In this paper, for given mass m>0, we focus on the existence and nonexistence of constrained minimizers of the energy functional
I(u):=a2∫R3|∇u|2dx+b4(∫R3|∇u|2dx)2−∫R3F(u)dx
on Sm:={u∈H1(R3):‖u‖22=m},where a,b>0 and F satisfies the almost optimal mass subcritical growth assumptions. We also establish the relationship between the normalized ground state solutions and the ground state to the action functional I(u)−λ2‖u‖22. Our results extend, nontrivially, the ones in Shibata (Manuscripta Math. 143 (2014) 221–237) and Jeanjean and Lu (Calc. Var. 61 (2022) 214) to the Kirchhoff type equations, and generalize and sharply improve the ones in Ye (Math. Methods. Appl. Sci. 38 (2015) 2603–2679) and Chen et al. (Appl. Math. Optim. 84 (2021) 773–806).
Citation: Jing Hu, Jijiang Sun. On constrained minimizers for Kirchhoff type equations with Berestycki-Lions type mass subcritical conditions[J]. Electronic Research Archive, 2023, 31(5): 2580-2594. doi: 10.3934/era.2023131
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In this paper, for given mass m>0, we focus on the existence and nonexistence of constrained minimizers of the energy functional
I(u):=a2∫R3|∇u|2dx+b4(∫R3|∇u|2dx)2−∫R3F(u)dx
on Sm:={u∈H1(R3):‖u‖22=m},where a,b>0 and F satisfies the almost optimal mass subcritical growth assumptions. We also establish the relationship between the normalized ground state solutions and the ground state to the action functional I(u)−λ2‖u‖22. Our results extend, nontrivially, the ones in Shibata (Manuscripta Math. 143 (2014) 221–237) and Jeanjean and Lu (Calc. Var. 61 (2022) 214) to the Kirchhoff type equations, and generalize and sharply improve the ones in Ye (Math. Methods. Appl. Sci. 38 (2015) 2603–2679) and Chen et al. (Appl. Math. Optim. 84 (2021) 773–806).
Multi-nominal data are common in scientific and engineering research such as biomedical research, customer behavior analysis, network analysis, search engine marketing optimization, web mining etc. When the response variable has more than two levels, the principle of mode-based or distribution-based proportional prediction can be used to construct nonparametric nominal association measure. For example, Goodman and Kruskal [3,4] and others proposed some local-to-global association measures towards optimal predictions. Both Monte Carlo and discrete Markov chain methods are conceptually based on the proportional associations. The association matrix, association vector and association measure were proposed by the thought of proportional associations in [9]. If there is no ordering to the response variable's categories, or the ordering is not of interest, they will be regarded as nominal in the proportional prediction model and the other association statistics.
But in reality, different categories in the same response variable often are of different values, sometimes much different. When selecting a model or selecting explanatory variables, we want to choose the ones that can enhance the total revenue, not just the accuracy rate. Similarly, when the explanatory variables with cost weight vector, they should be considered in the model too. The association measure in [9],
To implement the previous adjustments, we need the following assumptions:
It needs to be addressed that the second assumption is probably not always the case. The law of large number suggests that the larger the sample size is, the closer the expected value of a distribution is to the real value. The study of this subject has been conducted for hundreds of years including how large the sample size is enough to simulate the real distribution. Yet it is not the major subject of this article. The purpose of this assumption is nothing but a simplification to a more complicated discussion.
The article is organized as follows. Section 2 discusses the adjustment to the association measure when the response variable has a revenue weight; section 3 considers the case where both the explanatory and the response variable have weights; how the adjusted measure changes the existing feature selection framework is presented in section 4. Conclusion and future works will be briefly discussed in the last section.
Let's first recall the association matrix
γs,t(Y|X)=E(p(Y=s|X)p(Y=t|X))p(Y=s)=α∑i=1p(X=i|Y=s)p(Y=t|X=i);s,t=1,2,..,βτY|X=ωY|X−Ep(Y)1−Ep(Y)ωY|X=EX(EY(p(Y|X)))=β∑s=1α∑i=1p(Y=s|X=i)2p(X=i)=β∑s=1γssp(Y=s) | (1) |
Our discussion begins with only one response variable with revenue weight and one explanatory variable without cost weight. Let
Definition 2.1.
ˆωY|X=β∑s=1α∑i=1p(Y=s|X=i)2rsp(X=i)=β∑s=1γssp(Y=s)rsrs>0,s=1,2,3...,β | (2) |
Please note that
It is easy to see that
Example.Consider a simulated data motivated by a real situation. Suppose that variable
1000 | 100 | 500 | 400 | 500 | 300 | 200 | 1500 | |||
200 | 1500 | 500 | 300 | 500 | 400 | 400 | 50 | |||
400 | 50 | 500 | 500 | 500 | 500 | 300 | 700 | |||
300 | 700 | 500 | 400 | 500 | 400 | 1000 | 100 | |||
200 | 500 | 400 | 200 | 200 | 400 | 500 | 200 |
Let us first consider the association matrix
0.34 | 0.18 | 0.27 | 0.22 | 0.26 | 0.22 | 0.27 | 0.25 | |||
0.13 | 0.48 | 0.24 | 0.15 | 0.25 | 0.24 | 0.29 | 0.23 | |||
0.24 | 0.28 | 0.27 | 0.21 | 0.25 | 0.24 | 0.36 | 0.15 | |||
0.25 | 0.25 | 0.28 | 0.22 | 0.22 | 0.18 | 0.14 | 0.46 |
Please note that
The correct prediction contingency tables of
471 | 6 | 121 | 83 | 98 | 34 | 19 | 926 | |||
101 | 746 | 159 | 107 | 177 | 114 | 113 | 1 | |||
130 | 1 | 167 | 157 | 114 | 124 | 42 | 256 | |||
44 | 243 | 145 | 85 | 109 | 81 | 489 | 6 | |||
21 | 210 | 114 | 32 | 36 | 119 | 206 | 28 |
The total number of the correct predictions by
total revenue | average revenue | |||
0.3406 | 0.456 | 4313 | 0.4714 | |
0.3391 | 0.564 | 5178 | 0.5659 |
Given that
In summary, it is possible for an explanatory variable
Let us further discuss the case with cost weight vector in predictors in addition to the revenue weight vector in the dependent variable. The goal is to find a predictor with bigger profit in total. We hence define the new association measure as in 3.
Definition 3.1.
ˉωY|X=α∑i=1β∑s=1p(Y=s|X=i)2rscip(X=i) | (3) |
Example. We first continue the example in the previous section with new cost weight vectors for
total profit | average profit | ||||
0.3406 | 0.3406 | 1.3057 | 12016.17 | 1.3132 | |
0.3391 | 0.3391 | 1.8546 | 17072.17 | 1.8658 |
By
We then investigate how the change of cost weight affect the result. Suppose the new weight vectors are:
total profit | average profit | ||||
0.3406 | 0.3406 | 1.7420 | 15938.17 | 1.7419 | |
0.3391 | 0.3391 | 1.3424 | 12268.17 | 1.3408 |
Hence
By the updated association defined in the previous section, we present the feature selection result in this section to a given data set
At first, consider a synthetic data set simulating the contribution factors to the sales of certain commodity. In general, lots of factors could contribute differently to the commodity sales: age, career, time, income, personal preference, credit, etc. Each factor could have different cost vectors, each class in a variable could have different cost as well. For example, collecting income information might be more difficult than to know the customer's career; determining a dinner waitress' purchase preference is easier than that of a high income lawyer. Therefore we just assume that there are four potential predictors,
total profit | average profit | ||||
7 | 0.3906 | 3.5381 | 35390 | 3.5390 | |
4 | 0.3882 | 3.8433 | 38771 | 3.8771 | |
4 | 0.3250 | 4.8986 | 48678 | 4.8678 | |
8 | 0.3274 | 3.7050 | 36889 | 3.6889 |
The first variable to be selected is
total profit | average profit | ||||
28 | 0.4367 | 1.8682 | 18971 | 1.8971 | |
28 | 0.4025 | 2.1106 | 20746 | 2.0746 | |
56 | 0.4055 | 1.8055 | 17915 | 1.7915 | |
16 | 0.4055 | 2.3585 | 24404 | 2.4404 | |
32 | 0.3385 | 2.0145 | 19903 | 1.9903 |
As we can see, all
In summary, the updated association with cost and revenue vector not only changes the feature selection result by different profit expectations, it also reflects a practical reality that collecting information for more variables costs more thus reduces the overall profit, meaning more variables is not necessarily better on a Return-Over-Invest basis.
We propose a new metrics,
The presented framework can also be applied to high dimensional cases as in national survey, misclassification costs, association matrix and association vector [9]. It should be more helpful to identify the predictors' quality with various response variables.
Given the distinct character of this new statistics, we believe it brings us more opportunities to further studies of finding the better decision for categorical data. We are currently investigating the asymptotic properties of the proposed measures and it also can be extended to symmetrical situation. Of course, the synthetical nature of the experiments in this article brings also the question of how it affects a real data set/application. It is also arguable that the improvements introduced by the new measures probably come from the randomness. Thus we can use
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1000 | 100 | 500 | 400 | 500 | 300 | 200 | 1500 | |||
200 | 1500 | 500 | 300 | 500 | 400 | 400 | 50 | |||
400 | 50 | 500 | 500 | 500 | 500 | 300 | 700 | |||
300 | 700 | 500 | 400 | 500 | 400 | 1000 | 100 | |||
200 | 500 | 400 | 200 | 200 | 400 | 500 | 200 |
0.34 | 0.18 | 0.27 | 0.22 | 0.26 | 0.22 | 0.27 | 0.25 | |||
0.13 | 0.48 | 0.24 | 0.15 | 0.25 | 0.24 | 0.29 | 0.23 | |||
0.24 | 0.28 | 0.27 | 0.21 | 0.25 | 0.24 | 0.36 | 0.15 | |||
0.25 | 0.25 | 0.28 | 0.22 | 0.22 | 0.18 | 0.14 | 0.46 |
471 | 6 | 121 | 83 | 98 | 34 | 19 | 926 | |||
101 | 746 | 159 | 107 | 177 | 114 | 113 | 1 | |||
130 | 1 | 167 | 157 | 114 | 124 | 42 | 256 | |||
44 | 243 | 145 | 85 | 109 | 81 | 489 | 6 | |||
21 | 210 | 114 | 32 | 36 | 119 | 206 | 28 |
total revenue | average revenue | |||
0.3406 | 0.456 | 4313 | 0.4714 | |
0.3391 | 0.564 | 5178 | 0.5659 |
total profit | average profit | ||||
0.3406 | 0.3406 | 1.3057 | 12016.17 | 1.3132 | |
0.3391 | 0.3391 | 1.8546 | 17072.17 | 1.8658 |
total profit | average profit | ||||
0.3406 | 0.3406 | 1.7420 | 15938.17 | 1.7419 | |
0.3391 | 0.3391 | 1.3424 | 12268.17 | 1.3408 |
total profit | average profit | ||||
7 | 0.3906 | 3.5381 | 35390 | 3.5390 | |
4 | 0.3882 | 3.8433 | 38771 | 3.8771 | |
4 | 0.3250 | 4.8986 | 48678 | 4.8678 | |
8 | 0.3274 | 3.7050 | 36889 | 3.6889 |
total profit | average profit | ||||
28 | 0.4367 | 1.8682 | 18971 | 1.8971 | |
28 | 0.4025 | 2.1106 | 20746 | 2.0746 | |
56 | 0.4055 | 1.8055 | 17915 | 1.7915 | |
16 | 0.4055 | 2.3585 | 24404 | 2.4404 | |
32 | 0.3385 | 2.0145 | 19903 | 1.9903 |
1000 | 100 | 500 | 400 | 500 | 300 | 200 | 1500 | |||
200 | 1500 | 500 | 300 | 500 | 400 | 400 | 50 | |||
400 | 50 | 500 | 500 | 500 | 500 | 300 | 700 | |||
300 | 700 | 500 | 400 | 500 | 400 | 1000 | 100 | |||
200 | 500 | 400 | 200 | 200 | 400 | 500 | 200 |
0.34 | 0.18 | 0.27 | 0.22 | 0.26 | 0.22 | 0.27 | 0.25 | |||
0.13 | 0.48 | 0.24 | 0.15 | 0.25 | 0.24 | 0.29 | 0.23 | |||
0.24 | 0.28 | 0.27 | 0.21 | 0.25 | 0.24 | 0.36 | 0.15 | |||
0.25 | 0.25 | 0.28 | 0.22 | 0.22 | 0.18 | 0.14 | 0.46 |
471 | 6 | 121 | 83 | 98 | 34 | 19 | 926 | |||
101 | 746 | 159 | 107 | 177 | 114 | 113 | 1 | |||
130 | 1 | 167 | 157 | 114 | 124 | 42 | 256 | |||
44 | 243 | 145 | 85 | 109 | 81 | 489 | 6 | |||
21 | 210 | 114 | 32 | 36 | 119 | 206 | 28 |
total revenue | average revenue | |||
0.3406 | 0.456 | 4313 | 0.4714 | |
0.3391 | 0.564 | 5178 | 0.5659 |
total profit | average profit | ||||
0.3406 | 0.3406 | 1.3057 | 12016.17 | 1.3132 | |
0.3391 | 0.3391 | 1.8546 | 17072.17 | 1.8658 |
total profit | average profit | ||||
0.3406 | 0.3406 | 1.7420 | 15938.17 | 1.7419 | |
0.3391 | 0.3391 | 1.3424 | 12268.17 | 1.3408 |
total profit | average profit | ||||
7 | 0.3906 | 3.5381 | 35390 | 3.5390 | |
4 | 0.3882 | 3.8433 | 38771 | 3.8771 | |
4 | 0.3250 | 4.8986 | 48678 | 4.8678 | |
8 | 0.3274 | 3.7050 | 36889 | 3.6889 |
total profit | average profit | ||||
28 | 0.4367 | 1.8682 | 18971 | 1.8971 | |
28 | 0.4025 | 2.1106 | 20746 | 2.0746 | |
56 | 0.4055 | 1.8055 | 17915 | 1.7915 | |
16 | 0.4055 | 2.3585 | 24404 | 2.4404 | |
32 | 0.3385 | 2.0145 | 19903 | 1.9903 |