} biggl( int _{Xi _{1}} frac{G(t,u_{n})}{ vert u_{n} vert ^{4}} vert omega _{n} vert ^{4},dt+ int _{Xi _{2}} frac{G(t,u_{n})}{ vert u_{n} vert ^{4}} vert omega _{n} vert ^{4},dt biggr)to +infty , end{aligned}$$

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which is in contradiction to (3.8). Thus, ({u_{n}}) is bounded, which implies that ({u_{n}}) possesses a convergent subsequence (named again ({u_{n} })) such that (u_{n}=u_{n}^{+}+u_{n}^{-}+u_{n}^{0}rightharpoonup u= u^{+}+u^{-}+u^{0}) and (u_{n}^{+}rightharpoonup u^{+}) in (E^{alpha}_{0}). Moreover, (u_{n}rightarrow u) and (u_{n}^{+}rightarrow u^{+}) uniformly in C. It should be mentioned that the dimensions of subspaces (E^{-}) and (E^{0}) are finite. Hence, (u_{n}^{-}rightarrow u^{-}) and (u_{n}^{0}rightarrow u^{0}) in (E^{alpha}_{0}). Furthermore, if (nto +infty ), one has

$$begin{aligned}& bigllangle Phi ‘(u_{n})-Phi ‘(u),u_{n}^{+}-u^{+} bigrrangle rightarrow 0, \& int _{0}^{T} b(t) bigl(u_{n}(t)-u(t) bigr) bigl(u_{n}^{+}(t)-u^{+}(t) bigr),dt rightarrow 0, \& sum_{j = 1}^{m} bigl(I_{1j} bigl(u_{n}(t_{j} ) bigr)-I_{1j} bigl(u(t_{j} ) bigr) bigr) bigl(u_{n}^{+}(t_{j} )-u^{+}(t_{j} ) bigr)to 0, \& sum_{j = 1}^{m} bigl(I_{2j} bigl(u_{n}(t_{j} ) bigr)-I_{2j} bigl(u(t_{j} ) bigr) bigr) bigl(u_{n}^{+}(t_{j} )-u^{+}(t_{j} ) bigr)to 0, \& int _{0}^{T} bigl(f bigl(t,u_{n}(t) bigr)-f bigl(t,u(t) bigr) bigr) bigl(u_{n}^{+}(t)-u^{+}(t) bigr),dtto 0, \& int _{0}^{T} bigl( biglvert {}_{0} D_{t}^{alpha}u_{n}(t) bigrvert ^{2}u_{n}(t)- biglvert {}_{0} D_{t}^{ alpha}u(t) bigrvert ^{2}u(t) bigr) bigl(u_{n}^{+}(t)-u^{+}(t) bigr),dtto 0 end{aligned}$$

and

$$begin{aligned}& int _{0}^{T} bigl( biglvert u_{n}(t) bigrvert ^{2}{}_{0} D_{t}^{alpha}u_{n}(t)- biglvert u(t) bigrvert ^{2}{}_{0} D_{t}^{alpha}u(t) bigr) bigl({}_{0} D_{t}^{alpha}u_{n}^{+}(t)-{}_{0} D_{t}^{ alpha}u^{+}(t) bigr),dt \& quad = int _{0}^{T} bigl( bigl( biglvert u_{n}(t) bigrvert ^{2}- biglvert u(t) bigrvert ^{2} bigr){}_{0} D_{t}^{alpha}u_{n}(t) – biglvert u(t) bigrvert ^{2} bigl({}_{0} D_{t}^{alpha}u_{n}(t)-{}_{0} D_{t}^{alpha}u(t) bigr) bigr) \& qquad {}times bigl({}_{0} D_{t}^{ alpha}u_{n}^{+}(t)-{}_{0} D_{t}^{alpha}u^{+}(t) bigr),dt \& quad = int _{0}^{T} biglvert u(t) bigrvert ^{2} biglvert {}_{0} D_{t}^{alpha}u_{n}^{+}(t)-{}_{0} D_{t}^{ alpha}u^{+}(t)) bigrvert ^{2} ,dt+o(1), end{aligned}$$

which implies that

$$begin{aligned}& bigllangle Phi ‘(u_{n})-Phi ‘(u),u_{n}^{+}-u^{+} bigrrangle rightarrow 0 \& quad = int _{0}^{T} biglvert {}_{0} D_{t}^{alpha}u_{n}^{+}(t)-{}_{0} D_{t}^{ alpha}u^{+}(t) bigrvert ^{2} ,dt+ int _{0}^{T} biglvert u(t) bigrvert ^{2} biglvert {}_{0} D_{t}^{alpha}u_{n}^{+}(t)-{}_{0} D_{t}^{alpha}u^{+}(t) bigrvert ^{2} ,dt+o(1). end{aligned}$$

Since the norm of (E_{0}^{alpha}) is equivalent to (|_{0} D_{t}^{alpha} u|_{L^{2}}), it is clear that (u_{n}^{+}rightarrow u^{+}) in (E^{alpha}_{0}). Thus, (u_{n}rightarrow u) in (E^{alpha}_{0}). Therefore, (Phi (u)) satisfies the (PS)-condition. □

Proof of Theorem 1.1

From Lemma 3.1, Lemma 3.2, and Lemma 3.3, Theorem 1.1 can be proven immediately by Lemma 2.7. □

Proof of Corollary 1.4

The proof is similar to Theorem 1.1. Therefore, we omit the detail. □

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