Abstract
In this work, we study the existence of infinitely many solutions to the following quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity:
where \(\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\), \(1< p< N\), \(\lambda \ge 0\), and \(1< m< p<2\alpha p<q<2\alpha p^{*}=\frac{2\alpha pN}{N-p}\). The functions \(V(x)\), \(h_{1}(x)\), and \(h_{2}(x)\) satisfy some suitable conditions. Using variational methods and some special techniques, we prove that there exists \(\lambda _{0}>0\) such that Eq. (0.1) admits infinitely many high energy solutions in \(W^{1,p}({\mathbb{R}}^{N})\) provided that \(\lambda \in [0,\lambda _{0}]\).
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1 Introduction and main result
In this paper, we are interested in the existence of infinitely many solutions to a class of quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity
where \(\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)(1< p< N)\) and \(\alpha >\frac{1}{2}\) is a parameter.
For the case \(p=2\), \(\alpha =1\), solutions of (1.1) are standing waves of the following Schrödinger equation:
where \(z: {\mathbb{R}}\times {\mathbb{R}}^{N} \to \mathbbm{ }\mathbb{C}\) and \(W:{\mathbb{R}}^{N}\to {\mathbb{R}}\) is a given potential, \(h_{1},g: {\mathbb{R}}^{+}\to {\mathbb{R}}\) are real functions.
It is well known that the standing wave solutions of the form \(z(t,x)=\exp (-i\omega t)u(x)\) satisfy (1.2) with \(g(s)=s\) if and only if the function \(u(x)\) solves the equation of elliptic type
where \(V(x)=W(x)-\omega \), \(\omega \in {\mathbb{R}}\) and \(h(u)\equiv h_{1}(|u|^{2})u\).
Quasilinear Schrödinger equations of form (1.2) appear naturally in mathematical physics and have been derived as models of several physical phenomena corresponding to various types of nonlinear term g. The case \(g(s)=s\) was used for the superfluid film equation in plasma physics by Kurihura in [11] (see also [12]). In the case \(g(s)=(1+s)^{1/2}\), Eq. (1.2) models the self-channeling of a high power ultra short laser in matter, see [7]. Equation (1.2) also appears in plasma physics and fluid mechanics [20], in mechanics [9], and in condensed matter theory [18]. More information on this subject can be found in [15] and the references therein.
For \(p=2\), several methods can be used to solve (1.1), e.g., the existence of positive ground state solution was proved in [17, 19] by using a constrained minimization argument; Eq. (1.1) was transformed to a semilinear one in [4–6, 10, 15] by a change of variables (dual approach); Nehari method was used to get the existence results of ground state solutions in [16, 22]. Especially, in [13, 15–17, 25], the existence of the ground state solutions for the following problem with a parameter \(\alpha (>\frac{1}{2})\):
was studied with subcritical nonlinearities \(g(x,u)\).
For (1.4), we find in the literature several types of potentials \(V(x)\) to obtain a solution. Wu in [25] studied Eq. (1.4) considering the subcritical case and a potential \(V(x)\), which is unbounded in \({\mathbb{R}}^{N}\) and satisfies the following assumption:
- \((A_{1})\):
-
The potential \(V(x)\in C({\mathbb{R}}^{N})\) and \(0< V_{0}:=\inf_{x\in {\mathbb{R}}^{N}}V(x)\), and for each \(M>0\), \(\operatorname{meas} (\{x\in {\mathbb{R}}^{N}: V(x)\le M\})<\infty \).
In [15–17], Liu et al. proved the existence of a positive solution to problem (1.4) with \(V(x)\in C({\mathbb{R}}^{N})\), \(\inf_{x\in {\mathbb{R}}^{N}}V(x)>0\) and the following conditions:
- \((A_{2})\):
-
\(\lim_{|x|\to \infty}V(x)=+\infty \);
- \((A_{3})\):
-
\(0< V_{0}:=\inf_{x\in {\mathbb{R}}^{N}}V(x)< \lim_{|x| \to \infty}V(x)=V_{\infty}=\|V\|_{L^{\infty}({\mathbb{R}}^{N})}< \infty \);
- \((A_{4})\):
-
V is radially symmetric, i.e., \(V(x)=V(|x|)\);
- \((A_{5})\):
-
V is periodic in each variable of \(x_{1}, \ldots , x_{N}\).
Similar assumptions also appeared in Severo [24], Ruiz and Siciliano [22], Fang and Szulkin [8]. By the variational principle in a suitable Orlicz space, do Ó and Severo in [3] established the existence of positive standing wave solutions for (1.4) with a concave-convex nonlinearity and the following condition:
- \((A_{6})\):
-
\(0< V_{0}\le V(x)\) in \({\mathbb{R}}^{N}\) and \(V^{-1}(x)\in L^{1}({\mathbb{R}}^{N})\).
Recently, Aires and Souto [1] considered (1.4) with \(\alpha =1\) and the vanishing potential \(V(x)\) at infinity.
Clearly, it is well known that assumption \((A_{1})\) or \((A_{2})\) guarantees that the embedding \(W^{1,2}({\mathbb{R}}^{N})\hookrightarrow L^{s}({\mathbb{R}}^{N})\) is compact for each \(2\le s<\frac{2N}{N-2}\). Similarly, the application of \((A_{3})\) in [2, 15, 24] shows that the solution is nontrivial.
It is worth pointing out that the aforementioned authors always assumed that the potential \(V(x)\) has some special characteristic. As far as we know, there are few papers that deal with a general bounded potential case for (1.1). Motivated by papers [1, 25], in the present paper we consider problem (1.1) with positive and more general bounded potential \(V(x)\) by a dual approach and establish the existence of infinitely many high energy solutions under a concave-convex nonlinearity and different type weight functions \(h_{1}(x)\), \(h_{2}(x)\). It is easy to verify that for a general continuous and bounded function \(V(x)\), assumptions \((A_{1})-(A_{6})\) fail to hold. We shall use mountain pass theorem under the Cerami condition to study Eq. (1.1).
Throughout this paper, we always assume the potential \(V(x)\in C({\mathbb{R}}^{N})\) and the weight function \(h_{2}(x)\ge 0\), ≢0 in \({\mathbb{R}}^{N}\). Furthermore, we let \(C, C_{1},C_{2},\ldots \) be positive generic constants that can change from line to line.
The main result in this paper is as follows.
Theorem 1.1
Assume:
- \((H_{0})\):
-
\(1< p< N\), \(1< m< p<2\alpha p<q<2\alpha p^{*}=\frac{2\alpha pN}{N-p}\);
- \((H_{1})\):
-
There exist the constants \(V_{0}, V_{1}>0\) such that \(V_{0}\le V(x)\le V_{1}\) for all \(x\in {\mathbb{R}}^{N}\);
- \((H_{2})\):
-
\(h_{1}\in L^{\sigma}({\mathbb{R}}^{N})\) with \(\sigma =\frac{2\alpha p}{2\alpha p-m}\);
In addition, suppose that one of the following two hypotheses holds:
- \((H_{3})\):
-
\(h_{2}\in L^{\gamma}({\mathbb{R}}^{N})\cap C_{\mathrm{loc}}({\mathbb{R}}^{N} \setminus \{0\})\) with \(\gamma =\frac{2\alpha p^{*}}{2\alpha p^{*}-q}\);
- \((H_{4})\):
-
\(h_{2}(x) \in L^{\gamma}_{\mathrm{loc}}({\mathbb{R}}^{N})\cap C_{\mathrm{loc}}({ \mathbb{R}}^{N}\setminus \{0\})\) with \(\gamma =\frac{2\alpha p^{*}}{2\alpha p^{*}-q}\), and \(h_{2}(x)\to 0\) as \(|x|\to \infty \);
Then there exists a constant \(\lambda _{0}>0\) such that for all \(\lambda \in [0, \lambda _{0}]\), Eq. (1.1) admits infinitely many high energy solutions in \(u_{n}\in W^{1,p}({\mathbb{R}}^{N})\) such that \(J(v_{n})\to \infty \) as \(n\to \infty \), where \(v_{n}=f^{-1}(u_{n})\) and \(f(t)\) is defined by (2.5) later.
Remark 1.2
Assumptions \((H_{3})-(H_{4})\) are independent. For example, let \(0<\tau <N/\gamma \) and \(k>N\), then the unbounded function
satisfies \((H_{3})\), but \(h_{2}(x)\not \to 0\) as \(|x|\to \infty \). On the other hand, the unbounded function \(h_{2}(x)=|x|^{-\tau}\), \(x\in {\mathbb{R}}^{N}\setminus \{0\}\) satisfies \((H_{4})\), but fails to verify \((H_{3})\).
Remark 1.3
When \(p=2\), \(\alpha =1\), \(\lambda =0\), and \(h_{2}=\mu >0\), problem (1.1) becomes
with \(4< q<22^{*}\). The authors [15] proved that for any \(\mu >0\), Eq. (1.6) has a positive solution under assumptions \((A_{2})-(A_{5})\). Fang and Szulkin [8] also established the existence of infinitely many solutions to (1.6) provided that \(V(x)\) satisfies \((A_{5})\). Clearly, if \(V(x)\) is continuous in \({\mathbb{R}}^{N}\) and verifies \((A_{5})\), then \(V(x)\) satisfies \((H_{1})\). Theorem 1.1 shows that there are infinitely many solutions to (1.6) if \((H_{1})\) is true.
This paper is organized as follows. In Sect. 2, with a convenient change of variable, we set up the variational framework for (1.1). In Sect. 3, we verify that the energy functional associated with (1.1) satisfies the Cerami condition. In Sect. 4, the geometric conditions of the mountain pass theorem are verified, and the proof of Theorem 1.1 is given.
2 Variational setting of the equation
Let \(E=W^{1,p}({\mathbb{R}}^{N})\) be the Sobolev spaces with the norm
By hypothesis \((H_{1})\), it is equivalent to the standard norm in E. It is well known that there is a constant \(S>0\) such that
From the approximation argument, we see that (2.2) holds on E.
We observe that the natural energy functional associated with Eq. (1.1) is given by
where
It should be pointed out that the functional I is not well defined in general in E. To overcome this difficulty, we employ an argument developed by Colin and Jeanjean [6] for the case \(p=2\) and Severo [24] for \(1< p\le N\). We make the change of variables \(u=f(v)\) or \(v=f^{-1}(u)\), where f is defined by
and by \(f(t)=-f(-t)\) on \((-\infty , 0]\). Then we have the following.
Lemma 2.1
The function \(f(t)\) satisfies the following properties:
\((f_{1})\) f is uniquely defined, odd, increasing, and invertible in \({\mathbb{R}}\);
\((f_{2})\) \(0< f'(t)\le 1\), \(\forall t\in {\mathbb{R}}\);
\((f_{3})\) \(|f(t)|\le |t|\), \(\forall t\in {\mathbb{R}}\);
\((f_{4})\) \(\frac{f(t)}{t}\to 1\) as \(t\to 0\);
\((f_{5})\) \(|f(t)|\le (2\alpha )^{1/2\alpha p}|t|^{1/2\alpha}\), \(\forall t\in { \mathbb{R}}\);
\((f_{6})\) \(\frac{1}{2}f(t)\le \alpha tf'(t)\le \alpha f(t)\), \(\forall t\in { \mathbb{R}}^{+}=[0, \infty )\) and \(\alpha f(t)\le \alpha tf'(t)\le \frac{1}{2}f(t)\), \(\forall t\in { \mathbb{R}}^{-}=(-\infty , 0]\);
\((f_{7})\) There exists \(a\in (0, (2\alpha )^{1/2\alpha p}]\) such that \(\frac{f(t)}{t^{1/2\alpha}}\to a\) as \(t\to +\infty \);
\((f_{8})\) There exists \(b_{0}>0\) such that
\((f_{9})\) For each \(\tau >0\), there exist \(C(\tau )=n\) if \(\tau =n\) and \(C(\tau )=n+1\) if \(\tau \in (n,n+1)\), \(n\in {\mathbb{N}}\) such that
Proof
The proof of properties \((f_{1})-(f_{8})\) can be found in [24](for the case \(1< p\le N\) and \(\alpha =1\)) and in [25] (for the case \(p=2\) and \(\frac{1}{2}<\alpha \le 1\)). For the case \(1< p< N\) and \(\alpha >\frac{1}{2}\), the proof of \((f_{1})-(f_{8})\) is similar and omitted. Here we prove \((f_{9})\). Note that
Then
For the second integral in (2.8), we take \(s=t+\xi \) and \(h(s)\ge (1+(2\alpha )^{p-1}|f(\xi )|^{p(2\alpha -1)})^{1/p}\). Thus,
Similarly, we have \(f(nt)\le nf(t)\) for \(t\ge 0\) and \(n\in \mathbb{N}\). Since \(f(t)\) is odd and increasing in \({\mathbb{R}}\), we obtain (2.6). □
So, after the change of variables, we can write \(I(u)\) as
which is well defined on E under assumptions \((H_{0})-(H_{4})\).
As in [24], we observe that if \(v\in W^{1,p}({\mathbb{R}}^{N})\cap L^{\infty}_{\mathrm{loc}}({\mathbb{R}}^{N})\) is a critical point of the functional J, that is, \(J'(v)\varphi =0\) for all \(\varphi \in W^{1,p}({\mathbb{R}}^{N})\), where
then v is a weak solution of the equation
and \(u = f(v)\) is a weak solution of (1.1). By using Theorem 1 in [23], we can conclude that v is locally bounded in \({\mathbb{R}}^{N}\). So, we consider the existence of solutions to (2.12) in E.
3 The boundedness of the Cerami sequences
To obtain the existence of solutions to problem (2.12), we need to prove that the functional J defined by (2.10) satisfies the Cerami condition.
We first recall that a sequence \(\{v_{n}\}\) in E is called a Cerami sequence of J if \(\{J(v_{n})\}\) is bounded and
The functional J satisfies the Cerami condition if any Cerami sequence possesses a convergent subsequence in E
Lemma 3.1
Assume \((H_{0})-(H_{2})\) and \(h_{2}\ge 0\) in \({\mathbb{R}}^{N}\). If \(\{v_{n}\}\subset E\) is a Cerami sequence, then \(\{v_{n}\}\) is bounded in E.
Proof
Without loss of generality, we assume \(v_{n}\neq0\) for all \(n\in \mathbb{N}\). Set \(\varphi _{n}(x)=\frac{f(v_{n}(x))}{f'(v_{n}(x))}\). Then, using \((f_{2})\) and \((f_{5})\) in Lemma 2.1, we have
Since \(\{v_{n}\}\) is a Cerami sequence in E, there is a constant \(C_{1}>0\) such that
This estimate and the assumption \(m\in (1,p)\) prove that \(\{\|\nabla v_{n}\|_{p}\}\) is bounded. Moreover,
where \(u_{n}=f(v_{n})\). Then \(\{\int _{{\mathbb{R}}^{N}}V|f(v_{n})|^{p}\,dx\}\) is bounded and so is \(\{A_{n}^{p}\}\), where
In the following, we show that there exists a constant \(C_{0}>0\) such that
We argue by contradiction and assume that, up to a subsequence, \(v_{n}\in E\) such that
Hence, \(\frac{A_{n}^{p}}{\|v_{n}\|_{E}^{p}}\to 0\) as \(n\to \infty \). Let \(\omega _{n}(x)=\frac{v_{n}(x)}{\|v_{n}\|_{E}}\), \(f_{n}(x)= \frac{|f(v_{n}(x))|^{p}}{\|v_{n}\|_{E}^{p}}\). Then
which shows
Moreover, since
we conclude
Similar to the idea of [25], we assert that for each \(\varepsilon >0\) there exists \(\alpha _{\varepsilon}\ge 1\) independent of n such that \(|\Omega _{n}|<\varepsilon \), where \(\Omega _{n}=\{x\in {\mathbb{R}}^{N}: |v_{n}(x)|\ge \alpha _{ \varepsilon}\}\) and \(|\Omega _{n}|=\operatorname{meas}(\Omega _{n})\). Otherwise, there are \(\varepsilon _{0}>0\) and subsequence \(\{v_{n_{k}}\}\subset \{v_{n}\}\) such that \(|\Omega _{n_{k}}|\ge \varepsilon _{0}\), where
By \((f_{8})\), one sees
as \(k\to \infty \). This is a contradiction. Hence the assertion is true. Denote \(\Omega _{n}^{c}={\mathbb{R}}^{N}\setminus \Omega _{n}\). For \(x\in \Omega _{n}^{c}\), we have \(|v_{n}(x)|\le \alpha _{\varepsilon}\). Using \((f_{8})\) and \((f_{9})\), we get
for some \(C_{2}>0\). Thus, as \(n\to \infty \),
On the other hand, from the integral absolute continuity, it follows that there is \(\delta >0\) such that whenever \(\Omega \subset {\mathbb{R}}^{N}\) and \(|\Omega |<\delta \),
For this \(\delta >0\), we have
Letting \(n\to \infty \), one sees from (3.11) and (3.17) that \(1\le \frac{1}{2}\). It is impossible. So, (3.6) is true and \(\{v_{n}\}\) is bounded in E. □
Since the sequence \(\{v_{n}\}\) given by Lemma 3.1 is a bounded sequence in E, there exist a constant \(M>0\) and \(v\in E\), and a subsequence of \(\{v_{n}\}\), still denoted by \(\{v_{n}\}\), such that \(\|v_{n}\|_{E}\le M\), \(\|v\|_{E}\le M \) and
Lemma 3.2
Assume \((H_{0})-(H_{2})\). If the sequence \(\{v_{n}\}\) satisfies (3.18), then
and
Proof
From (3.18), we have \(f(v_{n}(x))\to f(v(x))\) a.e. in \({\mathbb{R}}^{N}\). Then
for any \(r>0\), where \(B_{r}=\{x\in {\mathbb{R}}^{N}:|x|< r\}\), \(B_{r}^{c}={\mathbb{R}}^{N} \setminus B_{r}\). On the other hand, we see from Hölder’s inequality and (2.2) that
as \(r\to \infty \). By Fatou’s lemma, we obtain
Then, the application of (3.21)–(3.23) gives that (3.19). Similarly, noticing that \((f_{6})\) and
we can derive (3.20). □
Lemma 3.3
Assume \((H_{0})-(H_{2})\) and one of hypotheses \((H_{3})\) and \((H_{4})\). If the sequence \(\{v_{n}\}\) satisfies (3.18), then
and
Proof
If \((H_{3})\) is satisfied, we use a similar argument in the proof of Lemma 3.2 to get limits (3.24) and (3.25). We now assume \((H_{4})\). Choose \(t\in (0,1)\) such that \(q=2\alpha (pt+(1-t)p^{*})\). Then
and
Moreover, it follows from (3.18) that for all \(r>0\),
Then the application of (3.26)–(3.28) yields (3.24). Similarly, from \((f_{6})\), it follows that
and
Then we get (3.25) from (3.29)–(3.31). Then the proof of Lemma 3.3 is completed. □
Lemma 3.4
Assume that all hypotheses in Theorem 1.1hold. Let \(\{v_{n}\}\) be a Cerami sequence and satisfy (3.18). Then the following statements hold:
\((i)\). For each \(\varepsilon >0 \), there exists \(r_{0}\ge 1\) such that \(r\ge r_{0}\),
and
(ii). The weak limit \(v\in E\) is a critical point for functional J.
Proof
\((i)\). In fact, for \(r>1\), we choose the function \(\eta =\eta (|x|)\in C^{1}({\mathbb{R}}^{N})\) such that
Since the sequence \(\{v_{n}\}\) is bounded in E, the sequence \(\{\eta \varphi _{n}\}\), where \(\varphi _{n}=\frac{f(v_{n})}{f'(v_{n})}\), is also bounded in E. Hence, we have \(J'(v_{n})(\eta \varphi _{n})=o_{n}(1)\), that is,
For assumptions \((H_{2})-(H_{4})\), we have from (3.22) and (3.26) that
Hence, limits (3.37) and (3.38) show that
This estimate concludes (3.32). Moreover, limit (3.32) gives
and consequently,
Since \(v_{n}\to v\) in \(L^{p}(B_{2r})\), we get
Then, for all \(\varepsilon >0\), limits (3.40)–(3.42) yield
and limit (3.34) holds.
In the following, we prove (3.35). We first note that \((f_{6})\) and (3.38) show
Then the fact \(J'(v_{n})(\eta v_{n})=o_{n}(1)\) implies that
This shows that there exists a constant \(r_{0}\ge 1\) such that
for \(r>r_{0}\). So,
and consequently
Since \(v_{n}\to v\) in \(L^{p}(B_{2r})\), we have
and then
for every \(\varepsilon >0\). Therefore, limit (3.35) is true. The proof of part \((i)\) is completed.
\((ii)\). From (3.18), one sees that as \(n\to \infty \)
As in the proof of \((i)\), we can derive as \(n\to \infty \)
and
Then, from (3.51), (3.52), and (3.53), it follows
By the dense \(C_{0}^{\infty}({\mathbb{R}}^{N})\) in E, we have \(J'(v)\varphi =0\), \(\forall \varphi \in E\). In particular, \(J'(v)v=0\). Hence, v is a critical point of J in E. This completes the proof of Lemma 3.4. □
Lemma 3.5
Assume that all hypotheses in Theorem 1.1hold. Let \(\{v_{n}\}\) be a Cerami sequence and satisfy (3.18). Then \(v_{n}\to v\) in E, that is, the functional J satisfies the Cerami condition in E.
Proof
From \(J'(v_{n})v_{n}=o_{n}(1)\) as \(n\to \infty \), we obtain
Using limits (3.20), (3.25), and (3.35) together with \(J'(v)v=0\), we obtain
The application of Brezis–Lieb lemma in [14] yields
As in the proof of (3.6), we see that
Clearly, it follows from (3.57) and (3.58) that, to conclude \(v_{n}\to v\) in E, it remains to prove
Indeed, by Fatou’s lemma, for any \(r>0\), we have
On the other hand, from (3.34), one sees
Noticing that the function \(\phi ''(t)>p(p-2\alpha )|f(t)|^{p-2}(f'(t))^{2}>0\) in \({\mathbb{R}}\setminus \{0\}\), we know that \(\phi (t)\) is convex and even in \({\mathbb{R}}\), where \(\phi (t)=|f(t)|^{p}\). Hence, by \((f_{9})\), it follows from (3.40) and (3.41) that
for large n. Since \(|f(v_{n}-v)|^{p}\le |v_{n}-v|^{p}\) and \(v_{n}\to v\) in \(L^{p}(B_{2r})\), we have \(\int _{B_{2r}} V(x)|f(v_{n}-v)|^{p}\,dx\to 0\) as \(n\to \infty \). Altogether, we get (3.59) and \(v_{n}\to v\) in E. This completes the proof of Lemma 3.5. □
4 Proof of Theorem 1.1
We need the following mountain pass theorem to prove our result.
Lemma 4.1
([21], Theorem 9.12). Let E be an infinite dimensional real Banach space, \(J\in C^{1}(E,{\mathbb{R}})\) be even and satisfy the Cerami condition, and \(J(0)=0\). If \(E=Y\oplus Z\), Y is finite dimensional and J satisfies
\((J_{1})\) There exist constants \(\rho , \tau >0\) such that \(J(u)\ge \tau \) on \(\partial B_{\rho}\cap Z\);
\((J_{2})\) For each finite dimensional subspace \(E_{0}\subset E\), there is \(R_{0}=R_{0}(E_{0})\) such that \(J(u)\le 0\) on \(E_{0}\setminus B_{R_{0}}\), where \(B_{r}=\{v\in E: \|v\|_{E}< r\}\).
Then J possesses an unbounded sequence of critical values.
Proof of Theorem 1.1
Clearly, the functional J defined by (2.10) is even in E. By Lemmas 3.1–3.5 in Sect. 3, the functional J satisfies the Cerami condition. Next, we prove that J satisfies \((J_{1})\) and \((J_{2})\).
From \((f_{5})\) and Hölder’s inequality, we deduce that
with some constant \(C_{1}>0\). Similarly, if \((H_{3})\) is true, then one sees that
If \((H_{4})\) holds, one has
Moreover, it follows from \((f_{3})\), \((f_{5})\) and Hölder’s inequality that
with some \(C_{2}>0\) and \(h_{0}=\|h_{2}\|_{L^{\infty}(B_{1}^{c})}\), \(q_{0}=pt+p^{*}(1-t)\), \(t= \frac{2\alpha p^{*}-q}{2\alpha p^{*}-p}\). Clearly, \(q_{0}>\frac{q}{2\alpha}\). Then (4.3) and (4.4) show that there is a constant \(C_{3}>0\) such that
As in the proof of (3.6), we can derive
Then, from (4.1),(4.2), and (4.5), we conclude that
where \(\beta _{1}=C_{1}\), \(\beta _{2}=\min \{C_{1},C_{3}\}\). Denote
Choose \(z_{1}\in (0,1)\) such that
This is possible since \(\frac{q}{2\alpha}>p\). Moreover, let
Then
So, it follows from (4.8), (4.10), and (4.11) that there exist \(\lambda _{0}, \tau , \rho >0\) such that \(J(v)\ge \tau \) with \(\rho =z_{1}=\|v\|_{E}\) and \(\lambda \in [0,\lambda _{0}]\). Thus condition \((J_{1})\) is satisfied.
We now verify \((J_{2})\). For any finite dimensional subspace \(E_{0}\subset E\), we assert that there exists a constant \(R_{0}>\rho \) such that \(J<0\) on \(E_{0}\setminus B_{R_{0}}\). Otherwise, there is a sequence \(\{v_{n}\}\subset E_{0}\) such that \(\|v_{n}\|_{E}\to \infty \) and \(J(v_{n})\ge 0\). Hence,
Set \(\omega _{n}=\frac{v_{n}}{\|v_{n}\|_{E}}\). Then up to a subsequence, we can assume \(\omega _{n}\rightharpoonup \omega \) in E, \(\omega _{n}(x)\to \omega (x)\) a.e. in \({\mathbb{R}}^{N}\). Denote \(\Omega =\{x\in {\mathbb{R}}^{N}: \omega (x)\neq0\}\). Assume \(|\Omega |>0\). Clearly, \(v_{n}(x)\to \infty \) in Ω. It follows from (4.1) that
On the other hand, from \((f_{7})\), we derive
Therefore,
But it is easy to see that
We have a contradiction from (4.12), (4.15), and (4.16). So, \(|\Omega |=0\) and \(\omega (x)=0\) a.e. on \({\mathbb{R}}^{N}\). By the equivalency of all norms in \(E_{0}\), there exists a constant \(\beta >0\) such that
Hence,
It is impossible. This shows that there is a constant \(R_{0}>0\) such that \(J<0\) on \(E_{0}\setminus B_{R_{0}}\). Therefore, the existence of infinitely many solutions \(\{v_{n}\}\) for problem (2.12) follows from Lemma 4.1, and so \(u_{n}=f(v_{n})\) is a solution of Problem (1.1) for \(n=1,2,\ldots \) . We finish the proof of Theorem 1.1. □
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The support was received from the Fundamental Research Funds for the Central Universities of China (2015B31014), and the National Natural Science Foundation of China (No.11571092), and from the China Postdoctoral Science Foundations(Grant No.2017M611664).
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Chen, L., Chen, C., Chen, Q. et al. Infinitely many solutions for quasilinear Schrödinger equation with concave-convex nonlinearities. Bound Value Probl 2024, 24 (2024). https://doi.org/10.1186/s13661-023-01805-3
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DOI: https://doi.org/10.1186/s13661-023-01805-3