Abstract
We consider a class of generalized quasilinear Schrödinger equations
where \(l(t): \mathbb{R}\to\mathbb{R}^{+}\) is a nondecreasing function with respect to \(|t|\), the potential function V is allowed to be sign-changing so that the Schrödinger operator \(-\Delta+V\) possesses a finite-dimensional negative space. We obtain existence and multiplicity results for the problem via the Symmetric Mountain Pass Theorem and Morse theory.
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1 Introduction
In our article, we study the generalized quasilinear Schrödinger problem as follows:
where \(N\geq1\), \(f\in C(\mathbb{R},\mathbb{R})\) and the function l satisfies the following assumptions:
- \((l)\):
\(l\in C^{2}(\mathbb{R},\mathbb{R}^{+})\), \(l(t)=l(-t)\), \(l(0)=1\), \(l'(t)\geq0\) for all \(t\geq0\), \(tl'(t)\leq l(t)\) for all \(t\in\mathbb{R}\) and \(l''(t)\geq0\) is strict on a subset of positive measure in \(\mathbb{R}\).
Solutions of (1.1) are related to the solitary wave solutions for the following quasilinear Schrödinger equations:
where \(z:\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{C}\), \(W:\mathbb {R}^{N}\to\mathbb{R}\) is a given potential, and f̃, g are real functions. The above problem (1.2) has been studied in several areas of physics corresponding to various types of g. For example, the case \(g(t)=t\) was used in [9] for the superfluid film equation in plasma physics. If \(g(t)=(1 + t)^{1/2}\), equation (1.2) models the self-channeling of a high-power ultrashort laser in matter (see [2] and [3]). Equation (1.2) also has relations with condensed matter theory (see [12]).
Taking \(z(x,t)=\exp(-iEt)u(x)\) in (1.2), with \(E>0\), we are led to investigate the following elliptic equation:
with \(V(x)=W(x)-E\). If we choose
then (1.3) turns into (1.1). In particular, if \(g(t)=t\), we have
For equation (1.4), to the best of our knowledge, the first results were proved by Poppenberg, Schmitt, and Wang in [13]. The idea in [13] is a constrained minimization argument. Subsequently, a general existence result for (1.4) was derived by Liu, Wang, and Wang [10]. The main existence results were obtained, through making a change of variable, reducing the quasilinear problem (1.4) to a semilinear one, and an Orlicz space framework was used to prove the existence of a positive solutions via Mountain Pass Theorem. The same method of variable change was also used by Colin and Jeanjean in [4], but the usual Sobolev space \(H^{1}(\mathbb{R}^{N})\) framework was chosen as the working space. We refer the readers to [5, 6, 11, 15, 17, 18, 20, 22, 23] for more results.
In all these papers, it is required that the potential V satisfies the positivity condition
With this assumption and suitable conditions on the nonlinearity f, one can show that \(u=0\) is a local minimizer of the energy functional associated with (1.4), which would then verify the mountain pass geometry and so Mountain Pass Theorem can be applied to produce a solution. However, from \(V(x)=W(x)-E\) we can see that, if the frequency E is large, then the potential \(V(x)\) in (1.4) could not satisfy (1.5).
In the literature (see [7]), there are some existence results which allow the potential V to be negative somewhere. The strategy is to write \(V = V^{+}-V^{-}\) with \(V^{\pm}=\max\{0,\pm V\} \). Then if \(V^{-}\) is in some sense small, the function
is still a norm on the function space. This is the key to verify that \(\bf0\) is a local minimizer of the corresponding energy functional.
Recently, Shi and Chen [19] studied problem (1.1) with a sign-changing potential V. Compared with [7], (1.6) fails to be a norm any more. To overcome this difficulty, the authors chose a constant \(V_{0}>0\) satisfying
and considered the equivalent problem
where \(\widetilde{f}(u)=f(u)+V_{0}u\). Unfortunately, from
in their condition \((f_{3})\), we cannot ensure
since \(\frac{tL(t)}{l(t)}-t^{2}\leq0\), where \(L(t)=\int _{0}^{t}l(s)\,ds\), \(F(t)=\int_{0}^{t}f(s)\,ds\), and \(\widetilde {F}(t)=\int_{0}^{t}\widetilde{f}(s)\,ds\).
Here we give an example to indicate why (1.7) fails. Consider
where \(L(t)=\int_{0}^{t} l(s)\,ds=\frac{1}{2\sqrt{2}}\ln(\sqrt{2}t+\sqrt {1+2t^{2}})+\frac{1}{2}t\sqrt{1+2t^{2}}\). Of course, we have \(F(t)=\int_{0}^{t}f(s)\,ds=\frac{1}{p}|L(t)|^{p}\). It is easy to find that \(f(t)\) and \(l(t)\) satisfy conditions \((f_{1})\)–\((f_{4})\) (see [19]) and \((l)\), respectively. Then we denote
Due to
this implies that, for some \(M\gg1\), there exist \(T_{1}\), \(T_{2}\) (\(1< T_{1}< T_{2}\)) such that
Since \(L(t)\) is continuous on \([T_{1},T_{2}]\), there exists \(K>0\) such that
Thus, for \(V_{0}\geq\frac{4K}{M}\), we have
Therefore, unlike what the authors declared at the beginning of [19], this new nonlinearity \(\widetilde{f}(t)\) does not satisfy their condition \((f_{3})\) any more. For this reason, their result may be valid for the case when the potential V is positive.
To the best of our knowledge, up to now in the literature there is only one research paper devoted to the situation that the quasilinear Schrödinger problems with “strongly” sign-changing potential. Recently, S. Liu et al. [11] proved the multiplicity of solutions of
where V is a sign-changing potential.
Our results extend and modify those obtained by S. Liu et al. [11] and H. Shi et al. [19]. Inspired by [11], we now present our hypotheses on the potential V and the nonlinearity f:
- \((V_{1})\):
\(V\in C(\mathbb{R}^{N},\mathbb{R})\) and \(\inf_{x\in\mathbb {R}^{N}} V(x)>-\infty\);
- \((V_{2})\):
\(\mu(V^{-1}(-\infty,M])<\infty\) for all \(M>0\), where μ is the Lebesgue measure;
- \((f_{1})\):
\(f\in C(\mathbb{R},\mathbb{R})\) and there exist \(C_{1}, C_{2}>0\) such that for all \(t\in\mathbb{R}\), \(p\in(2,2^{\ast})\),
$$ \bigl\vert f(t) \bigr\vert \leq C_{1} l(t) \bigl\vert L(t) \bigr\vert +C_{2} l(t) \bigl\vert L(t) \bigr\vert ^{p-1}; $$- \((f_{2})\):
there exists \(\mu>2\) such that for \(t\neq0\),
$$ 0< \mu l(t)F(t)\leq L(t)f(t); $$- \((f_{3})\):
\(f(t)=o(t)\) as \(t\to0\);
- \((f_{4})\):
\(f(t)=-f(-t)\).
We now summarize our main results:
Theorem 1.1
Assume\((l)\), \((V_{1})\), \((V_{2})\), \((f_{1})\), \((f_{2})\), and\((f_{4})\)hold. Then problem (1.1) has infinitely many solutions\(\{u_{n}\}\)inXwith\(I(u_{n})\to\infty\) (Xand\(I(\cdot)\)will be defined in Sect. 2).
Remark 1.1
From \((V_{1})\), the potential \(V(x)\) is allowed to be sign-changing.
Remark 1.2
Since l satisfies \(l(t)=l(-t)\), \(l'(t)\geq0\) for all \(t\geq0\), and \(tl'(t)\leq l(t)\) for all \(t\in\mathbb{R}\), we can easily obtain
for some constants \(C_{3}, C_{4}, C_{5}>0\).
Theorem 1.2
Assume\((l)\), \((V_{1})\), \((V_{2})\), and\((f_{1})\)–\((f_{3})\)hold. If 0 is not an eigenvalue of (2.2), then problem (1.1) possesses at least one nontrivial solution.
This paper is organized as follows. In Sect. 2, we describe the main preliminaries which we will use in this paper. Theorems 1.1 and 1.2 are proved in Sect. 3 and Sect. 4, respectively.
Notation. In this paper we use the following notations:
\(|u|_{s}= (\int_{\mathbb{R}^{N}}|u|^{s}\,dx )^{1/s}\) denotes the usual norm in \(L^{s}\)-space.
\(C, C_{1}, C_{2},\dots\) denote different positive constants.
We denote the weak and strong convergence in X, as \(n\to \infty\), by \(u_{n}\rightharpoonup u\) and \(u_{n}\to u\), respectively.
2 Preliminaries
Since \(V(x)\) is bounded from below, there exists \(V_{0}>0\) satisfying
We now introduce the working space. Set
which is a Hilbert space with the inner product
and the corresponding norm
From condition \((V_{2})\), we have a compact embedding \(X\hookrightarrow \hookrightarrow L^{s}(\mathbb{R}^{N})\) for \(s\in[2,2^{\ast})\) (see Bartsch–Wang [1]).
Applying the spectral theory of self-adjoint compact operators, let
be the sequence of eigenvalues of
where each eigenvalue is repeated according to its multiplicity, and let \(e_{1},e_{2},\dots\) be the corresponding orthonormal eigenfunctions in \(L^{2}(\mathbb{R}^{N})\).
Problem (1.1) is the Euler–Lagrange equation of the following energy functional:
But \(I(u)\) may be ill-behaved in X. To overcome this difficulty, we make a change of variables introduced in Shen and Wang [16], as
Firstly, we give some properties for L and \(L^{-1}\) which are defined in Sect. 1.
Lemma 2.1
([19])
The functions\(L(t)\)and\(L^{-1}(s)\)satisfy the following properties:
- (1)
Lis odd, from class\(C^{2}\)and invertible;
- (2)
\(\lim_{|s|\to0}\frac{L^{-1}(s)}{s}=1\);
- (3)$$ \lim_{s\to+\infty}\frac{L^{-1}(s)}{s}=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{l(\infty)}, &\textit{if } l \textit{ is bounded},\\ o(1), &\textit{if } l \textit{ is unbounded}; \end{array}\displaystyle \right . $$
- (4)
\(L(t)\leq l(t)t\), for all\(t\geq0\);
- (5)
\(L^{-1}(s)\leq s\), for all\(s\geq0\);
- (6)
\(\frac{L^{-1}(s)}{s}\)is nonincreasing, for all\(s\geq0\);
- (7)
iflis unbounded, then\(\lim_{s\to+\infty}\frac {|L^{-1}(s)|^{2}}{s}=\frac{2}{l'(\infty)}\);
- (8)
\(0\leq\frac{t}{l(t)}l'(t)\leq1\), for all\(t\in\mathbb{R}\);
- (9)
there exists a positive constant\(C_{6}\)such that
$$ \bigl\vert L^{-1}(s) \bigr\vert \geq \left \{ \textstyle\begin{array}{l@{\quad}l} C_{6} \vert s \vert , &\textit{if } \vert s \vert \leq1,\\ C_{6} \vert s \vert ^{1/2}, &\textit{if } \vert s \vert \geq1; \end{array}\displaystyle \right . $$ - (10)
\(|L^{-1}(\alpha s)|^{2}\leq C(\alpha)|L^{-1}(s)|^{2}\), for all\(\alpha>0\)and\(C(\alpha)>0\)depends onα.
Thus, after the change of variables, we obtain the functional \(J(v)\) in the following form:
which is well defined in X. By Lemma 2.1, we know that \(J\in C^{1}\), and the critical points of J are the weak solutions of our problem (1.1) (see [16]). Hence, to prove our main results, we should find critical points of the functional J.
Secondly, we set
and rewrite J in the following form:
where \(\widetilde{V}(x)=V(x)+V_{0}\). Note that by \((f_{2})\), the new nonlinearity f̃ satisfies
It is easy to see that the nonlinearity \(\widetilde{f}(t)\) does not satisfy Ambrosetti–Rabinowitz condition any more, hence the boundedness of Palais–Smale sequences seems hard to verify. For this reason, we will show the functional J satisfies the Cerami condition.
Thirdly, recall that a \((C)_{c}\)-sequence \(\{v_{n}\}\) in X at the level c is such that
Then J is said to satisfy the Cerami condition if any \((C)_{c}\)-sequence has a convergent subsequence in X.
Lemma 2.2
Under assumptions\((V_{1})\), \((V_{2})\), \((l)\), \((f_{1})\), and\((f_{2})\), Jsatisfies the Cerami condition.
Proof
Let \(\{v_{n}\}\) be a Cerami sequence of J, i.e.,
for some \(c\in\mathbb{R}\).
Step 1. We prove that
If this conclusion is not true, we can suppose \(\rho_{n}\to+\infty\). Consider the sequence \(\{h_{n}\}\), defined by
Since \(l(t)\geq1\), we obtain
Passing to a subsequence, we may assume that
Subsequently, by (2.3) and Lemma 2.1(4), we get
After multiplying both sides of the above equation by \(\rho_{n}^{-2}\), for large n, we have
Since \(h_{n}\to h\) in \(L^{2}(\mathbb{R}^{N})\) and \(\mu>2\), it implies that \(h\neq0\). Thus the set \(\varTheta=\{x\in\mathbb{R}^{N}: h(x)\neq0\} \) has a positive Lebesgue measure.
Due to our assumption \((f_{2})\), it implies that
Noticing that \(\widetilde{F}(t)\geq0\) and by Fatou’s lemma, we obtain
Hence, we get
which is a contradiction. Thus, we obtain
Step 2. We prove that there exists a constant \(C_{7}>0\) such that
Indeed, we may assume \(v_{n}\not\equiv0\) (otherwise, the conclusion is trivial). If (2.6) is incorrect, passing to a subsequence, we suppose
Setting
one has
From (2.7), as \(n\to\infty\), we have
and
We claim that for each \(\varepsilon>0\), there exists a constant \(C_{8}>0\) independent of n, such that \(\operatorname{meas}(\varOmega _{n})<\varepsilon\), where \(\varOmega_{n}:=\{x\in\mathbb{R}^{N}: |v_{n}(x)|\geq C_{8}\}\). If this claim is not true, there is an \(\varepsilon_{0}>0\) and a subsequence \(\{v_{n_{k}}\}\) of \(\{v_{n}\}\) such that for each positive integer k, \(\operatorname{meas}(\{x\in\mathbb{R}^{N}: |v_{n_{k}}(x)|\geq k\} )\geq\varepsilon_{0}>0\). Set \(\varOmega_{n_{k}}:=\{x\in\mathbb{R}^{N}: |v_{n_{k}}(x)|\geq k\}\). By Lemma 2.1(9), we obtain
On the other hand, if \(|v_{n}(x)|< C_{8}\), by Lemma 2.1(9)–(10), one has
Therefore, there exists a constant \(C_{10}>0\) such that
By the absolutely continuity of the Lebesgue integral, there exist \(\varepsilon>0\) and \(n_{0}>0\) for \(n>n_{0}\) we have \(\operatorname{meas}(\varOmega _{n})<\varepsilon\) and \(\int_{\varOmega_{n}}\widetilde {V}(x)w^{2}_{n}\,dx<\frac{1}{2}\). For this ε, taking \(n\to +\infty\), we get
From (2.8), (2.9), and (2.10), we get a contradictory inequality \(1\leq\frac{1}{2}\). Thus, summing up the above arguments, we prove that the Cerami sequence \(\{v_{n}\}\) in (2.4) is bounded in X.
Step 3. We prove that \(v_{n}\to v\) in X.
From the boundedness of the sequence \(\{v_{n}\}\) and the compactness of the embedding \(X\hookrightarrow\hookrightarrow L^{s}(\mathbb{R}^{N})\), up to subsequence, we may assume
By the growth condition \((f_{1})\), the properties of \(L^{-1}\) described in Lemma 2.1 and Hölder inequality, we have
On the other hand, we claim that there exists \(C_{12}>0\) such that
There is no harm in supposing \(v_{n}\not\equiv v\) (otherwise, the conclusion is trivial). Denote
If (2.11) is not true, we can assume that
By Lemma 2.1(8), this implies
and \(\frac{d}{ds} (\frac {L^{-1}(s)}{l(L^{-1}(s))} )\mid_{s=0}=1\). Moreover, for each \(\delta>0\) there exists \(C_{\delta}>0\) such that
Therefore, we deduce that \(d_{n}(x)\) is positive and
By the argument of proving Lemma 3.11 in [8], we can obtain a contradiction.
Consequently,
We deduce that \(v_{n}\to v\) in X. □
3 Proof of Theorem 1.1
Since 0 is not an eigenvalue of
we can assume that there exists an integer \(d\geq0\) such that \(0\in (\lambda_{d},\lambda_{d+1})\). For \(d\geq1\), we denote
Specially, if \(d=0\), we set \(X^{-}=\{0\}\) and \(X^{+}=X\). Then \(X^{-}\) and \(X^{+}\) are the negative and positive spaces of the quadratic form
respectively. Furthermore, for some \(\eta>0\) we get
Since the principle part \(Q(v)=\frac{1}{2}\int_{\mathbb{R}^{N}} (|\nabla v|^{2}+V(x)|L^{-1}(v)|^{2} )\,dx\) of \(J(v)\) is a \(C^{2}\)-functional on X with derivatives given by
for all \(v, \phi, \psi\in X\), in particular, since \(L^{-1}(0)=0\), \(l(0)=1\) and \(|l'(t)|\leq C_{5}\), we have
and
Applying Taylor’s formula, we have
To prove Theorem 1.1, we will apply the following Symmetric Mountain Pass Theorem due to Ambrosetti–Rabinowitz [14].
Proposition 3.1
LetXbe an infinite-dimensional Banach space, \(X=Y\bigoplus Z\)with\(\dim Y<+\infty\). If\(J\in C^{1}(X,\mathbb{R})\)satisfies Cerami condition and
- (1)
\(J(0)=0\), \(J(-u)=J(u)\)for all\(u\in X\);
- (2)
there are constants\(\rho,\alpha>0\)such that\(J|_{\partial B_{\rho}\cap Z}\geq\alpha\);
- (3)
for any finite-dimensional subspace\(W\subset X\), there is an\(R=R(W)\)such that\(J\leq0\)on\(W\setminus B_{R(W)}\),
thenJhas a sequence of critical values\(c_{j}\to+\infty\).
Lemma 3.1
Assume that\((V_{1})\), \((V_{2})\), \((l)\), \((f_{1})\), \((f_{2})\)hold, andWis a finite-dimensional subspace ofX. If\(v\in W\), then
Proof
For any \(\{v_{n}\}\subset W\) with \(\|v_{n}\|\to+\infty\), consider
Then \(\{a_{n}\}\) is a bounded sequence in W. Since dim \(W<\infty\), there exists \(a\in W\setminus\{0\}\) such that
For \(x\in\{a\neq0\}\), we have
and, using Lemma 2.1(9), obtain
Thus, from \((f_{2})\), for \(x\in\{a\neq0\}\),
By Fatou’s Lemma, we get
Furthermore,
□
Lemma 3.2
Under assumptions\((V_{1})\), \((V_{2})\), \((l)\), and\((f_{1})\), one has
where\(Z_{k}:=\overline{\operatorname{span}}\{e_{k},e_{k+1},\dots\}\), ρ, k, αare positive constants, and\(k>d\).
Proof
By condition \((f_{1})\), there exist positive constants \(C_{1}\) and \(C_{2}\) such that
For \(i\geq d\), denote \(Z_{i}=\overline{\operatorname{span}}\{ e_{i},e_{i+1},\dots\}\). Then, similar to Lemma 3.8 in [21], we have the following fact:
Let \(Y=\operatorname{span}\{e_{1},\dots,e_{k-1}\}\) and \(Z_{k}=\overline {\operatorname{span}}\{e_{k},e_{k+1},\dots\}\), where \(k>d\) and k will be determined, then \(Z_{k}\subset X^{+}\). For \(v\in Z_{k}\) and \(\|v\|\) small enough, using (3.1) and Taylor’s expansion, we have
where we used \(p>2\). Noticing that \(\beta_{k}\to0\) as \(k\to\infty\), we then take k large enough such that \(\eta-C_{1}\beta_{k}^{2}>0\). □
Proof of Theorem 1.1
Obviously, using Lemmas 2.2, 3.1, 3.2 and \((f_{4})\), all conditions of Proposition 3.1 are satisfied. Therefore, J possesses a sequence of critical points \(\{v_{n}\}\) with \(J(v_{n})\to+\infty\). Letting \(u_{n}=L^{-1}(v_{n})\), we obtain that \(\{u_{n}\}\) is a sequence of weak solutions for problem (1.1) with \(I(u_{n})\to+\infty\). □
4 Proof of Theorem 1.2
In this section, by employing Morse theory, we prove the existence of one nontrivial solution for problem (1.1).
Let X be a real Banach space. For a given \(J\in C^{1}(X)\), we use the following notation:
U is a neighborhood of \(u\in\mathcal{K}\), where u is an isolated critical point of J with \(J(u)=c\). Then the qth critical group of J at an isolated critical point u is defined by
where \(H_{q}(\cdot,\cdot)\) is a qth singular relative homology group with integer coefficients. If J satisfies the Cerami condition and \(a<\inf_{u\in\mathcal{K}} J(u)\), then the critical groups of J at infinity are defined by
In Morse theory, the functional J is always required to satisfy the so-called deformation condition.
Definition 4.1
The functional J satisfies deformation condition if for every \(\varepsilon>0\) small enough, \(c\in\mathbb{R}\) and any neighborhood \(\mathcal{N}\) of \(\mathcal{K}_{c}\), there is a continuous deformation \(\eta: [0,1]\times X\to X\) such that
- (i)
\(\eta(t,v)=v\) for either \(t=0\) or \(v\notin J^{-1}[c-\varepsilon ,c+\varepsilon]\);
- (ii)
\(J(\eta(t,v))\) is nonincreasing in t for any \(v\in X\);
- (iii)
\(\eta(J^{c+\varepsilon}\setminus\mathcal{N})\subset J^{c-\varepsilon}\).
We note that if the functional J satisfies the (PS)-condition or the Cerami condition, then J satisfies the deformation condition.
Morse theory tells us that if J satisfies the Cerami condition, \(v=0\) is a critical point of J and \(\mathcal{K}=\{0\}\), then \(C_{q}(J,\infty)\cong C_{q}(J,0)\) for all \(q\in\mathbb{N}\). It follows that if \(C_{q}(J,\infty)\ncong C_{q}(J,0)\) for some \(q\in \mathbb{N}\) then J must have a nontrivial critical point. So one has to compute these groups to get the nontrivial critical point.
4.1 Critical groups at zero
In this section, we will use the following proposition to compute the critical groups of J at zero.
Proposition 4.1
Suppose\(J\in C^{1}(X,\mathbb{R})\)has a local linking at zero with respect to the decomposition\(X=Y\oplus Z\), i.e., for some\(\varepsilon>0\).
where\(B_{\varepsilon}=\{u\in X: \|u\|<\varepsilon\}\). If\(d=\dim Y<\infty\), then\(C_{d}(J,0)\neq0\).
Lemma 4.1
Under assumptions\((V_{1})\), \((V_{2})\), \((l)\), \((f_{1})\), and\((f_{3})\), functionalJhas a local linking at zero with respect to decomposition\(X=X^{-}\oplus X^{+}\), where\(X^{-}\), \(X^{+}\)are defined in Sect. 3and\(d=\dim X^{-}\).
Proof
From \((f_{1})\) and \((f_{3})\), for all \(\varepsilon>0\), there exists \(C_{\varepsilon}>0\), such that
Hence, we get
Using this and (3.2), we obtain
From this and (3.1), one obtains that J has a local linking property at zero. Then it follows from Proposition 4.1 that \(C_{d}(J,0)\ncong0\). □
4.2 Critical groups at infinity
Lemma 4.2
Suppose that\((f_{1})\)–\((f_{3})\), \((V_{1})\), \((V_{2})\), and\((l)\)hold. For any\(q\in\mathbb{N}\), \(C_{q}(J,\infty)\cong0\).
Proof
Let \(S^{\infty}\) be the unit sphere in X. Firstly, we will establish the following fact:
Due to \(|L^{-1}(sw)|\leq|sw|\) and \((f_{2})\), we deduce
Secondly, we will show that the following claim is true.
Claim
There exists \(A>0\) such that if \(J(v)\leq-A\) then
If this claim is false, there exists a sequence \(\{v_{n}\}\subset X\) such that
By the same argument as in Lemma 2.2, we deduce
Set
By the similar arguments of Lemma 2.2, we obtain
Then
Multiplying both sides of (4.5) by \(\rho_{n}^{-2}\), we obtain \(h\neq0\). By the assumption \(\langle J'(v_{n}),v_{n}\rangle\geq0\), we have
This is impossible, thus the conclusion of the claim must be true.
Hence, by this claim and (4.4), for any fixed \(a>A\) and \(v\in S^{\infty}\), there exists a unique \(T:=T(v)>0\) such that
By the implicit function theorem, this implies
Therefore the deformation retract \(\eta:[0,1]\times(X\setminus B^{\infty})\to X\) defined by
satisfies \(\eta(0,v)=v\), \(\eta(1,v)\in J^{-a}\) for a large enough, where \(B^{\infty}=\{v\in X:\|v\|\leq1\}\). It follows that
Proof of Theorem 1.2
We have verified that J satisfies the Cerami condition. By Lemma 4.1, J has a local linking at zero with respect to the decomposition \(X=X^{-}\oplus X^{+}\), hence, by Proposition 4.1, for \(d=\dim X^{-}\), we have
On the other hand, Lemma 4.2 says that for all \(q\in\mathbb {N}\), \(C_{q}(J,\infty)=0\). Hence, J has a nontrivial critical point v. Now \(u=L^{-1}(v)\) is a nontrivial solution of problem (1.1). □
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Huang, C. Multiplicity and existence of solutions for generalized quasilinear Schrödinger equations with sign-changing potentials. Bound Value Probl 2020, 73 (2020). https://doi.org/10.1186/s13661-020-01369-6
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DOI: https://doi.org/10.1186/s13661-020-01369-6