In order to apply the variational setting, we assume the solutions of (1.1) belong to the following subspace of \(\mathcal {D}^{2,p}({\mathbb{R}}^{N})\):
$$ E=\biggl\{ u\in\mathcal{D}^{2,p}\bigl({ \mathbb{R}}^{N}\bigr)\Big| \int_{{\mathbb {R}}^{N}}|\Delta u|^{p}+|\nabla u|^{p}+V(x)|u|^{p} \,dx< \infty\biggr\} $$
(2.1)
endowed with the norm
$$ \|u\|_{E}=\biggl( \int_{{\mathbb{R}}^{N}}\bigl(|\Delta u|^{p}+|\nabla u|^{p}+V(x)|u|^{p}\bigr)\, dx\biggr)^{1/p}, $$
(2.2)
where \(\mathcal{D}^{2,p}({\mathbb{R}}^{N})=\{u\in L^{p_{*}}({\mathbb {R}}^{N}) |\Delta u\in L^{p}({\mathbb{R}}^{N})\}\), \(\|\cdot\|_{s}\) means the norm in \(L^{s}({\mathbb{R}}^{N})\).
We denote by \(S_{*}\) the Sobolev constant, that is,
$$ S_{*}=\inf_{u\in\mathcal{D}^{2,p}\setminus\{0\}}\frac{\int _{{\mathbb{R}}^{N}}|\Delta u|^{p}\,dx}{(\int_{{\mathbb {R}}^{N}}|u|^{p_{*}}\, dx)^{p/p_{*}}} $$
(2.3)
and
$$ S_{*}\biggl( \int_{{\mathbb{R}}^{N}}|u|^{p_{*}}\, dx\biggr)^{p/p_{*}}\leq \int_{{\mathbb {R}}^{N}}|\Delta u|^{p}\,dx,\quad \forall u\in \mathcal{D}^{2,p}\bigl({\mathbb{R}}^{N}\bigr), $$
(2.4)
where \(S_{*}\) is obtained by a positive and radially symmetric function; see for instance [14].
Definition 2.1
A function \(u\in E\) is said to be a weak solution of (1.1) if, for any \(\varphi\in E\), we have
$$\begin{aligned}& \int_{{\mathbb{R}}^{N}}\bigl(|\Delta u|^{p-2}\Delta u\Delta\varphi+| \nabla u|^{p-2}\nabla u\varphi+V|u|^{p-2}u\varphi\bigr)\, dx \\& \quad = \int_{{\mathbb{R}} ^{N}}\bigl(\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u \bigr)\varphi\, dx. \end{aligned}$$
(2.5)
Let \(J(u):E\to{\mathbb{R}}\) be the energy functional associated with problem (1.1) defined by
$$ J(u)=\frac{1}{p}\|u\|^{p}_{E}- \frac{\lambda}{m} \int_{{\mathbb{R}} ^{N}}h_{1}|u|^{m}\,dx-\frac{1}{q} \int_{{\mathbb{R}}^{N}}h_{2}|u|^{q}\,dx. $$
(2.6)
From the embedding inequality (2.4) and the assumptions in Theorem 1.1, we see the functional \(J\in C^{1}(E,{\mathbb{R}})\) and its Gateaux derivative is given by
$$\begin{aligned} J'(u)\varphi =& \int_{{\mathbb{R}}^{N}}\bigl(|\Delta u|^{p-2}\Delta u\Delta \varphi +|\nabla u|^{p-2}\nabla u\varphi+V(x)|u|^{p-2}u\varphi\bigr)\, dx \\ &{}- \int_{{\mathbb{R}}^{N}}\bigl(\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u \bigr)\varphi\, dx. \end{aligned}$$
(2.7)
To prove the existence of infinitely many solutions to problem (1.1), we need to prove that the functional J defined by (2.6) satisfies the \((\mathit{PS})\) condition. Recall that a sequence \(\{u_{n}\}\) in E is called a \((\mathit{PS})_{c}\) sequence of J if
$$ J(u_{n})\to c, \qquad J'(u_{n}) \to0 \quad \mbox{in }E^{*}\mbox{ as }n\to\infty. $$
(2.8)
The functional J satisfies the \((\mathit{PS})\) condition if any \((\mathit{PS})_{c}\) sequence possesses a convergent subsequence in E.
Lemma 2.1
Assume (V), (H1), and (H2) hold. If
\(\{u_{n}\}\subset E\)
is a
\((\mathit{PS})_{c}\)
sequence of
J, then
\(\{u_{n}\}\)
is bounded in
E.
Proof
It follows from Hölder’s inequality that
$$\begin{aligned} \int_{{\mathbb{R}}^{N}}|h_{1}||u_{n}|^{m}\,dx \leq&V_{0}^{-\frac{m}{p}}\biggl( \int _{{\mathbb{R}} ^{N}}|h_{1}|^{\sigma}\, dx \biggr)^{\frac{1}{\sigma}}\biggl( \int_{{\mathbb{R}} ^{N}}V|u_{n}|^{p}\,dx \biggr)^{\frac{m}{p}} \\ \leq&a_{1}\|u_{n}\|^{m}_{E}, \end{aligned}$$
(2.9)
where \(a_{1}=V_{0}^{-{m}/{p}}\|h_{1}\|_{\sigma}\). Choose \(t\in(0,1)\) such that \(q=pt+(1-t)p_{*}\), then
$$\begin{aligned} \int_{{\mathbb{R}}^{N}}h_{2}|u_{n}|^{q}\,dx \leq&\biggl( \int_{{\mathbb {R}}^{N}}V|u_{n}|^{p}\,dx \biggr)^{t}\biggl( \int_{{\mathbb{R}} ^{N}}|u_{n}|^{p_{*}}h_{2}^{-{\frac{1}{1-t}}}V^{-\frac{t}{1-t}} \, dx\biggr)^{1-t} \\ \leq& a_{2}\biggl( \int_{{\mathbb{R}}^{N}}V|u_{n}|^{p}\,dx \biggr)^{t}\|\Delta u_{n}\| _{p}^{{(1-t)}p_{*}}\leq a_{2}\|u_{n}\|^{q}_{E}, \end{aligned}$$
(2.10)
where \(a_{2}=S_{*}^{-{p_{*}(q-p)}/{p(p_{*}-p)}}V_{0}^{-t}\|h_{2}\|_{\infty}\). Thus,
$$\begin{aligned} c+1+\|u_{n}\|_{E} \geq& J(u_{n})-q^{-1}J'(u_{n})u_{n} \\ \geq&\biggl(\frac{1}{p}-\frac{1}{q}\biggr)\|u\|_{E}^{p}- \lambda\biggl(\frac{1}{m}-\frac {1}{q}\biggr) \int_{{\mathbb{R}}^{N}}|h_{1}||u|^{m}\,dx \\ \geq&\biggl(\frac{1}{p}-\frac{1}{q}\biggr)\|u_{n} \|^{p}_{E}-\lambda\biggl(\frac {1}{m}-\frac{1}{q} \biggr)a_{1}\|u_{n}\|^{m}_{E}. \end{aligned}$$
(2.11)
Since \(1< m< p< q\), we conclude that \(\|u\|_{E}\) is bounded and the proof is complete. □
In the following, we shall show that \(\{u_{n}\}\) has a convergent subsequence in E. Since the sequence \(\{u_{n}\}\) given by (2.8) is a bounded sequence in E, there exist a subsequence of \(\{u_{n}\}\) (still denoted by \(\{u_{n}\}\)) and \(v\in E\) such that \(\|u_{n}\|_{E}\leq M\), \(\|v\|_{E}\leq M\), and
$$\begin{aligned}& u_{n}\rightharpoonup v \quad \text{weakly in }E, \\& u_{n}\to v \quad \text{in } L^{s}_{\mathrm{loc}}\bigl({ \mathbb{R}}^{N}\bigr), 1< s< p_{*}, \\& u_{n}(x)\to v(x)\quad \text{a.e. in } {\mathbb{R}}^{N}. \end{aligned}$$
(2.12)
Lemma 2.2
Assume (V), (H1), and (H
2) hold. If the sequence
\(\{u_{n}\}\)
is bounded in
E
satisfying (2.12), then
-
(i)
\(\lim_{n\to\infty}\int_{{\mathbb{R}} ^{N}}h_{1}(x)|u_{n}|^{m}\,dx=\int_{{\mathbb{R}}^{N}}h_{1}(x)|v|^{m}\,dx\), \(\lim_{n\to \infty}\int_{{\mathbb{R}}^{N}}h_{1}(x)|u_{n}-v|^{m}\,dx=0\);
-
(ii)
\(\lim_{n\to\infty}\int_{{\mathbb{R}} ^{N}}h_{2}(x)|u_{n}|^{q}\,dx=\int_{{\mathbb{R}}^{N}}h_{2}(x)|v|^{q}\,dx\), \(\lim_{n\to \infty}\int_{{\mathbb{R}}^{N}}h_{2}(x)|u_{n}-v|^{q}\,dx=0\).
Proof
(i) In fact, from \(h\in L^{\sigma}({\mathbb{R}}^{N})\) and (2.12), we obtain, for any \(r>0\),
$$ \int_{B_{r}}h_{1}(x)|u_{n}|^{m}\,dx \to \int_{B_{r}}h_{1}(x)|v|^{m}\,dx \quad \mbox{as }n\to \infty, $$
(2.13)
where and in the sequel \(B_{r}=\{x\in{\mathbb{R}}^{N}:|x|< r\}\), \(B^{c}_{r}={\mathbb{R}}^{N}\setminus \overline{B}_{r}\). On the other hand, we see from the Hölder inequality that
$$\begin{aligned} \begin{aligned}[b] \int_{B^{c}_{r}}|h_{1}||u_{n}|^{m}\,dx& \leq V_{0}^{-\frac{m}{p}}\biggl( \int _{B^{c}_{r}}|h_{1}|^{\sigma}\, dx \biggr)^{\frac{1}{\sigma}}\biggl( \int _{B^{c}_{r}}V|u_{n}|^{p}\,dx \biggr)^{\frac{m}{p}} \\ &\leq V_{0}^{-\frac{m}{p}}\|h_{1}\|_{L^{\sigma}(B^{c}_{r})} \|u_{n}\|^{m}_{E}\leq V_{0}^{-\frac{m}{p}}M^{m} \|h_{1}\|_{L^{\sigma}(B^{c}_{r})}\to0 \end{aligned} \end{aligned}$$
(2.14)
as \(r\to\infty\). By Fatou’s lemma, we see that, as \(n\to\infty\),
$$ \int_{B^{c}_{r}}|h_{1}||v|^{m}\,dx\leq\liminf _{n\to\infty} \int _{B^{c}_{r}}|h_{1}||u_{n}|^{m}\,dx \leq V_{0}^{-\frac{m}{p}}M^{m}\|h_{1} \|_{L^{\sigma}(B^{c}_{r})}\to0. $$
(2.15)
Then, the application of (2.13)-(2.15) gives the first limit of (i). Furthermore, by the Brezis-Lieb lemma in [15], we have the second limit of (i).
(ii) To prove the conclusion (ii), we follow the argument used in [10–12]. Here, we give a detailed proof for the reader’s convenience.
Since \(p< p< p_{*}\), it easy to see that, for any small \(\varepsilon>0\), there exist \(S_{0}>s_{0}>0\) such that \(|s|^{q}<\varepsilon|s|^{p}\) if \(|s|\leq s_{0}\) and \(|s|^{q}\leq\varepsilon|s|^{p_{*}}\), if \(|s|\geq S_{0}\). This shows that
$$ |s|^{q}\leq\varepsilon\bigl(|s|^{p}+|s|^{p_{*}} \bigr)+\chi _{[s_{0},S_{0}]}\bigl(\vert s\vert \bigr)|s|^{q}, \quad \forall s\in{\mathbb{R}}. $$
(2.16)
Denote \(A_{n}=\{x\in{\mathbb{R}}^{N};s_{0}\leq|u_{n}(x)|\leq S_{0}\}\). It follows from (2.16), (2.4), and (2.12) that
$$\begin{aligned} \int_{B^{c}_{r}}|h_{2}||u_{n}|^{q}\,dx \leq&\|h_{2}\|_{\infty}\int_{B^{c}_{r}} \bigl(\varepsilon\bigl(|u_{n}|^{p}+|u_{n}|^{p_{*}} \bigr)+\chi_{[s_{0},S_{0}]}\bigl(\vert u_{n}\vert \bigr)|u_{n}|^{q} \bigr)\, dx \\ \leq&\varepsilon\|h_{2}\|_{\infty}\biggl( \int_{{\mathbb{R}} ^{N}}V_{0}^{-1}V(x)|u_{n}|^{p} \,dx+S^{-\frac{p_{*}}{p}}\|\Delta u\|^{p_{*}}_{p}\biggr) \\ &{} +S_{0}^{q}\|h_{2}\|_{\infty}\operatorname{meas}\bigl(A_{n}\cap B^{c}_{r}\bigr) \\ \leq& M_{1}\varepsilon+S_{0}^{q} \|h_{2}\|_{\infty}\operatorname{meas}\bigl(A_{n}\cap B^{c}_{r}\bigr) \end{aligned}$$
(2.17)
with some constant \(M_{1}>0\), and
$$ |s_{0}|^{p_{*}}|A_{n}|\leq \int_{{\mathbb{R}}^{N}}|u_{n}|^{p_{*}}\, dx\leq M_{1},\quad \forall n\in {\mathbb{N}}, $$
(2.18)
where \(|A_{n}|=\operatorname{meas}(A_{n})\). Equation (2.18) implies that \(\sup_{n\in{\mathbb{N}}}|A_{n}|\leq M_{1}|s_{0}|^{-p_{*}}<\infty\), so it is easy to see that
$$ \lim_{r\to\infty}\operatorname{meas} \bigl(A_{n}\cap B_{r}^{c}\bigr)=0,\quad \mbox{for all } n\in{\mathbb{N}}. $$
(2.19)
In the following, we show that \(\lim_{r\to\infty }\operatorname{meas}(A_{n}\cap B_{r}^{c})=0\) uniformly in \(n\in{\mathbb{N}}\).
In fact, it follows from (2.12) that \(v\in L^{p}({\mathbb{R}}^{N})\) and \(u_{n}(x)\to v(x)\) a.e. \({\mathbb{R}}^{N}\). Therefore, for any small \(\varepsilon >0\), there exists \(r_{0}>1\) such that \(r\geq r_{0}\),
$$\int_{B_{r}^{c}}|v|^{p}\,dx\leq\varepsilon. $$
For this ε, we choose \(t_{1}=r_{0}\), \(t_{j}\uparrow\infty\) such that \(D_{j}=B^{c}_{t_{j}}\setminus\overline{B}^{c}_{t_{j+1}}\), \(B^{c}_{r_{0}}=\bigcup^{\infty}_{j=1}D_{j}\) and
$$\int_{D_{j}}|v|^{p}\, dx\leq\frac{\varepsilon}{2^{j}},\quad \forall j\in{\mathbb{N}}. $$
Obviously, for every fixed \(j\in N\), \(D_{j}\) is a bounded domain and \(D_{j}\cap D_{i}=\emptyset\) (\(j\neq i\)). Furthermore, \(s_{0}\leq|u_{n}|\leq S_{0}\) in \(D_{j}\cap A_{n}\). By Fatou’s lemma, we have, for every \(j\in {\mathbb{N}}\),
$$\limsup_{n\to\infty} \int_{D_{j}\cap A_{n}}|u_{n}|^{p}\,dx\leq \int _{D_{j}}\limsup_{n\to\infty}|u_{n}|^{p} \,dx\leq \int _{D_{j}}|v|^{p}\,dx\leq\frac{\varepsilon}{2^{j}}. $$
Then, for \(s_{1}=2^{1-q}s_{0}^{q}\), we obtain
$$\begin{aligned} s_{1}\limsup_{n\to\infty}\bigl\vert A_{n}\cap B_{r_{0}}^{c}\bigr\vert \leq&\limsup _{n\to\infty} \int_{B_{r_{0}}^{c}\cap A_{n}}|u_{n}|^{p}\,dx \\ =&\limsup _{n\to\infty}\sum^{\infty}_{j=1} \int_{D_{j}\cap A_{n}}|u_{n}|^{p}\,dx \\ \leq&\sum^{\infty}_{j=1}\limsup _{n\to\infty} \int _{D_{j}\cap A_{n}}|u_{n}|^{p}\,dx \\ \leq&\sum ^{\infty}_{j=1} \int _{D_{j}}|v|^{p}\,dx\leq\sum ^{\infty}_{j=1}\frac{\varepsilon }{2^{j}}=\varepsilon. \end{aligned}$$
(2.20)
Notice that, for any \(r\geq r_{0}\) and \(n\in{\mathbb{N}}\), we have \((A_{n}\cap B^{c}_{r})\subset(A_{n}\cap B_{r_{0}}^{c})\). Therefore, the application of (2.19) and (2.20) yields \(\lim_{r\to\infty}|A_{n}\cap B_{r}^{c}|=0\) uniformly in \(n\in{\mathbb{N}}\). Thus, for any \(\varepsilon>0\), there exists \(r_{0}\geq1\) such that \(\operatorname{meas}(A_{n}\cap B^{c}_{r})<\frac{\varepsilon}{S_{0}^{q}\|h_{2}\| _{\infty}}\), for \(r\geq r_{0}\). Then it follows from (2.17) that
$$ \int_{B^{c}_{r}}h_{2}|u_{n}|^{q}\,dx \leq\max\{M_{1},1\}\varepsilon,\quad \forall n\in {\mathbb{N}}, r\geq r_{0} $$
(2.21)
and
$$ \int_{B^{c}_{r}}h_{2}|v|^{q}\,dx\leq\liminf _{n\to\infty} \int _{B^{c}_{r}}h_{2}|u_{n}|^{q}\,dx \leq\max\{M_{1},1\}\varepsilon, \quad r\geq r_{0}. $$
(2.22)
Moreover, we derive from (2.12) that
$$ \int_{B_{r}}h_{2}(x)|u_{n}|^{q}\,dx \to \int_{B_{r}}h_{2}(x)|v|^{q}\,dx. $$
(2.23)
Therefore, using (2.21) and (2.22), and the application of Brezis-Lieb lemma in [15] we conclude the second limit of (ii). Then the proof is complete. □
Lemma 2.3
Let
\(\{u_{n}\}\)
be a
\((\mathit{PS})_{c}\)
sequence satisfying (2.12), then
\(u_{n}\to v\)
in
E, that is, the functional
J
satisfies the
\((\mathit{PS})\)
condition.
Proof
Denote
$$\begin{aligned} P_{n} =&J'(u_{n}) (u_{n}-v) \\ =& \int_{{\mathbb{R}}^{N}} \bigl(|\Delta u_{n}|^{p-2}\Delta u_{n} \Delta (u_{n}-v)+|\nabla u_{n}|^{p-2} \nabla u_{n}\nabla(u_{n}-u) \\ &{}+V(x)|u_{n}|^{p-2}u_{n}(u_{n}-v) \bigr)\, dx \\ &{} - \int_{{\mathbb{R}}^{N}}\bigl(\lambda h_{1}(x)|u_{n}|^{m-2}u_{n}+h_{2}(x)|u_{n}|^{q-2}u_{n} \bigr) (u_{n}-v) \, dx. \end{aligned}$$
Then the fact \(J'(u_{n})\to0\) in \(E^{*}\) shows that \(P_{n}\to0\) as \(n\to \infty\). Moreover, the fact \(u_{n}\rightharpoonup v\) in E implies \(Q_{n}\to0\), where
$$ Q_{n}= \int_{{\mathbb{R}}^{N}}\bigl(|\Delta v|^{p-2}\Delta v \Delta (u_{n}-v)+|\nabla u|^{p-2}\nabla u\nabla(u_{n}-u)+V(x)|v|^{p-2}v(u_{n}-v) \bigr)\, dx. $$
It follows from the Hölder inequality and the limit (i) in Lemma 2.2 that
$$\begin{aligned} \int_{{\mathbb{R}}^{N}}\bigl\vert h_{1}(x)\bigr\vert \vert u_{n}\vert ^{m-1}\vert u_{n}-v\vert \, dx \leq&\biggl( \int _{{\mathbb{R}} ^{N}}\bigl\vert h_{1}(x)\bigr\vert \vert u_{n}-v\vert ^{m}\,dx\biggr)^{\frac{1}{m}}\biggl( \int_{{\mathbb{R}} ^{N}}\bigl\vert h_{1}(x)\bigr\vert \vert u_{n}\vert ^{m}\,dx\biggr)^{\frac{m-1}{m}} \\ \to&0. \end{aligned}$$
(2.24)
Similarly, we can derive from the limit (ii) in Lemma 2.2 that
$$\begin{aligned} \int_{{\mathbb{R}}^{N}}h_{2}(x)|u_{n}|^{q-1}|u_{n}-v| \, dx \leq&\biggl( \int_{{\mathbb{R}} ^{N}}h_{2}(x)|u_{n}-v|^{q} \,dx\biggr)^{\frac{1}{q}}\biggl( \int_{{\mathbb {R}}^{N}}h_{2}(x)|u_{n}|^{q}\,dx \biggr)^{\frac{q-1}{q}} \\ \to&0. \end{aligned}$$
(2.25)
Then (2.24) and (2.25) show that as \(n\to\infty\)
$$\begin{aligned} o_{n}(1) =&P_{n}-Q_{n} \\ =& \int_{{\mathbb{R}}^{N}}\bigl(\bigl(|\Delta u_{n}|^{p-2} \Delta u_{n}-|\Delta v|^{p-2}\Delta v\bigr) \Delta(u_{n}-v) \\ &{}+\bigl(|\nabla u_{n}|^{p-2}\nabla u_{n}-|\nabla v|^{p-2}\nabla v\bigr) \nabla (u_{n}-v) \\ &{}+V(x) \bigl(|u_{n}|^{p-2}u_{n}-|v|^{p-2}v\bigr) (u_{n}-v)\bigr)\, dx. \end{aligned}$$
(2.26)
Then we have \(\|u_{n}-v\|_{E}\to0\) as \(n\to\infty\). Thus \(J(u)\) satisfies the \((\mathit{PS})\) condition on E and the proof is completed. □