# Enriching Frame Representations with Distributionally Induced Senses

###### Abstract

We introduce a new lexical resource that enriches the Framester knowledge
graph, which links Framnet, WordNet, VerbNet and other resources, with
semantic features from text corpora. These features are extracted from
distributionally induced sense inventories and subsequently linked to the
manually-constructed frame representations to boost the performance of frame
disambiguation in context. Since Framester is a frame-based knowledge graph,
which enables full-fledged OWL querying and reasoning, our resource paves the
way for the development of novel, deeper semantic-aware applications that
could benefit from the combination of knowledge from text and complex symbolic
representations of events and participants. Together with the resource we also
provide the software we developed for the evaluation in the task of Word Frame
Disambiguation (WFD).
Keywords: distributional semantics, linked open data, frame semantics.

languageresourceLanguage Resources

Enriching Frame Representations with Distributionally Induced Senses

Stefano Faralli†, Alexander Panchenko‡, Chris Biemann‡, Simone Paolo Ponzetto†
---
†Data and Web Science Group, School of Business Informatics and Mathematics,
Universität Mannheim, Germany
‡ Language Technology Group, Department of Informatics, Universität Hamburg,
Germany
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

Abstract content

##  1\. Introduction

Recent years have witnessed an impressive amount of work on the automatic
construction of wide-coverage knowledge resources. Web-scale information
extraction systems like NELL [Carlson et al., 2010] or Knowledge Vault [Dong
et al., 2014] can acquire massive amounts of machine-readable knowledge from
the Web, whereas projects like DBpedia [Bizer et al., 2009], YAGO [Rebele et
al., 2016] or BabelNet [Navigli and Ponzetto, 2012] have turned
collaboratively-generated content into large knowledge bases. However, all of
these resources are entity-centric in that they are primarily built around the
notion of _entities_ , as either provided by an external resource (e.g.,
Wikipedia pages) or automatically discovered from text (e.g., by clustering
entity mentions). The entities are most commonly represented by nouns, noun
phrases, and named entities. Entities lie at the heart of the Semantic Web and
are central to enable semantic technologies on a large scale [Dietz et al.,
2017]. Besides, they also provide the foundation for more complex semantic
representations like event templates or frames [Fillmore, 1968], which are in
the focus of our interest in this paper.

Recent work looked at ways to populate the Linked Open Data (LOD) cloud with
wide-coverage information about semantic frames on the basis of Framester
[Gangemi et al., 2016a], a frame-centric resource that is meant to act as a
hub between several other linguistic resources that are already part of the
LOD cloud. In parallel, researchers looked at ways to combine knowledge graphs
of this kind with distributional semantics to include human-readable meaning
representations based on semantic vector spaces within the Semantic Web
ecosystem [Faralli et al., 2016]. Consequently, in this paper we bring these
two lines of research together and present a new resource that combines a
wide-coverage symbolic repository of frames with human-readable distributional
semantic representations from text.

Figure 1: The overall resource construction workflow of our approach:
distributionally induced sense representations extracted from a corpus (PCZ)
are used to enrich the Framester symbolic knowledge graph.

In Table 3, we show an excerpt from a distributional-based sense inventory, a
proto-conceptualization (PCZ) proposed by ?) and further developed by ?). The
senses in this resource are induced automatically from text via clustering of
graphs of semantically related words and linked to BabelNet and WordNet
lexical resources. In particular, in the resource, each sense (e.g.,
_read#VB#1_) is defined by its related senses (_quote#VB#0_ , _recite#VB#0_ ,
and so on) which have been observed to co-occur with the sense. Sense
inventories with related senses as well as other features, such as weighted
hypernymy senses and context clues, can provide rich and human-readable
knowledge to disambiguate word meanings in context, e.g., the meaning for the
verb _to read_ as ‘read aloud’ versus ‘make sense’ as in:

When we hear someone read a text, our understanding of what we hear is usually
spontaneous – the rules by which we interpret meaning.

Here, the reference to the ‘Reading_aloud’ frame from
FrameNet111https://framenet.icsi.berkeley.edu/ could be triggered on the basis
of the occurrences of the verbs in the sentence: _hear_ , _understanding_ and
_interpret_. Such connections, in turn, could be provided by a hybrid resource
where distributional representations from text have been explicitly linked to
semantic knowledge repositories, since hybrid resources of this kind have been
shown in the past to improve performance on lexical understanding [Panchenko
et al., 2017] as well as taxonomy learning and cleaning [Faralli et al.,
2017].

In this paper, we bridge the gap between distributional and frame semantics by
linking distributional semantic representations to Framester, a knowledge
graph that acts as a hub between resources like FrameNet, BabelNet and
DBpedia, among others. As a result of this, we introduce a new lexical
resource that enriches the Framester knowledge graph with distributional
features extracted from text, and show how this hybrid resource yields better
results on the task of recognizing frames in running text.

BabelNet Synset ID |  English Word Senses |  Gloss
---|---|---
bn:00085007v |  quote, cite |  ‘To repeat someone’s exact words.’, ‘To quote; to repeat, as a passage from a book, or the words of another.’
bn:00090740v |  observe, mention, note, remark |  ‘She observed that his presentation took up too much time.’, ‘They noted that it was a fine day to go sailing.’
Table 1: Excerpt of two BabelNet synsets. Frame ID | Related BabelNet Synsets
---|---
Evidence | bn:00085007v, bn:00084633v, …
Telling | bn:00085007v, bn:00083488v, …
Communication | bn:00085007v, bn:00090740v, …
Table 2: Excerpt of the Framester TransX profile.

Joining distributional and frame semantics builds upon and extends our
framework for combining symbolic and statistical meaning representations
[Biemann et al., 2018]: the main objective of this line of research is to
‘join forces’ across heterogeneous knowledge and semantic models in order to
mutually enrich them and combine the strengths of the lexicographic tradition
that describes linguistic information manually with the coverage and
versatility of the corpus and data-driven approaches that derive meaning
representations directly from the data. Bringing together the ‘best of both
worlds’ has the potential to combine the benefits of wide-coverage symbolic
[Gangemi et al., 2016b] and statistical [Chen et al., 2014] semantics for
frame parsing, which, in turn, can be exploited for many different
applications ranging from sentiment analysis [Recupero et al., 2015] all the
way to content-based recommendations [De Clercq et al., 2014].

The contributions of this paper are the following ones:

  1. 1.

We present the LOaDing lexical resource, an extension of Framester that adds
distributional-based features [Faralli et al., 2016, Biemann et al., 2018] to
frame representations (see Section 2.);

  2. 2.

We evaluate the resource in the task of Word Frame Disambiguation (WFD), i.e.,
on the identification of frames in context [Gangemi et al., 2016b], and show
significant improvements over the original Framester frame representations
(see Section 3.).

Both resources and software produced in this work are available under a CC-BY
4.0 license.222http://web.informatik.uni-mannheim.de/joint/

PCZ Sense ID | BabelNet Synset ID |  Related Senses |  Context Clues
---|---|---|---
quote#VB#0 | bn:00085007v |  cite#NN#1$[1.0]$, interview#VB#0$[0.8]$, mention#VB#1$[0.7]$, publish#VB#0$[0.6]$, review#VB#0$[0.6]$, … |  in#IN#pcomp$[24799.3]$, Register#NP#prep_on$[21282.9]$, Track#NP#-vmod$[16808.9]$, …
mention#VB#1 | bn:00090740v |  attest#VB#0$[1.0]$, describe#VB#0$[0.8]$, document#VB#0$[0.7]$, quote#VB#0$[0.6]$, … |  Register#NP#prep_on$[45477.7]$, Track#NP#-vmod$[35850.4]$, say#VB#prepc_as$[17041.8]$, …
Table 3: Excerpt from a proto-conceptualization (PCZ) of two distributional-based senses. Numbers behind related senses are normalized similarity scores, numbers behind context clues are association scores. Frame ID | Related BabelNet Synsets | Related PCZ Senses
---|---|---
Evidence | bn:00084633v$[29.0]$, bn:00085007v$[13.0]$, … | quote#VB#0, …
Telling | bn:00085007v$[12.0]$, bn:00083488v$[8.0]$, … | quote#VB#0, …
Communication | bn:00090740v$[18.0]$, bn:00085007v$[15.0]$, … | mention#VB#1, quote#VB#0, …
Table 4: Excerpt of a LOaDing TransX profile. Numbers behind related synsets
$bs$ are the distributional semantics-based weights $w(F,bs)$ we assign in
Step 2.4 of our LOaDing approach.

##  2\. Enriching Framester with Symbolic Distributional Sense Representations

In Figure 1 we show the diagram describing the workflow used to add
distributional features to the Framester profiles to enable a better word
frame disambiguation.

###  2.1. Resources and Datasets

Our approach makes use of three linguistic resources:

BabelNet [Navigli and Ponzetto, 2012]: a multilingual encyclopedic dictionary
that connects concepts and named entities in a very large network of semantic
relations (see Table 1).

Framester [Gangemi et al., 2016a]: a linguistic LOD hub that provides links
from FrameNet’s frames to semantically related BabelNet senses (see Table 2).
The current version consists of six different profiles: Base contains only
manually curated links, whereas the other five (DirectX, FrameBase, Fprofile,
TransX and XWFN) are automatically built extensions or subsets of the Base
profile.

Proto-Conceptualizations (PCZs) [Faralli et al., 2016]: a fully disambiguated
sense inventory automatically induced from text, providing human-readable
distributional semantic representations. In this resource, each sense is
provided with (see Table 3): a) a weighted list of semantically-related and
hypernymy senses; b) links to existing knowledge bases (i.e., BabelNet); c)
context clues used to disambiguate the senses in the context.

###  2.2. Combination of Resources

The goal of our method is a combination of the three resources mentioned in
the previous section as illustrated in Table 4. This is made possible by using
BabelNet as a pivot, since Framester provides links from FrameNet to BabelNet
(Table 2) while our PCZs are also linked to BabelNet (Table 3). As a first
result of such a combination we created LOaDing, an extension of Framester’s
profiles where we provide weights for each frame-related BabelNet synset. We
show in Table 4 an excerpt of the resulting extended Framester’s TransX
profile shown in Table 2.

The enriched frame representations are built in two main steps as illustrated
in Figure 1:

Step 1: Learning a proto-conceptualization (PCZ). We apply the methodology
from ?), to produce disambiguated sense inventories with entries for nouns and
verbs.

Step 2: LOaDing a PCZ into Framester. For each frame $F$, e.g. Communication,
and related BabelNet sysnet $bs$, e.g. bn:00085007v, we assign a weight
$w(F,bs)$. In order to compute the weight, we create three bag-of-word
representations for each of the three resources listed in the previous
section:

  1. 1.

$B_{bn}=BoW(bs)$ to represent $bs$ including the counts for all the content
words from the synset word senses and the glosses;

  2. 2.

From the PCZ we collect all the entries $PCZ(bs)$ matching $bs$, for instance
$PCZ($bn:00085007v$)=\\{$quote#VB#0$,\dots\\}$, and create a bag of words
$B_{z}=BoW(PCZ(bs))$ by collecting all the weights of content words from the
senses $s\in PCZ(bs)$ and the corresponding related senses.

  3. 3.

We create a BoW from $B_{f}$ by collecting all the content words from the
description of the frame $F$.

  4. 4.

Finally, we compute a relatedness score between a frame $F$ and a BabelNet
synset $bs$:

$w(F,bs)=\sum_{w\in I}{B_{z}.c(w)\times(B_{f}(F).c(w)+B_{bn}.c(w))},$

where $I=B_{z}(bs)\cap(B_{bn}(bs)\cup B_{f}(F))$ and $A.c(w)$ equals the
number of occurrences of the word $w$ in $A$. For instance, with respect to
the excerpts of Tables 3 and 4 we obtain $w($Communication,
bn:00085007v$)=15.0$.

##  3\. Using the Enriched Representations for Word Frame Disambiguation

We evaluate our extensions of Framester profiles following the experimental
setting of ?), and compare the extended and the original profiles in a task of
Word Frame Disambiguation (WFD).

###  3.1. Dataset: FrameNet Full Text Documents

To create a silver standard we processed all 108 documents from the FrameNet
1.7 dataset [Baker et al., 1998] with BabelFy [Moro et al.,
2014]333http://babelfy.org. By combining the original frame annotations with
the automatically generated entity links we collected a total of 81,706
annotations, which we use in our experimental setting as a silver standard.

Figure 2: The distribution of synset counts per number of candidate frames (step 1 of our WFD approach, see Section 3.2.). The y axis counts the synsets $bs$ for which $|CF(bs)|=x$.  | Base | FrameBase | XWFN | TransX | DirectX | Fprofile
---|---|---|---|---|---|---
oracle | 67.60 | 67.74 | 57.85 | 77.82 | 73.74 | 68.16
original (cond) | 53.13 | 52.73 | 45.44 | 45.21 | 46.30 | 53.04
wiki-n30-1400k (cond) | 53.23 | 52.82 | 45.42 | 44.75 | 46.58 | 53.46
wiki-n30-1400k (inv) | 55.52 | 55.06 | 46.73 | 28.24 | 45.39 | 55.69
Table 5: F1 performances on Word Frame Disambiguation across profiles and
ranking methodologies.

###  3.2. Word Frame Disambiguation

Following the WFD approach described in ?) we implemented a simple word frame
disambiguator, where for each provided annotation in our silver standard we
try to predict a frame label only on the basis of the BabelNet synsets
generated through BabelFy. In order to provide the most suitable frame label
$F$ for the provided BabelNet synset label $bs$:

  1. 1.

Creating the set of candidate frames $CF(bs)$, by collecting all the frame IDs
for which $bs$ is a semantically related BabelNet synsets (e.g.,
$CF($bn:00085007v)$=\\{$Evidence,Telling, Communication$\\}$ see Table 4);

  2. 2.

Ranking all the candidate frames $f\in CF(bs)$ by computing the following
scoring functions:

     * •

the score is equal to the conditional probability of a frame given a synset
(_cond_):

$score_{cond}(f,bs)=\frac{w(f,bs)}{\sum_{b\in relatedSynsets(f)}{w(f,b)}};$

     * •

the score is equal to the number of related synsets divided by the weight of
the synsets (_inv_):

$score_{inv}(f,bs)=\frac{|relatedSynsets(f)|}{w(f,bs)}.$

Such a scoring function promotes the candidate frame that relates with the
highest number of synsets $|relatedSynsets(f)|$, and it also penalizes the
selection of candidates which are triggered by synsets having higher weights,
in other words promoting candidates triggered by synsets in the ‘long tail’;

  3. 3.

Finally, we select the candidate frame with the highest score.

We experimented with a variety of scoring functions based on weights $w(F,bs)$
and $|relatedSynsets(F)|$, and present only the two best performing ones here.

In Figure 2 we draw the distribution of the counts of synsets across the
number of frames that are triggered in the phase of candidate selection (step
1 of our WFD approach), i.e., in the XWFN profile, we counted ten thousand
synsets $bs$ for which $|CF(bs)|=1$.

###  3.3. Results

In Table 5, we show the resulting word frame disambiguation performances in
terms of the F-score and compare across profiles and ranking methodologies. To
better understand the limits of our ‘silver standard’ and the limits of
ranking methodologies we introduced a so called ‘oracle’, which selects the
labeled frame if present in the list of selected candidates independently from
its position in the ranking. The results indicate that:

  * •

For Base, FrameBase, XWFN and Fprofile profiles, adding distributional-
semantic-based features lead to 1 to 2 points improvements of F1 in the WFD
task;

  * •

Our approach lowers the performance of DirectX profile and even drastically
lowers the performances of the TransX profile;

  * •

Overall, we noticed that the former two profiles are very dense by means of
related senses per frame and lead to a potential high recall (see the oracle
performances on the two profiles at 73.74%, 77.82% respectively) while
introducing more noisy senses;

  * •

Finally, the presence of many frequent and noisy senses let the $score_{inv}$
function outperform other function based on weights $w(F,bs)$ and
$|relatedSynsets(F)|$.

To summarize this experiment: when using our induced weighting with the
$score_{inv}$ ranking, we achieve a boost on Framester profiles that showed a
good performance in the original setting, while losing performance on
inferior, more noisier profiles. The overall best performance was reached by
our extension of the high-quality, manually created Base profile, followed by
our extension to the Fprofile.

###  3.4. Error analysis

An in-depth analysis of the errors made during the experiments (see Section
3.3.) by our WFD approach (see Section 3.2.) allowed us to identify the
following categories of errors:

  * •

Misaligned annotations: our silver standard is automatically created by
collecting BabelFy annotations for a total of 81,706 annotations from the
original 93,358 FrameNet 1.7 annotations;

  * •

Entity linking errors: we estimated that around 15% of the 81,706 BabelFy
annotations444A sample of 1000 annotations has been manually annotated as
correct or wrong by two annotators, and a third annotator provided a final
judgment only for the annotations in disagreement. assigned a wrong sense,
eventually triggering the selection of a wrong frame;

  * •

Ranking error: errors where an inferior ranking leads to bad performance.
These are a consistent part of the WFD errors and may be improved with more
complex weighting/ranking approaches;

  * •

Profile errors: related to a specific profile. In this category we identify
errors due to: i) the slightly difference between some frame meaning (e.g.,
many errors are due to the prediction of the frame Measure duration instead of
Calendric unit); ii) the frequent absence of related senses so that is not
possible to trigger any frame, even for correct BabelFy annotations.

##  4\. Conclusions

In this paper, we presented LOaDing, a novel lexical resource that connects
distributional sense representations induced from text to symbolic frame
representations. Hybrid semantic representations have been shown to enable
complex semantic tasks like, for instance, end-to-end taxonomy induction
[Faralli et al., 2017]: they could provide an additional signal to improve
state-of-the-art semantic parsing (as illustrated by our running example). As
we have shown for word sense disambiguation for nouns [Panchenko et al.,
2017], distributional information is able to considerably alleviate the
sparsity inherent in knowledge-based methods. Consequently, in future work we
would like to study the contribution and the potential of our proto-
conceptualizations into more complex frame-centered tasks such as verb frame
induction [Vulić et al., 2017]. Our vision in the longer term is to exploit
hybrid statistical and symbolic approaches to go beyond the vocabulary and
relations of entity- and encyclopedic-centric resources to produce novel
semantic representations of event templates and frames with a large coverage.

## Acknowledgements

This work was funded by the Deutsche Forschungsgemeinschaft (DFG) within the
JOIN-T project.

##  5\. Bibliographical References

## References

  * Baker et al., 1998 Baker, C. F., Fillmore, C. J., and Lowe, J. B.  (1998).  The Berkeley Framenet project.  In Proceedings of the 36th Annual Meeting of the Association for Computational Linguistics and 17th International Conference on Computational Linguistics-Volume 1, pages 86–90. Association for Computational Linguistics.
  * Biemann et al., 2018 Biemann, C., Faralli, S., Panchenko, A., and Ponzetto, S. P.  (2018).  A framework for enriching lexical semantic resources with distributional semantics.  Natural Language Engineering, pages 1–48.
  * Bizer et al., 2009 Bizer, C., Lehmann, J., Kobilarov, G., Auer, S., Becker, C., Cyganiak, R., and Hellmann, S.  (2009).  DBpedia – A crystallization point for the web of data.  JWS, 7(3):154–165.
  * Carlson et al., 2010 Carlson, A., Betteridge, J., Kisiel, B., Settles, B., Hruschka, E. R., and Mitchell, T. M.  (2010).  Toward an architecture for never-ending language learning.  In Proceedings of AAAI, pages 1306–1313, Atlanta, GA, USA. AAAI Press.
  * Chen et al., 2014 Chen, Y. N., Wang, W. Y., and Rudnicky, A. I.  (2014).  Leveraging frame semantics and distributional semantics for unsupervised semantic slot induction in spoken dialogue systems.  In Proceedings of IEEE Spoken Language Technology Workshop (SLT), pages 584–589, South Lake Tahoe, NV, USA. IEEE.
  * De Clercq et al., 2014 De Clercq, O., Schuhmacher, M., Hoste, V., and Ponzetto, S. P.  (2014).  Exploiting FrameNet for content-based book recommendation.  In Proceedings of CBRecSys Workshop at RecSys, pages 14–20, Foster City, CA, USA. ACM.
  * Dietz et al., 2017 Dietz, L., Kotov, A., and Meij, E.  (2017).  Utilizing knowledge graphs in text-centric information retrieval.  In Proceedings of WSDM, pages 815–816. ACM.
  * Dong et al., 2014 Dong, X., Gabrilovich, E., Heitz, G., Horn, W., Lao, N., Murphy, K., Strohmann, T., Sun, S., and Zhang, W.  (2014).  Knowledge vault: A web-scale approach to probabilistic knowledge fusion.  In Proceedings of KDD, pages 601–610, New York, NY, USA. ACM.
  * Faralli et al., 2016 Faralli, S., Panchenko, A., Biemann, C., and Ponzetto, S. P.  (2016).  Linked disambiguated distributional semantic networks.  In Proceedings of ISWC, pages 56–64, Kobe, Japan. Springer International.
  * Faralli et al., 2017 Faralli, S., Panchenko, A., Biemann, C., and Ponzetto, S. P.  (2017).  The ContrastMedium algorithm: Taxonomy induction from noisy knowledge graphs with just a few links.  In Proceedings of EACL, pages 590–600, Valencia, Spain. Association for Computational Linguistics.
  * Fillmore, 1968 Fillmore, C. J.  (1968).  The case for case.  In Universals in Linguistic Theory, pages 2–88. Holt, Rinehart and Winston, New York, NY, USA.
  * Gangemi et al., 2016a Gangemi, A., Alam, M., Asprino, L., Presutti, V., and Recupero, D. R.  (2016a).  Framester: A wide coverage linguistic linked data hub.  In Proceedings of EKAW, pages 239–254, Bologna, Italy.
  * Gangemi et al., 2016b Gangemi, A., Alam, M., and Presutti, V.  (2016b).  Word frame disambiguation: Evaluating linguistic linked data on frame detection.  In Proceedings of LD4IE Workshop at ISWC, pages 23–31, Kobe, Japan. Springer International.
  * Moro et al., 2014 Moro, A., Raganato, A., and Navigli, R.  (2014).  Entity Linking meets Word Sense Disambiguation: a Unified Approach.  Transactions of the Association for Computational Linguistics (TACL), 2:231–244.
  * Navigli and Ponzetto, 2012 Navigli, R. and Ponzetto, S. P.  (2012).  BabelNet: The automatic construction, evaluation and application of a wide-coverage multilingual semantic network.  ArtInt., 193:217–250.
  * Panchenko et al., 2017 Panchenko, A., Faralli, S., Ponzetto, S. P., and Biemann, C.  (2017).  Using linked disambiguated distributional networks for word sense disambiguation.  In Proceedings of SENSE Workshop at EACL, pages 1–7, Valencia, Spain. Association for Computational Linguistics.
  * Rebele et al., 2016 Rebele, T., Suchanek, F. M., Hoffart, J., Biega, J., Kuzey, E., and Weikum, G.  (2016).  YAGO: A multilingual knowledge base from Wikipedia, Wordnet, and Geonames.  In Proceedings of ISWC, pages 177–185, Kobe, Japan. Springer International.
  * Recupero et al., 2015 Recupero, D. R., Presutti, V., Consoli, S., Gangemi, A., and Nuzzolese, A. G.  (2015).  Sentilo: Frame-based sentiment analysis.  Cognitive Computation, 7(2):211–225.
  * Vulić et al., 2017 Vulić, I., Mrkšić, N., and Korhonen, A.  (2017).  Cross-lingual induction and transfer of verb classes based on word vector space specialisation.  In Proceedings of EMNLP, pages 2536–2548, Copenhagen, Denmark. Association for Computational Linguistics.


# Standing wave solutions and instability for the Logarithmic Klein–Gordon
equation

Lijia Han111 E–mail address<EMAIL_ADDRESS>(L. Han), Yue Qiu, Xiaohong
Wang
North China Electric Power University, Beijing 102206, China

Abstract. In this paper, we study the standing wave solutions of Klein–Gordon
equation with logarithmic nonlinearity. The existence of the standing wave
solution related to the ground state $\phi_{0}(x)$ is obtained. Further, we
prove the instability of solutions around $\phi_{0}(x)$.

Key words and phrases. Logarithmic Klein–Gordon equations; Standing wave
solutions; Variational method; instability.

##  1 Introduction

In this paper, we study the Cauchy problem for the logarithmic Klein–Gordon
equation given by

$\displaystyle u_{tt}-\Delta u+u=|u|^{p-1}u\ln|u|^{2},$ (1.1)

with initial data

$\displaystyle u(0,x)=u_{0}(x)\in H_{r}^{1},\quad u_{t}(0,x)=u_{1}(x)\in
L_{r}^{2},$ (1.2)

where, $H_{r}^{1}=\left\\{\phi\mid\right.$ radially symmetric functions on
${R}^{3}\\}$, $L_{r}^{2}=\left\\{\phi\mid\right.$ radially symmetric functions
on ${R}^{3}\\}$, $u$ is a pair of real–valued functions of $(t,x)$
$\in{R}\times{R}^{3}$, $\Delta$ is the Laplacian on ${R}^{3}$, $2<p<4$.

Equation (1) has been applied in many branches of physics, such as nuclear
physics [1], optics [2] and geophysics [3]. It is also introduced in quantum
field [4]. Logarithmic nonlinearity appears naturally in inflation cosmology
and in supersymmetric field theories [5-7]. Equation (1) has conservation
quantum $E(t)$ as

$\displaystyle E(t)$
$\displaystyle=\frac{1}{2}\int_{R^{3}}\left|u_{t}\right|^{2}dx+\frac{1}{2}\int_{R^{3}}|\nabla
u|^{2}dx+\frac{1}{2}\int_{R^{3}}|u|^{2}dx+\frac{2}{(p+1)^{2}}\int_{R^{3}}|u|^{p+1}dx$
(1.3)
$\displaystyle\quad-\frac{1}{p+1}\int_{R^{3}}|u|^{p+1}\operatorname{In}|u|^{2}dx.$
(1.4)

There are many results about equation (1). When $p=1$, [1] and [3] proved the
existence of classical solutions and weak solutions for the logarithmic
Klein–Gordon equation. By using Galerkin method and compactness criterion, [8]
proved the existence of global solutions and the blow up of solutions for the
logarithmic Klein–Gordon equation. In [9], the authors studied two regularized
finite difference methods for the logarithmic Klein–Gordon equation. In [10],
the authors studied the orbital stability of the standing wave for the
logarithmic Klein–Gordon equation. In [11], the author considered the
logarithmic Klein–Gordon equation with damping term $u_{t}$, and obtained the
existence of global solutions by using the potential well method. When $p>1$,
[12] studied the existence of global solutions and finite–time blow up
solutions for logarithmic Klein–Gordon equation. In [13], the author
considered equation (1.5),

$u_{tt}-\Delta
u+u=-|u|^{\frac{4}{n-2}}u\log^{\gamma}\left(\log\left(10+|u|^{2}\right)\right),$
(1.5)

for $n=3,4,5$, $\gamma>0$ is a constant, the blow up behavior and the global
existence of solution were obtained. The logarithmic nonlinearity was also
studied in many other equations, such as wave equations and so on. [14]
considered the global existence and the finite time blow up of solutions for
the semilinear logarithmic wave equation $u_{tt}-\Delta
u=|u|^{p}\operatorname{In}|u|$, where $1<p<\infty$ if $n=1,2$ and
$2<p<\frac{4}{n-2}$ if $n=3,4,5$. [15] studied the energy decay estimates and
infinite blow up phenomena for a strongly damped semilinear logarithmic wave
equation, $u_{tt}-\Delta u-\Delta u_{t}=u\log|u|^{2}$. [16] concerned with the
existence of positive solutions for logarithmic elliptic equation,
$-\varepsilon^{2}\Delta u+V(x)u=u\log|u|^{2}$, where $\varepsilon>0$,
$N\geqslant 3$ and $V$ is a continuous function with a global minimum.

In the past decades, the classical Klein–Gordon system ($p>1$)

$\displaystyle u_{tt}-\Delta u+u=|u|^{p-1}u,\quad(t,x)\in R\times R^{N},$
(1.6) $\displaystyle u(x,0)=u_{0}(x),u_{t}(x,0)=u_{1}(x),\quad x\in R^{N},$
(1.7)

had been widely studied by many researchers. For the results about the
existence of local and global solutions, blow up behavior and scattering
behavior of (1.6), we could refer to [17-25] and the references therein.

There are also some results about the standing wave solutions of Klein–Gordon
equation. [26] considered the instability of standing wave solutions for
nonlinear Klein–Gordon equation (1.6). Especially, [27] studied the
instability of standing wave solutions for nonlinear Klein–Gordon equation
(1.6) in the critical case in one dimension. [28] studied the stability of the
standing waves of least energy for nonlinear Klein–Gordon equations (1.8),

$\displaystyle u_{tt}-\Delta u+u+f(|u|)\arg u=0,\quad x\in R^{N},N>2,$ (1.8)

where $f$ satisfy
$\underset{\eta\rightarrow\infty}{lim}f(\eta)/\eta^{l}\geqslant 0$. By using
variational method, the sufficient conditions of the minimum energy standing
wave solutions stability were obtained for (1.8). Based on the method in [28],
[30] considered the standing wave solutions of nonlinear Klein–Gordon equation
(1.9),

$\displaystyle u_{tt}-\Delta u+u=K(x)|u|^{\frac{N}{4}}u,$ (1.9)

and discussed the instability of standing wave solutions by establishing some
sharp conditions for blow up and global existence of the solutions, where
$K(x)>0$ is a given $C^{1}$ function such that there is
$\frac{N^{2}}{4}<q<\infty$ such that $K(x)\in L^{q}\left(R^{N}\right)$. [31]
studied the stability of standing wave solutions for the coupled nonlinear
Klein–Gordon equation (1.10)

$\begin{cases}\phi_{tt}-\Delta\phi+\phi=(p+1)|\phi|^{p-1}|\psi|^{q+1}\phi,&t>0,x\in
R^{N},\\\
\psi_{tt}-\Delta\psi+\psi=(q+1)|\psi|^{p-1}|\phi|^{q+1}\psi,&t>0,x\in
R^{N},\end{cases}$ (1.10)

where $(\phi,\psi)$ is a pair of real–valued functions, $N>2$,
$0<p,q<\frac{2}{N-2}$ and $p+q<\frac{4}{N}$. In [32], the authors studied the
instability of standing wave solutions for nonlinear Klein–Gordon equation
with a damping term, $u_{tt}-\Delta u+m^{2}u+\alpha u_{t}=|u|^{p-2}u$, where
$p>2$, $m\neq 0$, $\alpha>0$. In [10], the authors studied the orbital
stability of standing wave solutions
$u(x,t)=e^{ict}e^{\frac{1}{2}+\frac{1-c^{2}}{p}}e^{-\frac{px^{2}}{4}}$ for the
logarithmic Klein–Gordon equation (1.11),

$\displaystyle u_{tt}-\Delta u+u=\mu\log\left(|u|^{p}\right)u,$ (1.11)

where $c$ is the frequency, $p>0$, $\mu>0$, $u$ :
$\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}$ is a complex valued
function.

As far as we know, there are no results about the existence and instability
for the standing wave solution related to ground state of logarithmic
Klein–Gordon equation (1.1). Study for the ground state of equation (1.1) is
very important. The ground state solution is the lowest energy state of a
system in quantum mechanics. A solution is said to be a ground state if it has
the least energy among all solutions. From physical point of view, it is the
most stable configuration of a quantum–mechanical system. Inspired by [28-29],
in this paper we study the existence for the standing wave solution of
equation (1.1) related to ground state by using the variational method,
Gagliardo–Nirenberg inequality, compactness theorem, Sobolev embedding
theorem, stretching transformation and so on. Further, we prove the
instability of solutions around the ground state.

This paper is organized as follows. In Section 2, we state some preliminaries
and the main results. In Section 3, we prove the existence of standing wave
solutions of equation (1.1) related to ground state $\phi_{0}(x)$ by using the
variational method. In Section 4, we prove the instability of solutions around
$\phi_{0}(x)$.

##  2 Preliminaries and statement of main results

###  2.1 Notations

We start by giving some definitions of $H_{1}^{r}$ and $L_{p}^{r}$,

$\displaystyle H_{r}^{1}=\left\\{u\in
H^{1}\left(R^{3}\right);u(x)=u(|x|)\right\\},\quad L_{r}^{p}=\left\\{u\in
L^{p}\left(R^{3}\right);u(x)=u(|x|),p>2\right\\},$

which are radially symmetric spaces with the norm

$\displaystyle\|u\|_{H_{r}^{1}}^{2}=\int_{R^{3}}|\nabla
u(x)|^{2}dx+\int_{R^{3}}|u(x)|^{2}dx<\infty,$
$\displaystyle\|u\|_{L_{r}^{p}}^{p}=\int_{R^{3}}|u(x)|^{p}dx<\infty.$

Next, we introduce the Gagliardo–Nirenberg inequality in $R^{N}$, $N>2$,

$\displaystyle\|u\|_{L^{r}}\leqslant C\|\nabla
u\|_{L^{p}}^{\theta}\|u\|_{L^{q}}^{1-\theta},$ (2.1)

where $1<p<N$, $\theta\in[0,1]$,
$\frac{1}{r}=\frac{\theta}{p^{*}}+\frac{1-\theta}{q}$, $p^{*}=\frac{Np}{N-p}$,
$C$ is the best constant. When $p=q=2$, $r=\alpha$, we get

$\displaystyle\left\|\phi_{k}\right\|_{L^{\alpha}}\leqslant
C_{1}\left\|\phi_{k}\right\|_{L^{2}}^{1-\frac{N(\alpha-2)}{2\alpha}}\left\|\nabla\phi_{k}\right\|_{L^{2}}^{\frac{N(\alpha-2)}{2\alpha}},$
(2.2)

for 2 ¡$\alpha<\frac{2N}{N-2}$.

In this paper, $C_{i}>0(i=1,2,3\ldots,20)$ is a constant. For $f(u)=o(u)$
means that $|f(u)/u|\rightarrow 0$ as $|u|\rightarrow 0$, for $f(u)=O(u)$
means that $|f(u)/u|$ is bounded as $|u|\rightarrow 0$.

###  2.2 The main results

Consider the nonlinear logarithmic Klein–Gordon equation (1.1), which has
nontrivial standing wave solutions

$\displaystyle u(x,t)=e^{i\omega t}\phi(x),$

where $\phi(x)$ is a ground state solution of the equation

$\displaystyle-\Delta\phi+\left(1-\omega^{2}\right)\phi=|\phi|^{p-1}\phi\operatorname{In}|\phi|^{2}.$
(2.3)

If there is a solution $\phi(x)$ such that the functional $J_{\omega}(\phi)$
is minimum, then the solution $\phi(x)$ is said to be a ground–state solution
of equation (2.3). Here we define the functional $J_{\omega}(\phi)$

$\displaystyle
J_{\omega}(\phi)=\frac{1}{2}\int_{R^{3}}|\nabla\phi|^{2}dx+\frac{1-\omega^{2}}{2}\int_{R^{3}}|\phi|^{2}dx+\frac{2}{(p+1)^{2}}\int_{R^{3}}|\phi|^{p+1}dx-\frac{1}{p+1}\int_{R^{3}}|\phi|^{p+1}\operatorname{In}|\phi|^{2}dx,$

and the functional $K_{\omega}(\phi)$

$\displaystyle K_{\omega}(\phi)$
$\displaystyle=\frac{1}{2}\int_{R^{3}}|\nabla\phi|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}|\phi|^{2}dx+\frac{6}{(p+1)^{2}}\int_{R^{3}}|\phi|^{p+1}dx$
$\displaystyle\quad-\frac{3}{p+1}\int_{R^{3}}|\phi|^{p+1}\operatorname{In}|\phi|^{2}dx.$

In addition, we define the set

$\displaystyle M_{\omega}=\left\\{\phi\in H_{r}^{1}\mid
K_{\omega}(\phi)=0,\phi\neq 0\right\\},$

then we consider the constrained variational problem

$\displaystyle d(\omega)=\inf_{\phi\in M_{\omega}}J_{\omega}(\phi).$

First, we have the existence of the standing wave solution related to the
ground state $\phi_{0}(x)$.

######  Theorem 2.1.

Let $2<p<4$, $N=3$ and $\omega\in[0,1]$,

(i) variational problem

$\displaystyle d(\omega)=\inf_{\phi\in M_{\omega}}J_{\omega}(\phi)$ (2.4)

are equivalent to minimization problems

$\displaystyle\widetilde{d(\omega)}=\inf_{\begin{subarray}{c}\phi\end{subarray}}\left\\{\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx,K_{\omega}(\phi)\leqslant
0,\phi\neq 0\right\\},$ (2.5)

(ii) there exists $\phi_{0}\in M_{\omega}$ such that
$d(\omega)=\underset{\phi\in
M_{\omega}}{\inf}J_{\omega}(\phi)=J_{\omega}\left(\phi_{0}\right)$,

(iii) $\phi_{0}$ is a solution of
$-\Delta\phi+\left(1-\omega^{2}\right)\phi=|\phi|^{p-1}\phi\ln|\phi|^{2}$.

Second, let

$\displaystyle X=\left\\{u\in
H_{r}^{1}\mid\int_{R^{3}}G(|u|)dx<\infty\right\\},$

where are equivalent to minimization problems

$\displaystyle
G(|u|)=\frac{2}{(p+1)^{2}}|u|^{p+1}-\frac{1}{p+1}|u|^{p+1}\operatorname{In}|u|^{2}.$
(2.6)

We define an invariant set

$\displaystyle R_{1}=\left\\{u\in X,u_{t}\in L_{r}^{p}\mid
E\left(u,u_{t}\right)<d(0),K_{0}(u)<0,u\neq 0\right\\},$

and obtain the instability of solutions around $\phi_{0}$.

######  Theorem 2.2.

Let $\phi(x)$ be a ground state solution of equation (2.3). For any
$\varepsilon>0$, there exists $\left(u_{0},u_{1}\right)\in R_{1}$ that
satisfies

$\displaystyle\left\|\left(u_{0},u_{1}\right)-\left(\phi,0\right)\right\|_{H_{r}^{1}\times
L_{r}^{2}}<\varepsilon.$ (2.7)

And if the solutions $u(t,x)$ of equation (1.1) satisfy
$\left(u(0,x),u_{t}(0,x)\right)=\left(u_{0},u_{1}\right)\in R_{1}$ and
radially symmetric. Then either

(i) the solutions exist only locally in time $[0,T_{0}),T_{0}<\infty$, with
$\left\\{t_{n}\right\\}$ such that
$\left\|u\left(t_{n}\right)\right\|_{H_{r}^{1}}\rightarrow\infty$ as
$t_{n}\rightarrow T_{0}$ for some sequence $\left\\{t_{n}\right\\}$; or

(ii) the solutions exist globally, then there exists a sequence
$\left\\{t_{n}\right\\}$ such that
$\left\|u\left(t_{n}\right)\right\|_{H_{r}^{1}}\rightarrow\infty$ as
$t_{n}\rightarrow\infty$.

##  3 Existence of Standing Wave Solutions

In this section, we prove Theorem 2.1, first, we give some lemmas.

######  Lemma 3.1.

$[34]$ Let $N>2$, under radially symmetric condition, $H^{1}\hookrightarrow
L^{p}$ is compact for $2<p<2+4/(N-2)$.

######  Lemma 3.2.

If $\phi_{\omega}\in H_{r}^{1}$ is a solution of
$-\Delta\phi+\left(1-\omega^{2}\right)\phi=|\phi|^{p-1}\phi\operatorname{In}|\phi|^{2}$,
then

$\displaystyle K_{\omega}\left(\phi_{\omega}\right)=0.$

###### Proof.

Let $\psi_{\beta}(x)=\phi_{\omega}\big{(}\frac{x}{\beta}\big{)}$, we have

$\displaystyle J_{\omega}\left(\psi_{\beta}\right)$
$\displaystyle=\frac{1}{2}\int_{R^{3}}\left|\nabla\psi_{\beta}\right|^{2}dx+\frac{1-\omega^{2}}{2}\int_{R^{3}}\left|\psi_{\beta}\right|^{2}dx$
$\displaystyle\quad+\frac{2}{(p+1)^{2}}\int_{R^{3}}\left|\psi_{\beta}\right|^{p+1}dx-\frac{1}{p+1}\int_{R^{3}}\left|\psi_{\beta}\right|^{p+1}\operatorname{In}\left|\psi_{\beta}\right|^{2}dx$
$\displaystyle=\frac{\beta}{2}\int_{R^{3}}\left|\nabla\phi_{\omega}\right|^{2}dx+\frac{\beta^{3}\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\phi_{\omega}\right|^{2}dx$
$\displaystyle\quad+\frac{2\beta^{3}}{(p+1)^{2}}\int_{R^{3}}\left|\phi_{\omega}\right|^{p+1}dx-\frac{\beta^{3}}{p+1}\int_{R^{3}}\left|\phi_{\omega}\right|^{p+1}\operatorname{In}\left|\phi_{\omega}\right|^{2}dx.$

Since $\phi_{\omega}$ is a solution of equation (2.3), then

$\displaystyle\delta
J_{\omega}\left(\phi_{\omega}\right)=\left.0\Rightarrow\frac{d\left(J_{\omega}\left(\psi_{\beta}\right)\right)}{d\beta}\right|_{\beta=1}=0,$

so

$\displaystyle\left.\frac{d\left(J_{\omega}\left(\phi_{\beta}\right)\right)}{d\beta}\right|_{\beta=1}$
$\displaystyle=\frac{1}{2}\int_{R^{3}}|\nabla\phi|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}|\phi|^{2}dx$
$\displaystyle\quad+\frac{6}{(p+1)^{2}}\int_{R^{3}}|\phi|^{p+1}dx-\frac{3}{p+1}\int_{R^{3}}|\phi|^{p+1}\operatorname{In}|\phi|^{2}dx$
$\displaystyle=K_{\omega}\left(\phi_{\omega}\right)=0.$

This completes the proof of the Lemma 3.2. ∎

Proof of Theorem 2.1. We divide the proof of Theorem 2.1 into three steps.
Step1: Let
$\widetilde{d(\omega)}=\inf\left\\{\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx,K_{\omega}(\phi)\leqslant
0,\phi\neq 0\right\\}$, we prove the equivalence of $d(\omega)$ and
$\widetilde{d(\omega)}$. By the definition of $d(\omega)$ and
$\widetilde{d(\omega)}$, we have $\widetilde{d(\omega)}\leqslant d(\omega)$.
For any $\phi\in H_{r}^{1}$ satisfying $K_{\omega}(\phi)<0$, let
$\psi_{\beta}(x)=\phi\big{(}\frac{x}{\beta}\big{)}$,

$\displaystyle K_{\omega}\left(\psi_{\beta}\right)$
$\displaystyle=\frac{1}{2}\int_{R^{3}}\left|\nabla\psi_{\beta}\right|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\psi_{\beta}\right|^{2}dx$
$\displaystyle\quad+\frac{6}{(p+1)^{2}}\int_{R^{3}}\left|\psi_{\beta}\right|^{p+1}dx-\frac{3}{p+1}\int_{R^{3}}\left|\psi_{\beta}\right|^{p+1}\operatorname{In}\left|\psi_{\beta}\right|^{2}dx$
$\displaystyle=\frac{\beta}{2}\int_{R^{3}}|\nabla\phi|^{2}dx+\frac{3\beta^{3}\left(1-\omega^{2}\right)}{2}\int_{R^{3}}|\phi|^{2}dx$
$\displaystyle\quad+\frac{6\beta^{3}}{(p+1)^{2}}\int_{R^{3}}|\phi|^{p+1}dx-\frac{3\beta^{3}}{p+1}\int_{R^{3}}|\phi|^{p+1}\operatorname{In}|\phi|^{2}dx.$

When $\beta=1$, $K_{\omega}\left(\psi_{1}\right)=K_{\omega}(\phi)<0$.
When $\beta\rightarrow 0$,

$\displaystyle K_{\omega}\left(\psi_{\beta}\right)$
$\displaystyle=\beta\left[\frac{1}{2}\int_{R^{3}}|\nabla\phi|^{2}dx+\frac{3\beta^{2}\left(1-\omega^{2}\right)}{2}\int_{R^{3}}|\phi|^{2}dx\right.$
$\displaystyle\quad\left.+\frac{6\beta^{2}}{(p+1)^{2}}\int_{R^{3}}|\phi|^{p+1}dx-\frac{3\beta^{2}}{p+1}\int_{R^{3}}|\phi|^{p+1}\operatorname{In}|\phi|^{2}dx\right]>0.$

So there exists $\beta^{*}\in(0,1)$ such that
$K_{\omega}\left(\psi_{\beta^{*}}\right)=0$ such as

$\displaystyle\frac{1}{3}\int_{R^{3}}\left|\nabla\psi_{\beta^{*}}\right|^{2}dx=\frac{\beta^{*}}{3}\int_{R^{3}}|\nabla\phi|^{2}dx<\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx,$

which implies $d(\omega)\leqslant\widetilde{d(\omega)}$. Thus,
$d(\omega)=\widetilde{d(\omega)}$ holds.
Step2: We prove there exists $\phi_{0}\in M_{\omega}$ such that

$\displaystyle d(\omega)=\inf_{\phi\in
M_{\omega}}J_{\omega}(\phi)=J_{\omega}\left(\phi_{0}\right).$

Let $\phi_{k}$ be a minimum sequence for the minimization problem (2.5), then
$\int_{R^{3}}\left|\nabla\phi_{k}\right|^{2}dx$ is bounded. Since
$K_{\omega}\left(\phi_{k}\right)\leqslant 0$, then

$\displaystyle K_{\omega}\left(\phi_{k}\right)$
$\displaystyle=\frac{1}{2}\int_{R^{3}}\left|\nabla\phi_{k}\right|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\phi_{k}\right|^{2}dx$
(3.1)
$\displaystyle\quad+\frac{6}{(p+1)^{2}}\int_{R^{3}}\left|\phi_{k}\right|^{p+1}dx-\frac{3}{p+1}\int_{R^{3}}\left|\phi_{k}\right|^{p+1}\operatorname{In}\left|\phi_{k}\right|^{2}dx\leqslant
0,$ (3.2)

this indicates

$\displaystyle\frac{1}{2}\int_{R^{3}}\left|\nabla\phi_{k}\right|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\phi_{k}\right|^{2}dx$
(3.3)
$\displaystyle\leqslant\frac{3}{p+1}\int_{R^{3}}\left|\phi_{k}\right|^{p+1}\operatorname{In}\left|\phi_{k}\right|^{2}dx+\frac{6}{(p+1)^{2}}\int_{R^{3}}\left|\phi_{k}\right|^{p+1}dx.$
(3.4)

Notice that $u\operatorname{In}u^{2}\leqslant C_{2}\left(1+u^{2}\right)$,
$u>0$, then

$\displaystyle\frac{1}{2}\int_{R^{3}}\left|\nabla\phi_{k}\right|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\phi_{k}\right|^{2}dx$
(3.5)
$\displaystyle\leqslant\frac{3C_{2}}{p+1}\int_{R^{3}}\left|\phi_{k}\right|^{p}dx+\frac{3C_{2}}{p+1}\int_{R^{3}}\left|\phi_{k}\right|^{p+2}dx+\frac{6}{(p+1)^{2}}\int_{R^{3}}\left|\phi_{k}\right|^{p+1}dx.$
(3.6)

By using the Gagliardo–Nirenberg inequality, we obtain

$\displaystyle\left\|\phi_{k}\right\|_{L_{r}^{p}}\leqslant
C_{3}\left\|\phi_{k}\right\|_{L_{r}^{2}}^{1-\frac{3(p-2)}{2p}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3(p-2)}{2p}},$
(3.7) $\displaystyle\left\|\phi_{k}\right\|_{L_{r}^{p+1}}\leqslant
C_{4}\left\|\phi_{k}\right\|_{L_{r}^{2}}^{1-\frac{3(p-1)}{2(p+1)}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3(p-1)}{2(p+1)}},$
(3.8) $\displaystyle\left\|\phi_{k}\right\|_{L_{r}^{p+2}}\leqslant
C_{5}\left\|\phi_{k}\right\|_{L_{r}^{2}}^{1-\frac{3p}{2(p+2)}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3p}{2(p+2)}},$
(3.9)

where $2<p<4$. Then from Young’s inequality in (3.7)–(3.9), we have

$\displaystyle\left\|\phi_{k}\right\|_{L_{r}^{p}}^{p}\leqslant
C_{6}\left\|\phi_{k}\right\|_{L_{r}^{2}}^{p-\frac{3(p-2)}{2}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3(p-2)}{2}}\leqslant
C_{6}\left(\varepsilon\left\|\phi_{k}\right\|_{L_{r}^{2}}^{\frac{m(6-p)}{2}}+\varepsilon^{-\frac{n}{m}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3n(p-2)}{2}}\right),$
(3.10) $\displaystyle\left\|\phi_{k}\right\|_{L_{r}^{p+1}}^{p+1}\leqslant
C_{7}\left\|\phi_{k}\right\|_{L_{r}^{2}}^{p+1-\frac{3(p-1)}{2}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3(p-1)}{2}}\leqslant
C_{7}\left(\varepsilon\left\|\phi_{k}\right\|_{L_{r}^{2}}^{\frac{m(5-p)}{2}}+\varepsilon^{-\frac{n}{m}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3n(p-1)}{2}}\right),$
(3.11) $\displaystyle\left\|\phi_{k}\right\|_{L_{r}^{+2}}^{p+2}\leqslant
C_{8}\left\|\phi_{k}\right\|_{L_{r}^{2}}^{p+2-\frac{3p}{2}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3p}{2}}\leqslant
C_{8}\left(\varepsilon\left\|\phi_{k}\right\|_{L_{r}^{2}}^{\frac{m(4-p)}{2}}+\varepsilon^{-\frac{n}{m}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3np}{2}}\right),$
(3.12)

where $\varepsilon$ is sufficiently small, $m$, $n$ satisfy $m>1$, $n>1$,
$\frac{1}{m}+\frac{1}{n}=1$. If $\left\|\phi_{k}\right\|_{L_{r}^{2}}<1$,
$\left\|\phi_{k}\right\|_{H_{r}^{1}}$ is obviously bounded. If
$\left\|\phi_{k}\right\|_{L_{r}^{2}}>1$, from (3.7)–(3.12), we have

$\displaystyle\frac{1}{2}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{2}+\frac{3\left(1-\omega^{2}\right)}{2}\left\|\phi_{k}\right\|_{L_{r}^{2}}^{2}-\frac{C_{9}\varepsilon\left\|\phi_{k}\right\|_{L_{r}^{2}}^{\frac{m(6-p)}{2}}}{p+1}-\frac{C_{9}\varepsilon\left\|\phi_{k}\right\|_{L_{r}^{2}}^{\frac{m(4-p)}{2}}}{p+1}-\frac{C_{9}\varepsilon\left\|\phi_{k}\right\|_{L_{r}^{2}}^{\frac{m(5-p)}{2}}}{(p+1)^{2}}$
(3.13)
$\displaystyle\leqslant\frac{C_{9}\varepsilon^{-\frac{n}{m}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3n(p-2)}{2}}}{p+1}+\frac{C_{9}\varepsilon^{-\frac{n}{m}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3np}{2}}}{p+1}+\frac{C_{9}\varepsilon^{-\frac{n}{m}}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3n(p-1)}{2}}}{(p+1)^{2}}.$
(3.14)

Choose $1<m<\frac{4}{6-p}$, we obtain

$\displaystyle\max\left\\{2,\frac{m(6-p)}{2},\frac{m(5-p)}{2},\frac{m(4-p)}{2}\right\\}=2,$

so (3.13) can be written as

$\displaystyle\frac{3\left(1-\omega^{2}\right)}{2}\left\|\phi_{k}\right\|_{L_{r}^{2}}^{2}\leqslant
C_{9}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3n(p-2)}{2}}+C_{9}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3np}{2}}+C_{9}\left\|\nabla\phi_{k}\right\|_{L_{r}^{2}}^{\frac{3n(p-1)}{2}}.$
(3.15)

Since $\int_{R^{3}}\left|\nabla\phi_{k}\right|^{2}dx$ is bounded, then
$\left\|\phi_{k}\right\|_{H_{r}^{1}}$ is bounded by (3.15), so there must be a
convergent subsequence, also denoted as $\phi_{k}$, such that

$\displaystyle\phi_{k}\rightarrow\phi_{0}\text{ weakly in }H_{r}^{1},$ (3.16)

from Lemma 3.1,

$\displaystyle\phi_{k}\rightarrow\phi_{0}\text{ strongly in }L_{r}^{p},\quad
2<p<6.$ (3.17)

By the lower semicontinuity of the limit, we have

$\displaystyle K_{\omega}\left(\phi_{0}\right)=$
$\displaystyle\frac{1}{2}\int_{R^{3}}\left|\nabla\phi_{0}\right|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\phi_{0}\right|^{2}dx$
(3.18)
$\displaystyle+\frac{6}{(p+1)^{2}}\int_{R^{3}}\left|\phi_{0}\right|^{p+1}dx-\frac{3}{p+1}\int_{R^{3}}\left|\phi_{0}\right|^{p+1}\operatorname{In}\left|\phi_{0}\right|^{2}dx$
(3.19) $\displaystyle\leqslant$
$\displaystyle\varliminf_{k\rightarrow\infty}\left(\frac{1}{2}\int_{R^{3}}\left|\nabla\phi_{k}\right|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\phi_{k}\right|^{2}dx\right.$
(3.20)
$\displaystyle\left.+\frac{6}{(p+1)^{2}}\int_{R^{3}}\left|\phi_{k}\right|^{p+1}dx-\frac{3}{p+1}\int_{R^{3}}\left|\phi_{k}\right|^{p+1}\operatorname{In}\left|\phi_{k}\right|^{2}dx\right)$
(3.21) $\displaystyle=$
$\displaystyle\varliminf_{k\rightarrow\infty}K_{\omega}\left(\phi_{k}\right)\leqslant
0,$ (3.22)

according to Step1, we can obtain $K_{\omega}\left(\phi_{0}\right)=0$. Next we
prove $\phi_{0}\neq 0$. By contradiction, if $\phi_{0}=0$, we get

$\displaystyle\phi_{k}\rightarrow 0\in H_{r}^{1},\quad\phi_{k}\rightarrow 0\in
L_{r}^{p},\quad 2<p<6.$ (3.23)

However, by Sobolev embedding theorem, when $2\leqslant p\leqslant 4$, we get

$\displaystyle\left\|\phi_{k}\right\|_{L_{r}^{p}}^{p}\leqslant
C_{10}\left\|\phi_{k}\right\|_{H_{r}^{1}}^{p},\quad\left\|\phi_{k}\right\|_{L_{r}^{p}}^{p+1}\leqslant
C_{11}\left\|\phi_{k}\right\|_{H_{r}^{1}}^{p+1},\quad\left\|\phi_{k}\right\|_{L_{r}^{p}}^{p+2}\leqslant
C_{12}\left\|\phi_{k}\right\|_{H_{r}^{1}}^{p+2}.$ (3.24)

So, from (3.5) we obtain

$\displaystyle\alpha\left\|\phi_{k}\right\|_{H_{r}^{1}}^{2}\leqslant\frac{C_{13}}{p+1}\left\|\phi_{k}\right\|_{H_{r}^{1}}^{p}+\frac{C_{13}}{p+1}\left\|\phi_{k}\right\|_{H_{r}^{1}}^{p+2}+\frac{C_{13}}{(p+1)^{2}}\left\|\phi_{k}\right\|_{H_{r}^{1}}^{p+1},$
(3.25)

where
$\alpha=\min\left\\{\frac{1}{2},\frac{3\left(1-\omega^{2}\right)}{2}\right\\}$.
Since $\phi_{k}\neq 0$ and $\left\|\phi_{k}\right\|_{H_{r}^{1}}$ is bounded,
then

$\displaystyle\left\|\phi_{k}\right\|_{H_{r}^{1}}^{p-2}+\left\|\phi_{k}\right\|_{H_{r}^{1}}^{p}+\left\|\phi_{k}\right\|_{H_{r}^{1}}^{p-1}\geqslant\frac{\alpha(p+1)}{C_{13}},$
(3.26)

which contradicts to (3.23). Therefore $\phi_{0}\neq 0$ and combine with
$K_{\omega}\left(\phi_{0}\right)=0$, we deduce that $\phi_{0}\in M_{\omega}$.
By the lower semicontinuity of the limit, we have

$\displaystyle d(\omega)=\inf_{\phi\in M_{\omega}}J_{\omega}(\phi)\leqslant
J_{\omega}\left(\phi_{0}\right)=\frac{1}{3}\int_{R^{3}}\left|\nabla\phi_{0}\right|^{2}dx\leqslant\varliminf_{k\rightarrow\infty}\frac{1}{3}\int_{R^{3}}\left|\nabla\phi_{k}\right|^{2}dx=d(\omega).$
(3.27)

Step3: We prove $\phi_{0}$ is a solution of
$-\Delta\phi+\left(1-\omega^{2}\right)\phi=|\phi|^{p-1}\phi\operatorname{In}|\phi|^{2}$.

By step2, we consider $\inf J_{\omega}(\phi)$ under condition
$K_{\omega}\left(\phi\right)=0$. Using lagrange multiplier method, we have

$\displaystyle\delta J_{\omega}\left(\phi_{0}\right)=\lambda\delta
K_{\omega}\left(\phi_{0}\right),$ (3.28)

that is

$\displaystyle-(1-\lambda)\Delta\phi_{0}+(1-3\lambda)\left(1-\omega^{2}\right)\phi_{0}-(1-3\lambda)\left|\phi_{0}\right|^{p}\operatorname{In}\left|\phi_{0}\right|^{2}=0.$
(3.29)

Let

$\displaystyle J_{1_{\omega}}(\phi)=$
$\displaystyle\frac{1-\lambda}{2}\int_{R^{3}}|\nabla\phi|^{2}dx+\frac{(1-3\lambda)\left(1-\omega^{2}\right)}{2}\int_{R^{3}}|\phi|^{2}dx$
(3.30)
$\displaystyle+\frac{2(1-3\lambda)}{(p+1)^{2}}\int_{R^{3}}|\phi|^{p+1}dx-\frac{1-3\lambda}{p+1}\int_{R^{3}}|\phi|^{p+1}\operatorname{In}|\phi|^{2}dx,$
(3.31)

similar to the proof of Lemma 3.2, we obtain

$\displaystyle\frac{1}{2}\int_{R^{3}}\left|\nabla\phi_{0}\right|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\phi_{0}\right|^{2}dx+\frac{6}{(p+1)^{2}}\int_{R^{3}}\left|\phi_{0}\right|^{p+1}dx$
(3.32)
$\displaystyle\quad-\frac{3}{p+1}\int_{R^{3}}\left|\phi_{0}\right|^{p+1}\operatorname{In}\left|\phi_{0}\right|^{2}dx$
(3.33)
$\displaystyle=\frac{\lambda}{2}\int_{R^{3}}\left|\nabla\phi_{0}\right|^{2}dx+\frac{9\lambda\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\phi_{0}\right|^{2}dx$
(3.34)
$\displaystyle\quad+\frac{18\lambda}{(p+1)^{2}}\int_{R^{3}}\left|\phi_{0}\right|^{p+1}dx-\frac{9\lambda}{p+1}\int_{R^{3}}\left|\phi_{0}\right|^{p+1}\operatorname{In}\left|\phi_{0}\right|^{2}dx.$
(3.35)

Thanks to $K_{\omega}\left(\phi_{0}\right)=0$ , we show that 3$\lambda
K_{\omega}\left(\phi_{0}\right)=0$,

$\displaystyle\frac{3\lambda}{2}\int_{R^{3}}\left|\nabla\phi_{0}\right|^{2}dx+\frac{9\lambda\left(1-\omega^{2}\right)}{2}\int_{R^{3}}\left|\phi_{0}\right|^{2}dx$
(3.36)
$\displaystyle+\frac{18\lambda}{(p+1)^{2}}\int_{R^{3}}\left|\phi_{0}\right|^{p+1}dx-\frac{9\lambda}{p+1}\int_{R^{3}}\left|\phi_{0}\right|^{p+1}\operatorname{In}\left|\phi_{0}\right|^{2}dx=0.$
(3.37)

From (3.32)–(3.36), we have

$\displaystyle-\lambda\int_{R^{3}}\left|\nabla\phi_{0}\right|^{2}dx=0.$

It implies $\lambda=0$, from (3.28) we obtain
$J_{\omega}\left(\phi_{0}\right)=0$, therefore $\phi_{0}$ is a solution of
$-\Delta\phi+\left(1-\omega^{2}\right)\phi=|\phi|^{p-1}\phi\operatorname{In}|\phi|^{2}$.
This completes the proof of Theorem 2.1.$\hfill\Box$

From Theorem 2.1, we get the following corollary.

######  Corollary 3.1.

Let $G_{\omega}$ be the solution set of (2.4), $G_{\omega}$ is also the
solution set of problem (3.38),

$\inf
J_{\omega}(\phi)=d(\omega),\quad\omega\in(0,1),\quad\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx=d(\omega).$
(3.38)

###### Proof.

Assuming that $\phi$ satisfies

$\displaystyle\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx=d(\omega),\quad
J_{\omega}(\phi)<d(\omega),$ (3.39)

then

$\displaystyle\frac{1}{3}K_{\omega}(\phi)=J_{\omega}(\phi)-\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx=J_{\omega}(\phi)-d(\omega)<0.$

According to Theorem 2.1,

$\displaystyle
d(\omega)=\inf\left\\{\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx,K_{\omega}(\phi)\leqslant
0,\phi\neq 0\right\\},$

we obtain $d(\omega)<\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx$, which
contradicts to (3.39), then

$\displaystyle\inf
J_{\omega}(\phi)=d(\omega),\quad\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx=d(\omega).$

Next, we prove that $G_{\omega}$ is also the solution set of problem (3.38).
Let $\phi$ be the solution to problem (3.38), then

$\displaystyle\delta
J_{\omega}(\phi)+\lambda\delta\int_{R^{3}}|\nabla\phi|^{2}dx=0.$ (3.40)

We calculate the (3.40) and obtain

$\displaystyle\int_{R^{3}}-(1+\lambda)\Delta\phi+\left(1-\omega^{2}\right)\phi-|\phi|^{p-1}\phi\operatorname{In}|\phi|^{2}dx=0.$

Let

$\displaystyle J_{2\omega}(\phi)=$
$\displaystyle\frac{1+\lambda}{2}\int_{R^{3}}|\nabla\phi|^{2}dx+\frac{1-\omega^{2}}{2}\int_{R^{3}}|\phi|^{2}dx$
(3.41)
$\displaystyle+\frac{2}{(p+1)^{2}}\int_{R^{3}}|\phi|^{p+1}dx-\frac{1}{p+1}\int_{R^{3}}|\phi|^{p+1}\operatorname{In}|\phi|^{2}dx,$
(3.42)

similar to the proof of Lemma 3.2, we obtain

$\displaystyle\frac{1+\lambda}{2}\int_{R^{3}}|\nabla\phi|^{2}dx+\frac{3\left(1-\omega^{2}\right)}{2}\int_{R^{3}}|\phi|^{2}dx$
(3.43)
$\displaystyle+\frac{6}{(p+1)^{2}}\int_{R^{3}}|\phi|^{p+1}dx-\frac{3}{p+1}\int_{R^{3}}|\phi|^{p+1}\mathrm{In}|\phi|^{2}dx=0.$
(3.44)

According to (3.43) and the definition of $J_{\omega}(\phi)$, we get

$\displaystyle
J_{\omega}(\phi)=\frac{2-\lambda}{6}\int_{R^{3}}|\nabla\phi|^{2}dx=d(\omega)=\frac{1}{3}\int_{R^{3}}|\nabla\phi|^{2}dx,$

therefore $\lambda=0$. From (3.43), we obtain $K_{\omega}(\phi)=0$, $\phi\in
G_{\omega}$. This completes the proof of the Corollary 3.1. ∎

##  4 The Instability of Standing Wave Solutions

In this section, we consider the Cauchy problem for equation (1.1), and define
the energy of equation (1.1) as

$\displaystyle
E\left(u,u_{t}\right)=\frac{1}{2}\int_{R^{3}}\left|u_{t}\right|^{2}dx+J_{0}(u).$

where

$\displaystyle
J_{0}(\phi)=\frac{1}{2}\int_{R^{3}}|\nabla\phi|^{2}dx+\frac{1}{2}\int_{R^{3}}|\phi|^{2}dx+\frac{2}{(p+1)^{2}}\int_{R^{3}}|\phi|^{p+1}dx-\frac{1}{p+1}\int_{R^{3}}|\phi|^{p+1}\operatorname{In}|\phi|^{2}dx.$

Fristly, we give the following lemmas.

######  Lemma 4.1.

If the solutions of Cauchy problem (1.1)–(1.2) satisfy
$E\left(u_{0},u_{1}\right)=E\left(u,u_{t}\right)$, then $R_{1}$ is an
invariant set under the flow of problems (1.1)–(1.2).

###### Proof.

Let $\left(u_{0},u_{1}\right)\in R_{1}$, according to the conservation of
energy, we obtain

$\displaystyle E\left(u_{0},u_{1}\right)=E\left(u,u_{t}\right)<d(0).$ (4.1)

Assume there exists a $t_{1}$ such that
$\left(u\left(t_{1}\right),u_{t}\left(t_{1}\right)\right)\notin R_{1}$, then
$K\left(u\left(t_{1}\right)\right)\geqslant 0$. From the lower semi–continuity
of $K(u(t))$, there is a maximum value $t_{0}$, which satisfies
$K\left(u\left(t_{0}\right)\right)\geqslant 0$. So for
$t\in\left(t_{0},+\infty\right)$, we have $K(u(t))<0$,

$\displaystyle\frac{1}{3}\int_{R^{3}}\left|\nabla
u\left(t_{0}\right)\right|^{2}dx\leqslant\varliminf_{\begin{subarray}{c}t\rightarrow
t_{0}\\\ t<t_{0}\end{subarray}}\frac{1}{3}\int_{R^{3}}|\nabla u(t)|^{2}dx$
(4.2) $\displaystyle\leqslant\varliminf_{\begin{subarray}{c}t\rightarrow
t_{0}\\\ t<t_{0}\end{subarray}}\left(\frac{1}{3}\int_{R^{3}}|\nabla
u(t)|^{2}dx+\frac{1}{3}K(u(t))\right)$ (4.3)
$\displaystyle=\varliminf_{\begin{subarray}{c}t\rightarrow t_{0}\\\
t<t_{0}\end{subarray}}J(u(t))\leqslant\varliminf_{\begin{subarray}{c}t\rightarrow
t_{0}\\\
t<t_{0}\end{subarray}}\left(J(u(t))+\frac{1}{2}\int_{R^{3}}\left|u_{t}(t)\right|^{2}dx\right)$
(4.4) $\displaystyle=\varliminf_{\begin{subarray}{c}t\rightarrow t_{0}\\\
t<t_{0}\end{subarray}}E\left(u,u_{t}\right)<d(0),$ (4.5)

which contradicts the definition of $d(0)$, so $R_{1}$ is an invariant set
under the flow of Cauchy problem (1.1)–(1.2). ∎

######  Lemma 4.2.

$[34]$ let $u\in H_{r}^{1}$, then

$\displaystyle|u(x)|\leqslant C_{N}|x|^{\frac{1-N}{2}}\|u\|_{H_{r}^{1}},$

where, $C_{N}$ only depends on $N$.

######  Lemma 4.3.

Let

$\displaystyle\phi_{m}(r,t)$
$\displaystyle=rH(m-r)+\frac{\operatorname{In}(m)r}{\operatorname{In}(r)}[H(r-m)-H(r-(t+2m))]$
$\displaystyle\ \
-\frac{\operatorname{In}(m)(t+2m)}{\operatorname{In}(t+2m)}[H(r-(t+2m))],$

where, if $s\geqslant 0$, $H(s)=1$, if $s<0$, $H(s)=0$. Then we obtain

$\displaystyle\operatorname{Re}\int_{R^{3}}u_{t}\phi_{m}\bar{u}_{r}dx-\operatorname{Re}\int_{R^{3}}u_{t}(0)\phi_{m}(r,0)\bar{u}_{r}(0)dx$
(4.6)
$\displaystyle=\int_{0}^{t}\int_{r<m}-\frac{3}{2}\left|u_{t}\right|^{2}+\frac{1}{2}|\nabla
u|^{2}+\frac{3}{2}|u|^{2}+3G(|u|)dxds$ (4.7)
$\displaystyle\quad+\frac{\operatorname{In}m}{\operatorname{In}r}\int_{0}^{t}\int_{m<r<2m+t}-\frac{3}{2}\left|u_{t}\right|^{2}+\frac{1}{2}|\nabla
u|^{2}+\frac{3}{2}|u|^{2}+3G(|u|)dxds$ (4.8)
$\displaystyle\quad+\frac{\operatorname{In}m}{(\operatorname{In}r)^{2}}\int_{0}^{t}\int_{m<r<2m+t}\frac{1}{2}\left|u_{t}\right|^{2}+\frac{1}{2}|\nabla
u|^{2}-\frac{1}{2}|u|^{2}-G(|u|)dxds$ (4.9)
$\displaystyle\quad+\int_{0}^{t}\int_{r>2m+t}-\operatorname{Re}e_{1m}\bar{u}_{r}u_{t}+e_{2m}\left[-\frac{1}{2}\left|u_{t}\right|^{2}+|\nabla
u|^{2}-|u|^{2}-2G(|u|)\right]dxds.$ (4.10)

###### Proof.

We multiply equation (1.1) by $\phi_{m}(r,t)\bar{u}_{r}$, then integrate on
$R^{3}$, take the real part, we obtain

$\displaystyle\frac{d}{dt}\operatorname{Re}\int_{R^{3}}u_{t}\phi_{m}\bar{u}_{r}dx-\operatorname{Re}\int_{R^{3}}u_{t}\frac{d}{dt}\left(\phi_{m}\right)\bar{u}_{r}dx-\operatorname{Re}\int_{R^{3}}u_{t}\phi_{m}\frac{d}{dt}\left(\bar{u}_{r}\right)dx$
(4.11) $\displaystyle-\operatorname{Re}\int_{R^{3}}\Delta
u\phi_{m}\bar{u}_{r}dx+\operatorname{Re}\int_{R^{3}}u\phi_{m}\bar{u}_{r}dx+\operatorname{Re}\int_{R^{3}}f(|u|)\frac{u}{|u|}\phi_{m}\bar{u}_{r}dx=0,$
(4.12)

where, $f(|u|)=-|u|^{p}\operatorname{In}|u|^{2}$.

Let $G^{\prime}(|u|)=f(|u|)$, we calculate equation (4.11) and obtain

$\displaystyle\frac{d}{dt}\operatorname{Re}$
$\displaystyle\int_{R^{3}}u_{t}\phi_{m}\bar{u}_{r}dx+\int_{r<m}\frac{3}{2}\left|u_{t}\right|^{2}-\frac{1}{2}|\nabla
u|^{2}-\frac{3}{2}|u|^{2}-3G(|u|)dx$
$\displaystyle+\frac{\operatorname{In}m}{\operatorname{In}r}\left[\int_{m<r<2m+t}\frac{3}{2}\left|u_{t}\right|^{2}-\frac{1}{2}|\nabla
u|^{2}-\frac{3}{2}|u|^{2}-3G(|u|)dx\right]$
$\displaystyle-\frac{\operatorname{In}m}{(\operatorname{In}r)^{2}}\left[\int_{m<r<2m+t}\frac{1}{2}\left|u_{t}\right|^{2}+\frac{1}{2}|\nabla
u|^{2}-\frac{1}{2}|u|^{2}-G(|u|)dx\right]$
$\displaystyle+\int_{r>2m+t}\operatorname{Re}e_{1m}\bar{u}_{r}u_{t}+e_{2m}\left[\frac{1}{2}\left|u_{t}\right|^{2}-|\nabla
u|^{2}+|u|^{2}+2G(|u|)\right]dx=0,$

where $e_{im}$ $(i=1,2)$ is a constant and $\left|e_{im}\right|\leqslant
C_{17}$ for $r>2m+t$. Integrating with respect to $t$ we obtain (4.6).

∎

Proof of Theorem 2.2. We prove Theorem 2.2 by contradiction. Statement (i) is
just a consequence of the local existence of the solution. Now we proof (ii).
Assuming $u(t,x)$ is bounded in $H_{r}^{1}\left(R^{3}\right)$, then there
exists an $\varepsilon>0$ such that $K_{0}(u(t))<-\varepsilon$ for any $t$.
Otherwise, since $R_{1}$ is an invariant set, there exists an sequence
$\left\\{t_{k}\right\\}$ that satisfies
$K_{0}\left(u\left(t_{k}\right)\right)\rightarrow 0$, so there is a convergent
subsequence $u\left(t_{k}\right)$,

$\displaystyle u\left(t_{k}\right)\rightarrow u^{*}\text{ weakly in
}H_{r}^{1}\left(R^{3}\right),$

from Lemma 3.1, we have

$\displaystyle u\left(t_{k}\right)\rightarrow u^{*}\text{ strongly in
}L_{r}^{p}\left(R^{3}\right),\quad 2<p<6,$

then we get

$\displaystyle\frac{1}{3}\int_{R^{3}}\left|\nabla u^{*}\right|^{2}dx$
$\displaystyle\leqslant\varliminf_{k\rightarrow\infty}\frac{1}{3}\int_{R^{3}}\left|\nabla
u\left(t_{k}\right)\right|^{2}dx$ (4.13)
$\displaystyle=\varliminf_{k\rightarrow\infty}\left[J_{0}\left(u\left(t_{k}\right)\right)-\frac{1}{3}K\left(u\left(t_{k}\right)\right)\right]$
(4.14)
$\displaystyle=\varliminf_{k\rightarrow\infty}J_{0}\left(u\left(t_{k}\right)\right)$
(4.15)
$\displaystyle\leqslant\varliminf_{k\rightarrow\infty}\left[J_{0}\left(u\left(t_{k}\right)\right)+\frac{1}{2}\int_{R^{3}}\left|u_{t}\left(t_{k}\right)\right|^{2}dx\right]$
(4.16) $\displaystyle=E\left(u_{0},u_{1}\right)<d,$ (4.17)

which contradicts the definition of $d(0)$. Therefore such an $\varepsilon>0$
exists. According to Lemma 4.2, we can get that

$\displaystyle|u(r,t)|\leqslant C_{N}\|u(t)\|_{H_{r}^{1}}r^{-1}<C_{18}r^{-1},$

when $r$ is sufficiently large, we have $|u(r,t)|\leqslant 1$, so

$\displaystyle
G(|u|)=\frac{2}{(p+1)^{2}}|u|^{p+1}-\frac{1}{(p+1)}|u|^{p+1}\operatorname{In}|u|^{2}>0.$

We estimate the (4.6) and obtain

$\displaystyle\int_{m<r<2m+t}\frac{\operatorname{In}m}{2\operatorname{In}r}|\nabla
u|^{2}dx$
$\displaystyle+\int_{m<r<2m+t}\frac{\operatorname{In}m}{2(\operatorname{In}r)^{2}}|\nabla
u|^{2}dx$ (4.18) $\displaystyle<\frac{1}{2}\int_{m<r<2m+t}|\nabla
u|^{2}dx+\frac{1}{2\operatorname{In}m}\int_{m<r<2m+t}|\nabla u|^{2}dx,$ (4.19)
$\displaystyle\int_{m<r<2m+t}\frac{3\operatorname{In}m}{2\operatorname{In}r}|u|^{2}dx-\int_{m<r<2m+t}\frac{\operatorname{In}m}{2(\operatorname{In}r)^{2}}|u|^{2}dx<\frac{3}{2}\int_{m<r<2m+t}|u|^{2}dx,$
(4.20)
$\displaystyle\int_{m<r<2m+t}\frac{3\operatorname{In}m}{\operatorname{In}r}G(|u|)dx-\int_{m<r<2m+t}\frac{\operatorname{In}m}{(\operatorname{In}r)^{2}}G(|u|)dx<3\int_{m<r<2m+t}G(|u|)dx.$
(4.21)

Finally, since the radially symmetric solution is strong outside the optical
cone containing the origin, at the base of the optical cone, we have the
following estimates,

$\displaystyle\int_{r>2m+t}-\operatorname{Re}e_{1m}\bar{u}_{r}u_{t}$
$\displaystyle+e_{2m}\left[-\frac{1}{2}\left|u_{t}\right|^{2}+|\nabla
u|^{2}-|u|^{2}-2G(|u|)\right]dx$ (4.22) $\displaystyle\leqslant
C_{19}\int_{r>2m+t}\left|\bar{u}_{r}u_{t}\right|-\frac{1}{2}\left|u_{t}\right|^{2}+|\nabla
u|^{2}-|u|^{2}-2G(|u|)dx$ (4.23) $\displaystyle\leqslant
C_{19}\int_{r>2m+t}\left|\bar{u}_{r}u_{t}\right|+\frac{1}{2}\left|u_{t}\right|^{2}+|\nabla
u|^{2}+|u|^{2}+2G(|u|)dx$ (4.24) $\displaystyle\leqslant
C_{19}\int_{r>2m+t}\left|u_{1}\right|^{2}+\left|\nabla
u_{0}\right|^{2}+\left|u_{0}\right|^{2}+2G\left(\left|u_{0}\right|\right)dx.$
(4.25)

From estimates (4.18)–(4.21), (4.6) can be written as

$\displaystyle\operatorname{Re}\int_{R^{3}}u_{t}\phi_{m}\bar{u}_{r}dx$
$\displaystyle-\operatorname{Re}\int_{R^{3}}u_{t}(0)\phi_{m}(r,0)\bar{u}_{r}(0)dx$
$\displaystyle=\int_{0}^{t}\int_{r<m}-\frac{3}{2}\left|u_{t}\right|^{2}+\frac{1}{2}|\nabla
u|^{2}+\frac{3}{2}|u|^{2}+3G(|u|)dxds$
$\displaystyle+\int_{0}^{t}\int_{m<r<2m+t}\frac{1}{2}|\nabla
u|^{2}+\frac{3}{2}|u|^{2}+3G(|u|)dxds$
$\displaystyle+C_{19}\int_{0}^{t}\int_{r>2m+t}\left|u_{1}\right|^{2}+\left|\nabla
u_{0}\right|^{2}+\left|u_{0}\right|^{2}+2G\left(\left|u_{0}\right|\right)dxds$
$\displaystyle+\int_{0}^{t}\int_{m<r<2m+t}\frac{1}{2\operatorname{In}m}|\nabla
u|^{2}dxds$ $\displaystyle=I_{1}+I_{2}+I_{3}+I_{4}.$

According to $K(u(t))<-\varepsilon$, we get $I_{1}<0$ and

$\displaystyle I_{2}$
$\displaystyle=\int_{0}^{t}\int_{m<r<2m+t}\frac{1}{2}|\nabla
u|^{2}+\frac{3}{2}|u|^{2}+3G(|u|)dxds$
$\displaystyle=\int_{0}^{t}K_{0}(u(s))ds\leqslant-\varepsilon t.$

When $m$ is large enough, we get

$\displaystyle
I_{3}=C_{19}\int_{0}^{t}\int_{r>2m+t}\left|u_{1}\right|^{2}+\left|\nabla
u_{0}\right|^{2}+\left|u_{0}\right|^{2}+2G\left(\left|u_{0}\right|\right)dxds\leqslant\int_{0}^{t}\frac{\varepsilon}{4}ds\leqslant\frac{\varepsilon
t}{4},$ $\displaystyle
I_{4}=\int_{0}^{t}\int_{m<r}\frac{1}{2\operatorname{In}m}|\nabla
u|^{2}dxds\leqslant\int_{0}^{t}\frac{\varepsilon}{4}ds\leqslant\frac{\varepsilon
t}{4},$

so

$\displaystyle\operatorname{Re}\int_{R^{3}}u_{t}\phi_{m}\bar{u}_{r}dx-\operatorname{Re}\int_{R^{3}}u_{t}(0)\phi_{m}(r,0)\bar{u}_{r}(0)dx\leqslant-\frac{\varepsilon
t}{2}.$ (4.26)

In addition, when $t$ is sufficiently large, we can get

$\displaystyle\left|\phi_{m}(r,t)\right|\leqslant\frac{C_{20}t}{\operatorname{In}t}.$
(4.27)

Thus, (4.26) can be written as

$\displaystyle\int_{R^{3}}\nabla u\cdot u_{t}dx\geqslant
C(\varepsilon)\operatorname{In}t.$ (4.28)

By H?lder’s inequality, we have

$\displaystyle\left(\int_{R^{3}}|\nabla
u|^{2}dx\right)^{\frac{1}{2}}\cdot\left(\int_{R^{3}}\left|u_{t}\right|^{2}dx\right)^{\frac{1}{2}}\geqslant
C(\varepsilon)\operatorname{In}t,$ (4.29)

which contradicts the assumption that $u(t,x)$ is bounded in
$H_{r}^{1}(R^{3})$, so there is a sequence $\left\\{t_{n}\right\\}$, when
$t_{n}\rightarrow\infty$,
$\left\|u\left(t_{n}\right)\right\|_{H_{r}^{1}}\rightarrow\infty$.

Take $u_{0}(x)=\phi\left(\frac{x}{\lambda}\right)$, $u_{1}(x)=0$, $\lambda>1$,
where $\phi(x)$ is the ground state solution of (2.3), we obtain

$\displaystyle\left\|\left(u_{0},u_{1}\right)-\left(\phi,0\right)\right\|_{H_{r}^{1}\times
L_{r}^{2}}<\varepsilon.$ (4.30)

Then we prove $\left(u_{0},u_{1}\right)\in R_{1}$, from the proof process of
Lemma 3.2, we can get that $J_{0}\left(u_{\lambda}(x)\right)$ is a decreasing
function of $\lambda$($\lambda>1$), so

$\displaystyle
E\left(u_{0},u_{1}\right)=E\left(u_{0},0\right)=J_{0}\left(u_{0}(x)\right)=J_{0}\left(\phi\left(\frac{x}{\lambda}\right)\right)<J_{0}(\phi(x))=d(0),$
(4.31) $\displaystyle\frac{1}{3}\int_{R^{3}}\left|\nabla
u_{0}\right|^{2}dx=\frac{1}{3}\int_{R^{3}}\left|\nabla\phi\left(\frac{x}{\lambda}\right)\right|^{2}dx=\frac{\lambda}{3}\int_{R^{3}}|\nabla\phi(x)|^{2}dx>\frac{1}{3}\int_{R^{3}}|\nabla\phi(x)|^{2}dx=d(0).$
(4.32)

(4.31) and (4.32) indicate $\left(u_{0},u_{1}\right)\in R_{1}$, so we get
Theorem 2.2. Then the instability of the standing wave solutions is obtained .

## Acknowledgment

The first author is supported in part by the National Science Foundation of
China, grant 11971166.

## References

  * [1] K. Bartkowski, P. Gorka, One-dimensional Klein–Gordon equation with logarithmic nonlinearities, Journal of Physics A: Mathematical and Theoretical. 41 (2008), no. 35, 201-355.
  * [2] I. Bialynicki, J. Mycielski, Wave equations with logarithmic nonlinearities, Theoretical Physics. 23 (1975), no. 4, 461-466.
  * [3] P. Gorka, Logarithmic Klein–Gordon equation, Acta Physica Polonica B. 40 (2009), no. 1, 60-66.
  * [4] G. Rosen, Dilatation covariance and exact solutions in local relativistic field theories, Physical Review. 183 (1969), no. 5, 1186.
  * [5] John D. Barrow, P. Parsons, Inflationary models with logarithmic potentials, Physical Review D. 52 (1995), no. 10, 5577-5587.
  * [6] K. Enqvist, J. McDonald Q-balls and baryogenesis in the MSSM, Physical Letters B. 425 (1998), 309-321.
  * [7] A. Linde, Strings, textures, inflation and spectrum bending, Physical Letters B. 284 (1992), 215-222.
  * [8] Y. Ye, L. Li, Global existence and blow-up of solutions for logarithmic Klein-Gordon equation, AIMS Mathematics. 6 (2021), no. 7, 6898-6914.
  * [9] J. Yan, H. Zhang, Regularized finite difference methods for the logarithmic Klein-Gordon equation, East Asian Journal on Applied Mathematics. 11 (2021), no. 1, 119-142.
  * [10] F. Natali, E. Cardoso, Orbital stability of periodic standing waves for the logarithmic Klein-Gordon equation, Journal of Mathematical Analysis and Applications. 482 (2020), no. 2, 1-42.
  * [11] Y. Ye. Global solution and blow–up of logarithmic Klein–Gordon equation, Bulletin of the Korean Mathematical Society. 57 (2020), no. 2, 281-294.
  * [12] M. Khuddush, K. Prasad, B. Bharathi, Global existence and blow up of solutions for a semilinear Klein–Gordon equation with the product of logarithmic and power–type nonlinearity, Annali dell universita di ferrara. 68 (2022), no. 1, 187-201.
  * [13] T. Roy, Global existence of solutions of a Loglog energy–supercritical Klein–Gordon equation, Journal of Dynamics and Differential Equations. (2022), 1-44.
  * [14] L. Wei, M. Salik, R. Xu, Global existence and blow up of solution for semi–linear Klein–Gordon hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Mathematica. 40 (2020), no. 1, 111-130.
  * [15] L. Ma, Z. Fang Energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source, Math Meth Appl Sci. (2018), 1-15.
  * [16] O. Alves and F. Morais, Existence and concentration of positive solutions for a Schrodinger logarithmic equation, Z. Angew. Math. Phys. (2018) 69:144.
  * [17] J. Zhang, Sharp conditions of global existence for nonlinear Schrodinger and Klein–Gordon equations, Nonlinear Analysis. 48 (2002), no. 2, 191-207.
  * [18] H. Pecher, Lp–Abschtzungen und Klassische Lsungen fur Nichtlineare Wellengleichungen, Manuscripta Mathematica. 20 (1977), no. 3, 227-244.
  * [19] N. Hayashi, P. Naumkin, Final state problem for the cubic nonlinear Klein–Gordon equation, Journal of mathematical physics. 50 (2009), no. 10, 2476-2488.
  * [20] S. Zhu, Sharp energy criteria of blow–up for the energy–critical Klein–Gordon equation, Journal of Inequalities and Applications. 2015 (2015), no. 1, 1-9.
  * [21] X. Luo, Blow up of solutions of initial boundary value problem for a class of nonlinear Klein-Gordon equations, Journal of Southwest University for Nationalities.Natural Science Edition. 36 (2010), no. 1, 22-24.
  * [22] J. Zhao, S. Li, Sharp condition for global existence and blow-up on Klein-Gordon equation, Mathematical Problems in Engineering. 2009 (2009), 1–9.
  * [23] X. Cheng, Z. Guo, S. Masaki, Scattering for the mass-critial nonlinear Klein-Gordon equations in three and higher dimensions, Analysis of PDEs. (2022), 1–34.
  * [24] S. Ibrahim, N. Masmoudi, K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Analysis and PDE. 4 (2011), no. 3, 1–34.
  * [25] K. Nakanishi, Scattering Theory for Nonlinear Klein–Gordon Equation with Sobolev Critical Power, International Mathematics Research Notices. 1999 (1999), no. 1, 139-147.
  * [26] O. Masahito, T.Grozdena, Strong instability of standing waves for nonlinear Klein–Gordon Equations, Discrete and Continuous Dynamical Systems. 12 (2005), no.2, 315-322.
  * [27] Y. Wu, Instability of the standing waves for the nonlinear Klein–Gordon equation in one dimension, Mathematics Subject Classification. (2018), 1-18.
  * [28] J. Shatah, Stable standing waves of nonlinear Klein–Gordon equations, Communications in Mathematical Physics. 91 (1983), no. 3, 313-327.
  * [29] J. Shatah, Unstable ground state of nonlinear Klein–Gordon equations, Transactions of the American Mathematical Society. 290 (1985), no.2, 701-710.
  * [30] Z. Gan, J. Zhang, Standing waves of the inhomogeneous Klein–Gordon equations with critical exponent, Acta Mathematica Sinica. 22 (2006), no. 2, 357-366.
  * [31] J. Zhang, Z. Gan, B. Guo, Stability of the standing waves for a class of coupled nonlinear Klein–Gordon equations, Acta Mathematicae Applicatae Sinica, English Series. 26 (2010), no. 3, 427-442.
  * [32] W. Huang, J. Zhang, Instability of the standing waves for nonlinear Klein–Gordon equations with damping term,Applied mathematics and computation. 213 (2009), no. 2, 522-528.
  * [33] H. Berestycki, P. L. Lions Nonlinear scalar field equations I, II, Communications in Arch. Rat. Mech. Anal, 55 (1983), no. 82, 313-375.
  * [34] W.A. Strauss, Existence of solitary waves in higher dimensions, Communications in Mathematical Physics, 55 (1977), no. 2, 149-162.