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Sign-changing solutions for Schrödinger–Kirchhoff-type fourth-order equation with potential vanishing at infinity
Journal of Inequalities and Applications volume 2021, Article number: 27 (2021)
Abstract
The purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation:
where \(5\leq N\leq 7\), \(\Delta ^{2}\) denotes the biharmonic operator, \(K(x), V(x)\) are positive continuous functions which vanish at infinity, and \(f(u)\) is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally, f is an odd function, we obtain that equation has infinitely many nontrivial solutions.
1 Introduction and main results
This article is concerned with the following fourth-order Kirchhoff-type equation:
where \(5\leq N\leq 7\), \(\Delta ^{2}\) denotes the biharmonic operator, and \(a,b\) are positive constants.
When \(a=1, b=0\), equation (1.1) becomes the following fourth-order equation (replace \(\mathbb{R}^{N}\) with Ω):
where Ω is an open subset of \(\mathbb{R}^{N}\). There are many results focused on the existence, multiplicity, and concentration of solutions to problem (1.2), see for instance [12, 13, 45, 48, 53–55, 58, 59] and the references therein.
Problem (1.1) stems from the following Kirchhoff equation:
where \(\Omega \subset \mathbb{R}^{N}\) is a bounded domain, \(a,b>0\).
Problem (1.3) comes from the following equation:
Because equation (1.4) is regarded as a good approximation for describing nonlinear vibrations of beams or plates, it is used to describe some phenomena that appear in different physical, engineering, and other sciences [5, 9].
Since it involves \((\int _{\mathbb{R}^{N}}|\nabla u|^{2}\,dx)\Delta u\) or \((\int _{\Omega }|\nabla u|^{2}\,dx)\Delta u\), problem (1.1) or (1.3) has become particularly interesting. By means of the fixed point theorems or variational approach, Ma [26–28] discussed positive solutions (or solutions) for problems similar to problem (1.3) when \(\mathbb{N}=1\). When \(\mathbb{N}\geq 2\), there have been many papers about solutions to problem (1.1) or (1.3), see, for example, [4, 29, 36, 44, 46, 50–52]. However, except [21, 57], there are very few papers considering sign-changing solutions. By combining constraint variation methods and deformation lemma, Zhang et al. [57] studied sign-changing solution to problem (1.1) when \(K(x)\equiv 1\). When \(K(x)\equiv 1\) and \(f(u)=|u|^{p-2}u, 4< p<2_{\ast }\) (2∗ defined below), by the minimization argument on the sign-changing Nehari manifold, Khoutir and Chen [21] discussed sign-changing solution to problem (1.1).
It is noticed that in the past decades many mathematicians have paid much of their attention to nonlocal problems. The appearance of nonlocal terms in the equations not only marks their importance in many physical applications but also causes some difficulties and challenges from a mathematical point of view. Certainly, this fact makes the study of nonlocal problems particularly interesting. In addition to the equations of Kirchhoff type, there are also some nonlocal problems, such as Schrödinger–Poisson systems, equations with the fractional Laplacian operator, and so on. Especially, these days there is a good trend of existence of solutions for fractional-order differential equations which are definitely the generalized study [1, 2, 15–20, 22, 32, 33].
Throughout this paper, as in [3], we say that \((V, K)\in \mathcal{K}\) if continuous functions \(V, K:\mathbb{R}^{N}\rightarrow \mathbb{R}\) satisfy the following conditions:
- \((VK_{0})\):
-
\(V(x),K(x)>0\) for all \(x\in \mathbb{R}^{N}\) and \(K\in L^{\infty }(\mathbb{R}^{N})\).
- \((VK_{1})\):
-
If \(\{A_{n}\}_{n}\subset \mathbb{R}^{N}\) is a sequence of Borel sets such that \(|A_{n}|\leq R\) for all \(n\in \mathbb{N}\) and some \(R>0\), then
$$ \lim_{r\rightarrow +\infty } \int _{A_{n}\cap B^{c}_{r}(0)}K(x)=0 \quad\text{uniformly in } n\in \mathbb{N}. $$One of the following conditions occurs:
- \((VK_{2})\):
-
\(K/V\in L^{\infty }(\mathbb{R}^{N})\);
or
- \((VK_{3})\):
-
There is \(p\in (2,2_{\ast })\) such that
$$ \frac{K(x)}{ \vert V (x) \vert ^{\frac{2_{\ast }-p}{2_{\ast }-2}}}\rightarrow 0 \quad\text{as } \vert x \vert \rightarrow \infty, $$where \(2_{\ast }=\frac{2N}{N-4}\) is the critical Sobolev exponent.
As for the function f, we assume \(f\in C(\mathbb{R},\mathbb{R})\) and satisfies the following conditions:
- \((f_{1})\):
-
$$ f(t)=o \bigl( \vert t \vert ^{3} \bigr) \quad\text{as } t \rightarrow 0 \text{ if } (VK_{2}) \text{ holds}. $$
- \((f_{2})\):
-
$$ \limsup_{t\rightarrow 0}\frac{f(t)}{ \vert t \vert ^{p-1}}< +\infty, \quad\text{if }(VK_{3})\text{ holds}. $$
- \((f_{3})\):
-
f has a “quasicritical growth”, that is,
$$ \limsup_{t\rightarrow +\infty }\frac{f(t)}{ \vert t \vert ^{2_{\ast }-1}}=0. $$ - \((f_{4})\):
-
\(\lim_{t\rightarrow \infty }\frac{F(t)}{t^{4}}=+\infty \), where \(F(t)=\int _{0}^{t}f(s)\,ds\).
- \((f_{5})\):
-
\(\frac{f(t)}{|t|^{3}}\) is an increasing function of \(t\in \mathbb{R}\backslash \{0\}\).
Let
be a Banach space endowed with the norm \(\Vert u \Vert _{A}:=(\int _{\mathbb{R}^{N}}(|\nabla u|^{2}+V(x)u^{2})\,dx)^{ \frac{1}{2}}\).
It follows from \((V, K)\in \mathcal{K}\) that the space B given by
with
is compactly embedded into the weighted Lebesgue space \(L^{q}_{K}(\mathbb{R}^{N})\) for some \(q\in (2,2_{\ast })\) (see Proposition 2.2), where \(L^{q}_{K}(\mathbb{R}^{N})\) given by
with
In this paper, we discuss our problem on the space
So, it is easy to see that E is a Hilbert space. Furthermore,
For problem (1.1), the energy functional is given by
where \(F(u)=\int ^{u}_{0}f(t)\,dt\).
It follows from the conditions of this paper that \(I_{b}(u)\) belongs to \(C^{1}\) and
for any \(u,v\in E\).
The solution of problem (1.1) is the critical point of the functional \(I_{b}(u)\). Furthermore, we say that u is a sign-changing solution if \(u\in E\) is a solution to problem (1.1) and \(u^{\pm }\neq 0\), where \(u^{+}=\max \{u(x),0\}, u^{-}=\min \{u(x),0\}\).
As pointed out in the article, since the nonlocal term \((\int _{\mathbb{R}^{N}}|\nabla u|^{2}\,dx)\Delta u\) is involved, there is an essential difference between problem (1.1) and problem (1.2) when we discussed the existence of sign-changing solutions, see [6–8, 10, 25, 37, 61].
Therefore, to study sign-changing solutions for problem (1.1), as in [6, 7, 10], we first obtain a minimizer of \(I_{b}\) over the constraint
The rest is to prove that the minimizer is a sign-changing solution of problem (1.1). It is noticed that there are some interesting results, for example, [11, 14, 23, 24, 30, 31, 34, 35, 38–43, 47, 56, 60], which considered sign-changing solutions for other nonlocal problems.
The main result can be stated as follows.
Theorem 1.1
Suppose that \((V,K)\in \mathcal{K}\) and \((f_{1})\)–\((f_{5})\) are satisfied. Then problem (1.1) has a least energy sign-changing solution u in E. If, additionally, f is an odd function, then problem (1.1) has infinitely many nontrivial solutions.
Remark 1.1
In this paper, the potential V vanishing at infinity means
which is also used to characterize one problem as zero mass.
The rest of this paper proceeds as follows. Sections 2 and 3 are devoted to our variational setting, and necessary lemmas are shown and proved, which shall be used in the proof of our main results in Sect. 4.
2 The variational framework and preliminary results
Proposition 2.1
([13])
Assume \((V,K)\in \mathcal{K}\). If \((VK_{2})\) holds, then B is continuously embedded in \(L^{q}_{K}(\mathbb{R}^{N})\) for every \(q\in [2,2_{\ast }]\); if \((VK_{3})\) holds, then B is continuously embedded in \(L^{p}_{K}(\mathbb{R}^{N})\).
Proposition 2.2
([13])
Assume \((V,K)\in \mathcal{K}\). If \((VK_{2})\) holds, then B is compactly embedded in \(L^{q}_{K}(\mathbb{R}^{N})\) for every \(q\in (2,2_{\ast })\); if \((VK_{3})\) holds, then B is compactly embedded in \(L^{p}_{K}(\mathbb{R}^{N})\).
Remark 2.1
Since E is continuously embedded in B, it is easy to see that Proposition 2.1 and Proposition 2.2 also hold if B is replaced with E.
Lemma 2.1
([13])
Suppose that \((V,K)\in \mathcal{K}\) and \((f_{1})\)–\((f_{2})\) are satisfied. Let \(\{v_{n}\}\) be a sequence such that \(v_{n}\rightharpoonup v\) in E. Then
Lemma 2.2
Assume that \((V,K)\in \mathcal{K}\) and \((f_{1})\)–\((f_{5})\) hold. Then, for any \(u\in E\backslash \{0\}\),
Proof
The proof is similar to that of Lemma 2.4 in [24], so we omit it here. □
Similarly, we have the following results.
Lemma 2.3
Assume that \((V,K)\in \mathcal{K}\) and \((f_{1})\)–\((f_{5})\) hold. Then, for any \(u\in E\backslash \{0\}\),
Lemma 2.4
Assume that \((V,K)\in \mathcal{K}\) and \((f_{1})\)–\((f_{5})\) hold. Then, for any \(u\in E\backslash \{0\}\),
Let
The following results are very important, because they allow us to overcome the non-differentiability of \(\mathcal{N}\) (see Lemma 2.5(iii)).
Lemma 2.5
Assume that \((V,K)\in \mathcal{K}\) and \((f_{1})\)–\((f_{5})\) hold. If \(u\in E\) with \(u^{\pm }\neq 0\), then
-
(i)
For each \(u\in E\backslash \{0\}\), let \(h_{u}:\mathbb{R}_{+}\rightarrow R\) be defined by \(h_{u}(t)=I_{b}(tu)\). Then there is unique \(t_{u}>0\) such that \(h'_{u}(t)>0\) in \((0,t_{u})\) and \(h'_{u}(t)<0\) in \((t_{u},\infty )\).
-
(ii)
There is \(\tau >0\) independent of u such that \(t_{u}\geq \tau \) for all \(u\in \mathcal{S}\), which is the unit sphere on E. Moreover, for each compact set \(Q\subset \mathcal{S}\), there is \(C_{Q}>0\) such that \(t_{u}\leq C_{Q}\) for all \(u\in Q\).
-
(iii)
The map \(\widehat{m}:E\backslash \{0\}\rightarrow \mathcal{N}\) given by \(\widehat{m}(u)=t_{u}u\) is continuous and \(m:=\widehat{m}_{|_{\mathcal{S}}}\) is a homeomorphism between \(\mathcal{S}\) and \(\mathcal{N}\). Moreover, \(m^{-1}(u)= u/\|u\|\).
Proof
If \((VK_{2})\) holds. From \((f_{1})\) and \((f_{3})\), for any \(\varepsilon >0\), there exists \(C_{\varepsilon }>0\) such that
So,
Let \(\varepsilon <\frac{1}{2}/|K/V|_{\infty }\), there is \(t_{0}>0\) sufficiently small such that
If \((VK_{3})\) holds. According to arguments in [13], there is \(C_{p}>0\) such that, for given \(\varepsilon \in (0,C_{p})\), there exists \(L>0\) satisfying
So, it follows from \((f_{2})\) and \((f_{3})\) that
According to (2.4), Hölder’s inequality, and \((VK_{0})\), one has that
Since \(p>2\) and \(2_{\ast }>2\), we have that (2.3) also holds.
On the other hand, thanks to \(F(s)\geq 0, \forall s\in \mathbb{R}\), we get
where \(D\subset supp u\) is a measurable set with finite and positive measures.
Hence, by combining Fatou’s lemma and \((f_{4})\), one has
So, there is \(\widetilde{t}_{0}>0\) sufficiently large so that
Therefore, by the continuity of \(h_{u}\) and \((f_{5})\), there is \(t_{u}>0\) which is a maximum global point of \(h_{u}\) with \(t_{u}u\in \mathcal{N}\).
We assert that \(t_{u}\) is the unique critical point of \(h_{u}\). In fact, suppose, by contradiction, that there are \(t_{1}>t_{2}>0\) such that \(h'_{u}(t_{1})=h'_{u}(t_{2})=0\). Then
which is absurd.
Next, we prove (ii).
For any \(u\in \mathcal{S}\), according to (i), there exists \(t_{u}>0\) such that
By (2.1), we have that
So, there exists \(\tau >0\), independent of u, such that \(t_{u}\geq \tau \).
On the other hand, let \(Q\subset \mathcal{S}\) be compact. Suppose that there exist \(\{u_{n}\}\subset Q, u\in Q\) such that \(t_{n}:=t_{u_{n}}\rightarrow \infty \), \(u_{n}\rightarrow u\) in E. So, it follows from (2.6) that
According to \((f_{5})\), we obtain that
is increasing when \(t>0\) and decreasing when \(t<0\). Hence we have, for each \(u\in \mathcal{N}\), that
Thanks to \(\{t_{u_{n}}u_{n}\}\subset \mathcal{N}\), replaced u with \(t_{u_{n}}u_{n}\) in (2.12), from (2.10) we have a contradiction. Therefore, (ii) holds.
Finally, we prove (iii). We assert that m, m̂, \(m^{-1}\) are well defined. Indeed, for each \(u\in E\backslash \{0\}\), by (i), there is unique \(m(u)\in \mathcal{N}\). If \(u\in \mathcal{N}\), then \(u\neq 0\), it is easy to see that \(m^{-1}(u)=u/\|u\|\in \mathcal{S}\). So, \(m^{-1}\) is well defined. Furthermore, we have that
Hence, m is bijective and \(m^{-1}\) is continuous.
In what follows, we prove \(\widehat{m}:E\backslash \{0\}\rightarrow \mathcal{N}\) is continuous. Suppose \(\{u_{n}\}\subset E\backslash \{0\}\) and \(u\in E\backslash \{0\}\) such that \(u_{n}\rightarrow u\) in E. According to (ii), there is \(t_{0}>0\) such that \(\|u_{n}\|t_{u_{n}}=t_{(\frac{u_{n}}{\|u_{n}\|})}\rightarrow t_{0}\). So, we have \(t_{u_{n}}\rightarrow \frac{t_{0}}{\|u\|}=:\widetilde{t}_{0}\). Thanks to \(t_{u_{n}}u_{n}\in \mathcal{N}\), one has that
From the above equality, let \(n\rightarrow \infty \), one has that
which indicates that \((t_{0}/\|u\|)u\in \mathcal{N}\) and \(t_{u}=t_{0}/\|u\|\). Therefore, \(\widehat{m}(u_{n} )\rightarrow \widehat{m}(u)\). So, the proof is completed. □
Define \(\widehat{\Psi }:E\rightarrow \mathbb{R}\) and \(\Psi:\mathcal{S}\rightarrow \mathbb{R}\) by
By Lemma 2.5 and the result from [37], one has the following.
Proposition 2.3
Assume that \((V,K)\in \mathcal{K}\) and \((f_{1})\)–\((f_{5})\) hold, then
(i) \(\widehat{\Psi }\in C^{1}(E\setminus \{0\},\mathbb{R})\) and
(ii) \(\Psi \in C^{1}(\mathcal{S},\mathbb{R})\) and
(iii) If \({u_{n}}\) is a \((PS)_{d}\) sequence for Ψ, then \(m({u_{n})}\) is a \((PS)_{d}\) sequence for \(I_{b}\). If \({u_{n}}\subset \mathcal{N}\) is a bounded \((PS)_{d}\) sequence for \(I_{b}\), then \({m^{-1}(u_{n})}\) is a \((PS)_{d}\) sequence for Ψ.
(iv) u is a critical point of Ψ if, and only if, \(m(u)\) is a nontrivial critical point of \(I_{b}\). Moreover, corresponding critical values coincide and
Proposition 2.4
If \((f_{1})\)–\((f_{5})\) hold, then
3 Technical lemmas
For \(u\in E\) with \(u^{\pm }\neq 0\), let \(\varphi _{u}(s,t):=I_{b}(su^{+}+tu^{-}), s>0, t>0\).
Lemma 3.1
Assume that \((V,K)\in \mathcal{K}\) and \((f_{1})\)–\((f_{5})\) hold. If \(u\in E\) with \(u^{\pm }\neq 0\), then
-
(i)
the pair \((s,t)\) of a critical point of \(\varphi _{u}(s,t)\) with \(s,t>0\) if and only if \(su^{+}+tu^{-}\in \mathcal{M}\),
-
(ii)
the map \(\varphi _{u}(s,t)\) has a unique critical point \((s_{+},t_{-})\), with \(s_{+}=s_{+}(u)>0\) and \(t_{-}=t_{-}(u)>0\), which is the unique maximum point of \(\varphi _{u}(s,t)\).
-
(iii)
The maps \(\alpha _{+}(r)=\frac{\partial \varphi _{u}}{\partial s}(r,t_{-})r\) and \(\alpha _{-}(r)=\frac{\partial \varphi _{u}}{\partial t}(s_{+},r)r\) are such that \(\alpha _{+}(r)>0\) if \(r\in (0,s_{+})\), \(\alpha _{-}(r)>0\) if \(r\in (0,t_{-})\), \(\alpha _{+}(r)<0\) if \(r\in (s_{+},\infty )\), and \(\alpha _{-}(r)<0\) if \(r\in (t_{-},\infty )\).
Proof
It is easy to see that
where
Hence, item (i) is obvious.
In the following, we prove (ii). Firstly, we assert that \(\mathcal{M}\neq \emptyset \). By (i), we only prove the existence of a critical point of \(\varphi _{u}(s,t)\). Let \(u\in E\) with \(u^{\pm }\neq 0\) and \(t_{0}\geq 0\) fixed, it follows from (3.1) that
Together with Lemma 2.4 and Lemma 2.2, one gets
Since \(g_{u}(s,t_{0})\) is continuous, there exists \(s_{0}>0\) such that \(g_{u}(s_{0},t_{0})=0\). We assert that \(s_{0}\) is unique. In fact, supposing by contradiction, there exist \(0< s_{1}< s_{2}\) such that \(g_{u}(s_{1},t_{0})=g_{u}(s_{2},t_{0})=0\), and then we have
So,
Therefore, it follows from \((f_{5})\) and \(0< s_{1}< s_{2}\) that we have a contradiction. That is, there exists unique \(s_{0}>0\) such that \(g_{u}(s_{0},t_{0})=0\).
Let \(\phi _{1}(t):=s(t)\), where \(s(t)\) satisfies the properties just mentioned previously, with t in the place of \(t_{0}\). Then the map \(\phi _{1}: \mathbb{R}_{+}\rightarrow (0,\infty )\) is well defined.
By definition, one has that \(\frac{\partial \varphi _{u}}{\partial s}(\phi _{1}(t),t)=0 t\geq 0\). Then
We assert that \(\phi _{1}(t)\) has some good properties.
(1) \(\phi _{1}(t)\) is continuous. To this end, let \(t_{n}\rightarrow t_{0}\) as \(n\rightarrow \infty \) and suppose, by contradiction, that there is a subsequence, still denoted by \(t_{n}\), such that \(\phi _{1}(t_{n})\rightarrow \infty \).
Obviously, \(\phi _{1}(t_{n})\geq t_{n}\) for n large enough. According to (3.3), one has that
In view of Lemma 2.2, we have a contradiction. So \(\phi _{1}(t_{n})\) is bounded. Therefore, there exists \(s_{0}\geq 0\) such that, passing to a subsequence,
Combining (3.3) with (3.5), we have that
that is,
Consequently, \(s_{0}=\phi _{1}(t_{0})\). That is, \(\phi _{1}\) is continuous.
(2) \(\phi _{1}(t)>0\). Suppose, by contradiction, that there is a sequence \(\{t_{n}\}\) such that \(\phi _{1}(t_{n})\rightarrow 0+\) as \(n\rightarrow \infty \). In view of (3.3) and Lemma 2.4, we have
which is absurd, and hence there is \(C>0\) such that \(\phi _{1}(t)\geq C\).
(3) \(\phi _{1}(t)< t\) for t large. Indeed, if there exists a sequence \(\{t_{n}\}\) with \(t_{n}\rightarrow \infty \) such that \(\phi _{1}(t_{n})\geq t_{n}\) for all \(n\in \mathbb{N}\), then arguing as in (3.4), we have a contradiction. Thus, \(\phi _{1}(t)< t\) for t large.
Similarly, according to definition of \(h_{u}(s,t)\), we can define a map \(\phi _{2}:\mathbb{R_{+}}\rightarrow (0, \infty )\) by \(\phi _{2}(s)=t(s)\) satisfying (1), (2), and (3).
By (3), there exists \(C_{1}>0\) such that \(\phi _{1}(t)\leq t\) and \(\phi _{2}(s)\leq s\) respectively when \(t,s>C_{1}\). Let \(C_{2}=\max \{\max_{t\in [0,C_{1}]}\phi _{1}(t),\max_{s\in [0,C_{1}]} \phi _{2}(s)\}\), \(C=\max \{C_{1},C_{2}\}\), define \(T:[0,C]\times [0,C]\rightarrow \mathbb{R}^{2}_{+}\) by
It is easy to see that \(T(s,t)\in [0,C]\times [0,C]\) for all \((s, t)\in [0,C]\times [0,C]\). Since T is continuous, using the Brouwer fixed point theorem, there exists \((s_{+},t_{-})\in [0,C]\times [0,C]\) such that
It follows from \(\phi _{i}>0\) that \(s_{+},t_{-}>0\). According to the definition, we have
We next shall prove the uniqueness of \(s_{+},t_{-}\). Suppose that \(\omega \in \mathcal{M}\), one has
which shows that \((1,1)\) is a critical point of \(\varphi _{\omega }\). Now, we need to prove that \((1,1)\) is the unique critical point of \(\varphi _{\omega }\) with positive coordinates. Let \((s_{0},t_{0})\) be a critical point of \(\varphi _{\omega }\) such that \(0< s_{0}\leq t_{0}\). So, one has that
and
Thanks to \(0< s_{0}\leq t_{0}\) and (3.8), we have that
On the other hand, for \(\omega \in \mathcal{M}\), we have
Combining (3.9) with (3.10), one has that
If \(t_{0}>1\), the left-hand side of the above inequality is negative, which is absurd because the right-hand side is positive by condition \((f_{5})\). Therefore, we obtain that \(0< s_{0}\leq t_{0}\leq 1\).
Similarly, by (3.7) and \(0< s_{0}\leq t_{0}\), we get
and from \((f_{5})\) this is absurd. Therefore, we have \(s_{0}\geq 1\). Consequently, \(s_{0}=t_{0}=1\), which indicates that \((1,1)\) is the unique critical point of \(\varphi _{\omega }\) with positive coordinates.
Let \(u\in E\), \(u^{\pm }\neq 0\) and \((s_{1},t_{1})\), \((s_{2},t_{2})\) be the critical points of \(\varphi _{u}\) with positive coordinates. In view of (i), one has that
So,
It follows from \(\omega _{1}\in E\) and \(\omega _{1}^{\pm }\neq 0\) that \((\frac{s_{2}}{s_{1}},\frac{t_{2}}{t_{1}})\) is a critical point of the map \(\varphi _{\omega _{1}}\) with positive coordinates. Thanks to \(\omega _{1}\in \mathcal{M}\), one has that
Hence, \(s_{1}=s_{2}\), \(t_{1}=t_{2}\).
Now, we prove that the unique critical point is the unique maximum point of \(\varphi _{u}\). In fact, using Lemma 2.3, we have that
Hence, the maximum point of \(\varphi _{u}(s,t)\) cannot be achieved on the boundary of \((\mathbb{R}_{+}\times \mathbb{R}_{+})\). Without loss of generality, we may assume that \((0,\bar{t})\) is a maximum point of \(\varphi _{u}(s,t)\). But, according to Lemma 2.4, it is obvious that
is an increasing function with respect to s if s is small enough. Hence, \((0,\bar{t})\) is not a maximum point of φ in \(\mathbb{R}_{+}\times \mathbb{R}_{+}\).
Finally, we prove (iii). From (i) of Lemma 2.5, we get \(\frac{\partial \varphi _{u}}{\partial s}(r,t_{-})>0\) if \(r\in (0,s_{+})\) and \(\frac{\partial \varphi _{u}}{\partial s}(s_{+},t_{-})=0\) and \(\frac{\partial \varphi }{\partial s}(r,t_{-})<0\) if \(r\in (t_{-},\infty )\). Therefore, \(\alpha _{+}\) and \(\alpha _{-}\) have the same behavior. □
Lemma 3.2
If \(\{u_{n}\}\in \mathcal{M}\) and \(u_{n}\rightharpoonup u\) in E, then \(u\in E^{\pm }\).
Proof
For any \(v\in \mathcal{M}\), we have that
Similar to (2.9), we obtain that there is \(\tau >0\) such that
So, if \(\{u_{n}\}\subset \mathcal{M}\), we have
Combining \(u_{n}\rightharpoonup u\) in E with Proposition 2.2, we have
which shows that \(u\in E^{\pm }\). □
Next, we consider the following minimization problem:
We claim
In fact, since \(\mathcal{M}\subset \mathcal{N}\), we have \(m\geq c_{b}\). On the other hand, for any \(v\in \mathcal{M}\), according to (i) of Lemma 2.5, there exist positive constants \(s_{+}\) and \(t_{-}\) such that \(s_{+}v^{+}, t_{-}v^{-}\in \mathcal{N}\). Therefore, from (i) and (ii) of Lemma 3.1, we have
Lemma 3.3
Assume that \((V,K)\in \mathcal{K}\) and \((f_{1})\)–\((f_{5})\) hold, then m is achieved.
Proof
Let \(\{u_{n}\}\) be a sequence in \(\mathcal{M}\) such that
We will show that \(u_{n}\) is bounded in E. In fact, suppose that there exists a subsequence that we still call \({u_{n}}\) such that
Now, we define \(v_{n}:=u_{n}/\|u_{n}\|\) for all \(n\in \mathbb{N}\). So, there exists \(v\in E\) such that
From Proposition 2.2, we conclude that, up to a subsequence,
Using (i) of Lemma 3.1, it follows from \(\{u_{n}\}\subset \mathcal{M}\) that \(s_{+}(v_{n})=t_{-}(v_{n})=\|u_{n}\|\). Therefore, using (i) of Lemma 3.1 again, we obtain
Let \(t\geq 1\) in (3.21), we have that
Suppose that \(v=0\). Hence, from (3.19) and Lemma 2.1 we have
By (3.17) and (3.23), passing to the limit as \(n\rightarrow \infty \) in (3.22), we have that
Thanks to \(\|v_{n}\|=1\), there exists a constant \(\alpha _{0}>0\) such that
So, we have a contradiction. Hence, \(v\neq0\).
On the other hand, we get
Thanks to \(v\neq0\), by using Lemma 2.3, we get
Therefore, by (3.17), (3.18), and (3.26), passing to the limit as \(n\rightarrow \infty \) in (3.25), we have a contradiction.
Hence, we deduce that \(\{u_{n}\}\) is bounded in E. Therefore, there exists \(u\in E\) such that \(u_{n}\rightharpoonup u,u_{n}^{\pm }\rightharpoonup u^{\pm }\).
From Lemma 3.2, we have that \(u\in E^{\pm }\). So, according to Lemma 3.1, there exist \(s_{+},t_{-}>0\) such that
We assert that
In fact, according to Lemma 2.1, one has that
Since \(u_{n}\rightharpoonup u\) in E, combining the continuous embedding \(E\hookrightarrow D^{1,2}(\mathbb{R}^{N})\) with the weak semicontinuity of the norm \(\|u\|_{D^{1,2}}=(\int _{\mathbb{R}^{N}}|\nabla u|^{2}\,dx)^{ \frac{1}{2}}\), we have
Thanks to \(\{u_{n}\}\subset \mathcal{M}\), using (3.29), (3.32), and weak semicontinuity of the norm in E, we have
Suppose that \(0< s_{+}\leq t_{-}\), then from (3.27) we have that
By (3.33), we have that
Combining (3.34) with (3.35), we have that
From the above inequality and \((f_{5})\), we have that \(0< t_{-}\leq 1\).
Now, we prove that \(I_{b}(s_{+}u^{+}+t_{-}u^{-})=m\).
Denoting \(\bar{u}:=\overline{s}u^{+}+\overline{t}u^{-}\). So, from (2.11), (3.27), (3.29), (3.30), and Fatou’s lemma, we have that
Consequently, \(\bar{s}=\bar{t}=1\). Thus, \(\bar{u}=u\) and \(I_{b}(u)=m\). □
4 The proof of the main results
In this section, we prove Theorem 1.1.
Proof
First, we prove that the minimizer u for (3.15) is indeed a sign-changing solution of problem (1.1). If \(I_{b}'(u)\neq 0\), then there exist \(\delta >0\) and \(\theta >0\) such that
Choose \(\sigma \in (0,\min \{1/2,\frac{\delta }{\sqrt{2}\|u\|}\})\). Let \(\Omega:=(1-\sigma,1+\sigma )\times (1-\sigma,1+\sigma )\) and \(\gamma (s,t)=su^{+}+tu^{-}\), \((s,t)\in \Omega \). It follows from Lemma 3.1 that
For \(\varepsilon:=\operatorname{min}\{(m-\bar{m})/2,\theta \delta /8\}\) and \(S_{\delta }:=B(u,\delta )\), according to Lemma 2.3 in [49], there is a deformation \(\eta \in C([0,1]\times E,E)\) such that
-
(a)
\(\eta (t,v)=v\) if \(v\notin I_{b}^{-1}([m-2\varepsilon,m+2\varepsilon ]\cap S_{2\delta })\);
-
(b)
\(\eta (1,I_{b}^{m+\varepsilon }\cap S_{\delta })\subset I_{b}^{m- \varepsilon }\);
-
(c)
\(I_{b}(\eta (1,v))\leq I_{b}(v)\) for all \(v\in E\);
-
(d)
\(\|\eta (t,v)-v\|\leq \delta \) for all \(v\in E, t\in [0,1]\).
Firstly, we need to prove that
In fact, it follows from Lemma 3.1 that \(I_{b}(\gamma (s,t))\leq m< m+\varepsilon \). That is,
On the other hand, we have
which shows that \(\gamma (s,t)\in S_{\delta }\) for all \((s,t)\in \bar{\Omega }\).
Therefore, according to \((b)\), we have \(I_{b}(\eta (1,\gamma (s,t)))< m-\varepsilon \). Hence, (4.2) holds.
In the following, we prove that \(\eta (1,\gamma (\Omega ))\cap \mathcal{M}\neq \varnothing \), which contradicts the definition of m.
Let \(\xi (s,t):=\eta (1,\gamma (s,t))\) and
According to (iii) of Lemma 3.1, the \(C^{1}\) function \(\varphi _{+}(s) = \varphi _{u}(s, 1)\) has a unique global maximum point \(s^{+} = 1\) (note that \(s\varphi _{+}'(s) = I'(\gamma (s, 1))su^{+}\)). According to density, given \(\varepsilon >0\) small enough, there exists \(\varphi _{+,\varepsilon }\in C^{\infty }([1-\sigma,1+\sigma ])\) satisfying \(\|\varphi _{+}-\varphi _{+,\varepsilon }\|_{C^{\infty }([1-\sigma,1+ \sigma ])}<\varepsilon \) with \(s^{+}=1\) being the unique maximum global point of \(\varphi _{+,\varepsilon }\) in \([1-\sigma,1+\sigma ]\). Hence, \(\|\varphi _{+}-\varphi _{+,\varepsilon }\|_{C^{\infty }([1-\sigma,1+ \sigma ])}<\varepsilon \), \(\varphi _{+,\varepsilon }'(1)=0\) and \(\varphi _{+,\varepsilon }''(1)<0\). Similarly, there exists \(\varphi _{-,\varepsilon }\in C^{\infty }([1-\sigma,1+\sigma ])\) satisfying \(\|\varphi _{-}-\varphi _{-,\varepsilon }\|_{C^{\infty }([1-\sigma,1+ \sigma ])}<\varepsilon \), \(\varphi _{-,\varepsilon }'(1)=0\) and \(\varphi _{-,\varepsilon }''(1)<0\), where \(\varphi _{-} (t)= \varphi _{u}(1, t)\).
Let \(\Upsilon _{\varepsilon }\in C^{\infty }(\Omega )\) be defined by \(\Upsilon _{\varepsilon }(s,t)=(s\varphi _{+,\varepsilon }'(s),t \varphi _{-,\varepsilon }'(t))\). Then we get \(\|\Upsilon _{\varepsilon }-\Upsilon _{0}\|_{C(\Omega )}< \frac{3\sqrt{2}\varepsilon }{2}\), \((0,0)\notin \Upsilon _{\varepsilon }(\partial \Omega )\), and \((0,0)\) is a regular value of \(\Upsilon _{\varepsilon }\) in Ω. On the other hand, \((1, 1)\) is the unique solution of equation \(\Upsilon _{\varepsilon }(t,s)=(0,0)\) in Ω. By using Brouwer’s degree, for ε small enough, we have
From
one has
where \(\operatorname{Jac}(\Upsilon _{\varepsilon })\) is the Jacobian determinant of \(\Upsilon _{\varepsilon }\) and sgn denotes the sign function.
On the other hand, according to (4.2), one has
Combining (4.4) with item (a), we have that \(\gamma =\xi \) on ∂Ω. Therefore, \(\Upsilon _{1}=\Upsilon _{0}\) on ∂Ω and
Therefore, we have \(\Upsilon _{1}(s,t)=(0,0)\) for some \((t,s)\in \Omega \).
We claim that
If (4.6) holds, by \((1,1)\in \Omega \), we have that \(\xi (1,1)=\eta (1,\gamma (1,1))\in \mathcal{M}\), this is \(\eta (1,\gamma (\Omega ))\cap \mathcal{M}\neq \varnothing \).
In what follows, we prove (4.6). If the zero \((t, s)\) of \(\Upsilon _{1}\) obtained above is equal to \((1,1)\), there is nothing to do. If \((s,t)\neq (1,1)\), let \(\delta _{1}=\max \{|t-1|, |s-1|\}\), \(\Omega _{1}=(1-\delta _{1}/2, 1+\delta _{1}/2)\times (1-\delta _{1}/2, 1+\delta _{1}/2)\). So, \((s,t)\in \Omega \backslash \Omega _{1}\) and for getting \((s_{1}, t_{1})\in \Omega _{1}\) such that \(\Upsilon _{1}(s_{1}, t_{1})=0\), we just repeat for \(\Omega _{1}\) as used in Ω. If \((s_{1}, t_{1})=(1,1)\), there is nothing to do. Otherwise, we can continue with the argument and find in the nth step that (4.6) holds, or produce a sequence \((s_{n}, t_{n})\) which converges to \((1,1)\) such that \(\Upsilon _{1}(s_{n}, t_{n})\) and \((s_{n},t_{n})\in \Omega _{n-1}\backslash \Omega _{n}\) for all \(n\in \mathbb{N}\) with \(\Omega _{0}=\Omega \). Let \(n\rightarrow \infty \) and, using the continuity of \(\Upsilon _{1}\), we have that (4.6) holds. That is,
So, we obtain that \(u:=u^{+} + u^{-}\) is a critical point of \(I_{b}\), that is, a sign-changing solution for problem (1.1).
Furthermore, if f is odd, the functional Ψ is even. Now we prove that Ψ satisfies the \((PS)\) condition. From (2.4) and (3.16), we have that Ψ is bounded from below in \(\mathcal{S}\). Suppose that \(\{u_{n}\}\subset \mathcal{S}\) is a \((PS)_{d}\) sequence of Ψ, according to (iii) of Lemma 2.3, we know \(\{v_{n}:=m(u_{n})\}\subset \mathcal{N}\) is a \((PS)_{d}\) sequence of \(I_{b}\) on \(\mathcal{N}\). Through the standard agrement at the beginning of this section, we know that \(v_{n}\) is bounded in E. So, there exists nonzero \(v\in E\) such that
Therefore, we have that
According to Lemma 2.1, we have that
We claim that
In fact, if \((VK_{2})\) holds, since
where χ is a character function. Hence \(\{\sqrt{K(x)}f(v_{n})\chi _{\{|v_{n}\leq 1|\}}\}\) is bounded in \(L^{2}(\mathbb{R}^{N})\).
Similarly, we have \(\sqrt{K(x)}v\in L^{2}(\mathbb{R}^{N})\). So, from \(v_{n}\rightarrow v \text{a.e. in }\mathbb{R}^{N}\), we can get
On the other hand, since \(|Kf(v_{n})\chi _{\{|v_{n}\geq 1|\}}|^{\frac{2_{\ast }}{2_{\ast }-1}} \leq Cv_{n}^{2_{\ast }}\) and \(v_{n}\rightarrow v\) a.e. on \(\mathbb{R}^{N}\), we get
If \((VK_{3})\) holds, according to Proposition 2.2, \(\int _{\mathbb{R}^{N}}K|v_{n}|^{p}\,dx<+\infty \) and
Then, by similar discussion, we have that (4.9) and (4.10) hold.
Therefore, from the above discussion, we have that (4.8) holds. Similarly, we have
Since \(v_{n}\rightharpoonup v\) in E and \(E\subset D^{1,2}(\mathbb{R}^{N})\), we get \(v_{n}\rightharpoonup v\) in \(D^{1,2}\). Then, by weak semicontinuity of the norm in \(D^{1,2}(\mathbb{R}^{N})\), we have that \(b(\int _{\mathbb{R}^{N}}|\nabla v_{n}|^{2}\,dx)(\int _{\mathbb{R}^{N}} \nabla v_{n}\cdot \nabla (v_{n}-v)\,dx)\geq 0\). Therefore, according to (4.7), we get \(v_{n}\rightarrow v\) in E. From Proposition 2.3, \(\{u_{n}:=m^{-1}(v_{n})\}\subset \mathcal{S}\) and \(u_{n}\rightarrow u=m^{-1}(v)\in \mathcal{S}\). That is, Ψ satisfies the Palais–Smale condition on \(\mathcal{S}\). So, from Lemma 2.5, Proposition 2.3, and [37], the functional \(I_{b}\) has infinitely many critical points. □
5 Conclusions
In this paper, by the minimization argument on the sign-changing Nehari manifold and the quantitative deformation lemma, we discussed the existence of least energy sign-changing solution for a class of Schrödinger–Kirchhoff-type fourth-order equations with potential vanishing at infinity. Our results improve and generalize some interesting known results. Since these days there is a good trend of existence of solutions for fractional-order differential equations which are definitely the generalized study, we will discuss some problems about fractional-order differential equations in the follow-up work.
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Guan, W., Zhang, HB. Sign-changing solutions for Schrödinger–Kirchhoff-type fourth-order equation with potential vanishing at infinity. J Inequal Appl 2021, 27 (2021). https://doi.org/10.1186/s13660-021-02552-8
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DOI: https://doi.org/10.1186/s13660-021-02552-8