The following Symmetric Mountain Pass Theorem is crucial in proving the existence of infinitely many solutions with 4-superlinear nonlinearities.

Proposition 2.1

(Symmetric Mountain Pass Theorem [21])

Let X be an infinitedimensional Banach space, (X=Yoplus Z), where Y is a finitedimensional space. If (Jin C^{1}(X,mathbf{R})) satisfies the ((Ce)) condition, and

((J_{1})):

(J(0)=0), (J(-u)=J(u)) for all (uin X);

((J_{2})):

there exist two constants (delta ,alpha >0) such that (J|_{partial B_{delta }cap Z}geq alpha );

((J_{3})):

for any finitedimensional subspace (widetilde{X}subset X), there exists (R=R(widetilde{X})>0) such that (J(u)leq 0) on (widetilde{X}setminus B_{R}),

then J possesses an unbounded sequence of critical values.

In [21], the Symmetric Mountain Pass Theorem is established under the ((PS)) condition. Since the Deformation Theorem is still valid under the ((Ce)) condition, we see that the Symmetric Mountain Pass Theorem also holds under the ((Ce)) condition. The following embedding lemma, which follows from [17] or [33], plays a significant role in recovering the compactness result.

Lemma 2.1

If V satisfies the condition ((V_{1})), then the embedding map from E into (l^{q}) is compact for (2leq qleq +infty ).

With the help of Lemma 2.1, we have the following compactness result.

Lemma 2.2

Under the assumption of Theorem 1.1, the functional I satisfies the ((Ce)) condition.

Proof

Suppose that ({u^{n}} ) is a ((Ce)) sequence,

$$ biglvert Ibigl(u^{n}bigr) bigrvert leq M,qquad bigl(1+ biglVert u^{n} bigrVert bigr)I’bigl(u^{n} bigr)rightarrow 0quad text{as } nrightarrow infty . $$

It suffices to prove that ({u^{n}}) has a converging subsequence in E. We first obtain that ({u_{n}} ) is bounded in E. Otherwise, (|u^{n}|rightarrow +infty ) as (nrightarrow infty ). Let (v^{n}=u^{n}/|u^{n}|). Moving, if necessary, to a subsequence, we assume (v^{n}rightharpoonup v) in E, by Lemma 2.1, (v^{n}rightarrow v) in (l^{q}) and (v_{k}^{n}rightarrow v_{k}) for any (kin mathbf{Z}) as (nrightarrow infty ). There are only two cases (v=0) or (vneq 0). If (v=0), by ((f_{3})), we have

$$ begin{aligned} frac{1}{ Vert u^{n} Vert ^{2}}(M+1)&geq frac{1}{ Vert u^{n} Vert ^{2}} biggl(I bigl(u^{n}bigr)-frac{1}{4} bigllangle I’ bigl(u^{n}bigr),u^{n} bigrrangle biggr) \ &geq frac{1}{4}min {a,1}+frac{1}{ Vert u^{n} Vert ^{2}}sum _{kin mathbf{Z}}biggl(frac{1}{4}f_{k} bigl(u^{n}_{k}bigr)u_{k}^{n}-F_{k} bigl(u^{n}_{k}bigr)biggr) \ &geq frac{1}{4}min {a,1}-alpha sum_{kin mathbf{Z}} biglvert v_{k}^{n} bigrvert ^{2}, end{aligned} $$

which implies that (frac{1}{4}min {a,1}leq 0). That is impossible. If (vneq 0), then (Omega _{1}:={kin mathbf{Z}| v_{k}neq 0}neq emptyset ). For any (kin Omega _{1}), we have (|u_{k}^{n}|rightarrow +infty ) as (nrightarrow infty ). By ((f_{2})), one obtains that for any (kin Omega _{1}),

$$ frac{F_{k}(u^{n}_{k})}{ vert u^{n}_{k} vert ^{4}} biglvert v_{k}^{n} bigrvert ^{4}rightarrow + infty quad text{as } nrightarrow infty , $$

combined with Fatou’s Lemma, which implies that

$$ sum_{kin Omega _{1}}frac{F_{k}(u^{n}_{k})}{ vert u^{n}_{k} vert ^{4}} biglvert v_{k}^{n} bigrvert ^{4} rightarrow + infty quad text{as } nrightarrow infty . $$

(2)

It follows from ((f_{2})) that there exists (L_{1}>0) such that

$$ F_{k}(t)geq 0 quadtext{for any } kin mathbf{Z} text{ and } vert t vert geq L_{1}. $$

(3)

By ((f_{1})), we obtain (|F_{k}(t)|leq Ct^{2} quadtext{for any } kin mathbf{Z} text{ and } |t| leq L_{1}). Combining with (3), we have

$$ F_{k}(t)geq -Ct^{2} quadtext{for any } kin mathbf{Z} text{ and } t in mathbf{R}. $$

Hence, we obtain

$$ begin{aligned} &sum_{kin mathbf{Z}setminus Omega _{1}} frac{F_{k}(u^{n}_{k})}{ Vert u^{n} Vert ^{4}} geq -frac{C}{ Vert u^{n} Vert ^{4}} sum_{kin mathbf{Z}setminus Omega _{1}} biglvert u^{n}_{k} bigrvert ^{2} geq -C frac{ Vert u^{n} Vert _{2}^{2}}{ Vert u^{n} Vert ^{4}}geq -frac{C}{ Vert u^{n} Vert ^{2}}, end{aligned} $$

which implies that

$$ begin{aligned} liminf_{ nrightarrow infty }sum _{kin mathbf{Z} setminus Omega _{1}}frac{F_{k}(u^{n}_{k})}{ Vert u^{n} Vert ^{4}}geq 0. end{aligned} $$

(4)

Note that

$$ Ibigl(u^{n}bigr)+sum_{kin mathbf{Z}}F_{k} bigl(u^{n}_{k}bigr)leq frac{1}{2}max {a,1} biglVert u^{n} bigrVert ^{2}+frac{b}{4} biglVert u^{n} bigrVert ^{4}. $$

Dividing by (|u^{n}|^{4}) on both sides and letting (nrightarrow infty ), we obtain

$$ begin{aligned} frac{b}{4}&geq limsup_{ nrightarrow infty } sum_{k in mathbf{Z}}frac{F_{k}(u^{n}_{k})}{ Vert u^{n} Vert ^{4}} \ &geq limsup_{ nrightarrow infty } biggl(sum_{kin mathbf{Z} setminus Omega _{1}} frac{F_{k}(u^{n}_{k})}{ Vert u^{n} Vert ^{4}}+sum_{k in Omega _{1}}frac{F_{k}(u^{n}_{k})}{ Vert u^{n} Vert ^{4}} biggr) rightarrow +infty , end{aligned} $$

via (2) and (4), which is impossible. In any case, we obtain a contradiction and hence ({u^{n}} ) is bounded in E.

Moving if necessary to a subsequence, we can assume (u^{n}rightharpoonup u) in E. It follows that

$$begin{aligned}& bigllangle I’bigl(u^{n} bigr)-I'(u),u^{n}-ubigrrangle \& quad =bigl(a+b biglVert Delta u^{n} bigrVert _{2}^{2}bigr)sum_{kin mathbf{Z}}Delta u^{n}_{k-1} Delta bigl(u^{n}_{k-1}-u_{k-1} bigr)+sum_{kin mathbf{Z}}V_{k}u^{n}_{k} bigl(u^{n}_{k}-u_{k}bigr) \& qquad {}-bigl(a+b Vert Delta u Vert _{2}^{2}bigr)sum _{kin mathbf{Z}}Delta u_{k-1} Delta bigl(u^{n}_{k-1}-u_{k-1}bigr)-sum _{kin mathbf{Z}}V_{k}u_{k} bigl(u^{n}_{k}-u_{k} bigr) \& qquad {}-sum_{kin mathbf{Z}}bigl(f_{k} bigl(u^{n}_{k}bigr)-f_{k}(u_{k}) bigr) bigl(u^{n}_{k}-u_{k}bigr) \& quad =bigl(a+b biglVert Delta u^{n} bigrVert _{2}^{2}bigr)sum_{kin mathbf{Z}} biglvert Delta bigl(u^{n}_{k-1}-u_{k-1}bigr) bigrvert ^{2}+ sum_{kin mathbf{Z}}V_{k} biglvert u^{n}_{k}-u_{k} bigrvert ^{2} \& qquad {}-bbigl( Vert Delta u Vert _{2}^{2}- biglVert Delta u^{n} bigrVert _{2}^{2}bigr)sum _{kin mathbf{Z}}Delta u_{k-1}Delta bigl(u^{n}_{k-1}-u_{k-1}bigr) \& qquad {}-sum_{kin mathbf{Z}}bigl(f_{k} bigl(u^{n}_{k}bigr)-f_{k}(u_{k}) bigr) bigl(u^{n}_{k}-u_{k}bigr) \& quad geq min {a,1} biglVert u^{n}-u bigrVert ^{2} -bbigl( Vert Delta u Vert _{2}^{2}- biglVert Delta u^{n} bigrVert _{2}^{2}bigr)sum _{kin mathbf{Z}}Delta u_{k-1}Delta bigl(u^{n}_{k-1}-u_{k-1}bigr) \& qquad {}-sum_{kin mathbf{Z}}bigl(f_{k} bigl(u^{n}_{k}bigr)-f_{k}(u_{k}) bigr) bigl(u^{n}_{k}-u_{k}bigr). end{aligned}$$

One has

$$ begin{aligned} min {a,1} biglVert u^{n}-u bigrVert ^{2} leq{}& bigllangle I’ bigl(u^{n}bigr)-I'(u),u^{n}-u bigrrangle \ &{}+sum_{kin mathbf{Z}}bigl(f_{k} bigl(u^{n}_{k}bigr)-f_{k}(u_{k}) bigr) bigl(u^{n}_{k}-u_{k}bigr) \ &{}+bbigl( Vert Delta u Vert _{2}^{2}- biglVert Delta u^{n} bigrVert _{2}^{2}bigr)sum _{kin mathbf{Z}}Delta u_{k-1}Delta bigl(u^{n}_{k-1}-u_{k-1}bigr). end{aligned} $$

(5)

By the boundedness of ({u^{n}}) and (u^{n}rightharpoonup u) in E, it is obvious that

$$ bigllangle I’_{n}bigl(u^{n} bigr)-I'(u),u^{n}-ubigrrangle rightarrow 0quad text{as } nrightarrow infty .$$

(6)

By ((f_{1})), Lemma 2.1 and Lebesgue’s dominated convergence theorem

$$ sum_{kin mathbf{Z}}bigl(f_{k} bigl(u^{n}_{k}bigr)-f_{k}(u_{k}) bigr) bigl(u^{n}_{k}-u_{k}bigr) rightarrow 0 quad text{as } nrightarrow infty . $$

(7)

Let us consider the functional (P: Erightarrow mathbf{R}),

$$ P(w)=sum_{kin mathbf{Z}}Delta u_{k-1}Delta w_{k-1}. $$

Since (|P(w)|leq |u||w|), we can deduce that P is a continuous linear functional on E. By (u^{n}rightharpoonup u) in E, we obtain

$$ Pbigl(u^{n}-ubigr)=sum_{kin mathbf{Z}}Delta u_{k-1}Delta bigl(u^{n}_{k-1}-u_{k-1} bigr) rightarrow 0 quad text{as } nrightarrow infty . $$

By the boundedness of ({u^{n}}) in E, we have

$$ bbigl( Vert Delta u Vert _{2}^{2}- biglVert Delta u^{n} bigrVert _{2}^{2} bigr)sum_{kin mathbf{Z}} Delta u_{k-1}Delta bigl(u^{n}_{k-1}-u_{k-1}bigr)rightarrow 0 quad text{as } nrightarrow infty . $$

(8)

It follows from (5)–(8) that (|u^{n}-u|rightarrow 0) as (nrightarrow infty ). Thus, (u^{n}rightarrow u) strongly in E as (nrightarrow infty ). □

Let ({e^{j}}) be an orthonormal basis of E and define (X_{j}=operatorname{span}{e_{j}}), (Y_{m}=bigoplus _{j=1}^{m}X_{j}) and (Z_{m}=overline{bigoplus _{j=m+1}^{infty }X_{j}}) for any (min mathbf{N}).

Lemma 2.3

Under the assumption ((V_{1})), for any (2leq qleq +infty ),

$$ beta _{m}(q):=sup_{uin Z_{m}, Vert u Vert =1} Vert u Vert _{q}rightarrow 0 quad textit{as } mrightarrow infty . $$

(9)

Proof

It is obvious that (0<beta _{m+1}(q)leq beta _{m}(q)), so that (beta _{m}(q)rightarrow beta (q)geq 0) as (mrightarrow infty ). For every (min mathbf{N}), there exists (u^{m}in mathbf{Z}_{m}) with (|u^{m}|=1) such that

$$ biglVert u^{m} bigrVert _{q}> frac{beta _{m}(q)}{2}. $$

(10)

For any (win E), (w=sum_{j=1}^{infty } c_{j}e_{j}), by the Cauchy–Schwarz inequality, we have

$$ begin{aligned} biglvert bigllangle u^{m},wbigrrangle bigrvert &= Bigglvert Biggllangle u^{m},sum _{j=1}^{ infty } c_{j}e_{j}Biggrrangle Biggrvert = Bigglvert Biggllangle u^{m},sum _{j=m}^{infty } c_{j}e_{j} Biggrrangle Biggrvert \ &leq biglVert u^{m} bigrVert BigglVert sum _{j=m}^{infty } c_{j}e_{j} BiggrVert = BigglVert sum_{j=m}^{ infty } c_{j}e_{j} BiggrVert rightarrow 0 end{aligned} $$

as (mrightarrow infty ), which implies that (u^{m}rightharpoonup 0) in E. The compact embedding of (Ehookrightarrow l^{q}), (qin [2,infty ]), implies that (u^{m}rightarrow 0) in (l^{q}). Let (mrightarrow infty ) in (10) and we obtain (beta _{m}(q)rightarrow 0) as (mrightarrow infty ). □

The proof of Theorem 1.1

We will make use of the Symmetric Mountain Pass Theorem and Proposition 2.1 to prove Theorem 1.1. It is easy to see that ((J_{1})) follows from the condition ((f_{4})). It follows from ((f_{1})) that

$$ F_{k}(t)leq c_{1}t^{2}+c_{2} vert t vert ^{p}quad text{for any } kin mathbf{Z} text{ and } tin mathbf{R}. $$

By Lemma 2.3, there exists (min mathbf{N}) large enough such that

$$ Vert u Vert ^{2}_{2}leq frac{min {a,1}}{4c_{1}} Vert u Vert ^{2} quad text{and}quad Vert u Vert ^{p}_{p} leq frac{min {a,1}}{4c_{2}} Vert u Vert ^{p} quadtext{for any } uin Z_{m}. $$

It follows from the above three inequalities that

$$ begin{aligned} I(u)&=frac{1}{2}min {a,1} Vert u Vert ^{2} -sum_{kin mathbf{Z}}F_{k} bigl(u^{n}_{k}bigr) \ &geq frac{1}{2}min {a,1} Vert u Vert ^{2}-c_{1} Vert u Vert _{2}^{2}-c_{2} Vert u Vert _{p}^{p} \ &geq frac{1}{4}min {a,1} Vert u Vert ^{2}bigl(1- Vert u Vert ^{p-2}bigr). end{aligned} $$

It follows from (p>2) that there exist (delta _{1}), (alpha >0) such that (I|_{partial B_{delta _{1}}cap Z_{m}}geq alpha ). Thus, ((J_{2})) holds.

It remains to prove ((J_{3})). Since all norms are equivalent in a finite-dimensional space, there exists (c_{4}) such that (|u|^{4}_{4}geq c_{4}|u|^{4}) for any (uin widetilde{E}).

By ((f_{2})), for any (M>frac{b}{4c_{4}}), there exists (L_{1}>0) such that (F_{k}(t)geq Mt^{4}) for (|t|geq L_{1}). It follows from ((f_{1})) that there exists (C_{1}>0) such that (F_{k}(t)geq -C_{1}t^{2}) for (|t|leq L_{1}). From the above two inequalities, it follows that for any (kin mathbf{Z}),

$$ F_{k}(t)geq Mt^{4}-C_{M}t^{2} quadtext{for any } kin mathbf{Z} { text{ and }} tin mathbf{R}, $$

where (C_{M}=C_{1}+ML_{1}^{2}). It follows that

$$ begin{aligned} I(u)&leq frac{1}{2}max {a,1} Vert u Vert ^{2}+frac{b}{4} Vert u Vert ^{4}-sum _{kin mathbf{Z}}F_{k}(u_{k}) \ &leq frac{1}{2}max {a,1} Vert u Vert ^{2}+ frac{b}{4} Vert u Vert ^{4}-M Vert u Vert _{4}^{4}+C_{M} Vert u Vert _{2}^{2} \ &leq C Vert u Vert ^{2}-biggl(Mc_{4}- frac{b}{4}biggr) Vert u Vert ^{4}, end{aligned} $$

for all (uin widetilde{E}). Hence, there exists a large (R=R(widetilde{E})) such that (I(u)leq 0) on (widetilde{E}setminus B_{R}). This completes the proof. □

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