The Existence of Least Energy Sign-Changing Solution for Kirchhoff-Type Problem with Potential Vanishing at Infinity

Xiao, Ting; Gan, Canlin; Zhang, Qiongfen

doi:10.1155/2021/6690204

Cited by 3 publications

(2 citation statements)

References 30 publications

(30 reference statements)

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“…It can be unpredictable weather events that disrupt these technologies. To improve the sustainability, some energy storage systems and modern controllers should be used to make a balance between consumption and germination [1,2].…”

Section: Introductionmentioning

confidence: 99%

A New Type-3 Fuzzy PID for Energy Management in Microgrids

Fan

Mohammadzadeh

Kausar

et al. 2022

Advances in Mathematical Physics

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More recently, type-3 (T3) fuzzy logic systems (FLSs) with better learning ability and uncertainty modeling have been presented. On other hand, the proportional-integral-derivative (PID) is commonly employed in most industrial control systems, because of its simplicity and efficiency. The measurement errors, nonlinearities, and uncertainties degrade the performance of conventional PIDs. In this study, for the first time, a new T3-FLS-based PID scheme with deep learning approach is introduced. In addition to rules, the parameters of fuzzy sets are also tuned such that a fast regulation efficiency is obtained. Unlike the most conventional approaches, the suggested tuning approach is done in an online scheme. Also, a nonsingleton fuzzification is suggested to reduce the effect of sensor errors. The proposed scheme is examined on a case-study microgrid (MG), and its good frequency stabilization performance is demonstrated in various hard conditions such as variable load, unknown dynamics, and variation in renewable energy (RE) sources.

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Section: Introductionmentioning

confidence: 99%

A New Type-3 Fuzzy PID for Energy Management in Microgrids

Fan

Mohammadzadeh

Kausar

et al. 2022

Advances in Mathematical Physics

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“…In recent years, many authors are concerned with nonlocal problems like (1.2) which can been seen as a generalization of the second-order semilinear elliptic equations, p-Laplacian equations and (p, q)-Laplacian equations (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,33] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Existence and multiplicity of solutions for a class of Kirchhoff type $(Φ_1,Φ_2)$-Laplacian system with locally super-linear condition in $\mathbb{R}^N$

Liu¹,

Zhang²

2022

Preprint

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We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N . We assume that the nonlinear term satisfies the locallywhich is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global|u| m 1 +|v| m 2 = +∞ for a.e. x ∈ R N . We obtain that system has at least one weak solution by using the classical Mountain Pass Theorem. To a certain extent, our theorems extend the results of Tang-Lin-Yu [Journal of Dynamics and Differential Equations, 2019, 31(1): 369-383]. Moreover, under the above naturally global restriction, we obtain that system has infinitely many weak solutions of high energy by using the Symmetric Mountain Pass Theorem, which is different from those results of Wang-Zhang-Fang [Journal of Nonlinear Sciences and Applications, 2017, 10(7): 3792-3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

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Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition

Liu

Zhang

2023

Advances in Nonlinear Analysis

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We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{\mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) \left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ G x\in G , where G G is a domain in R N {{\mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ R N x\in {{\mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

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The Existence of Least Energy Sign-Changing Solution for Kirchhoff-Type Problem with Potential Vanishing at Infinity

Abstract: In this paper, we study the Kirchhoff-type equation: − a + b ∫ … Show more

Cited by 3 publications

References 30 publications

A New Type-3 Fuzzy PID for Energy Management in Microgrids

A New Type-3 Fuzzy PID for Energy Management in Microgrids

Existence and multiplicity of solutions for a class of Kirchhoff type $(Φ_1,Φ_2)$-Laplacian system with locally super-linear condition in $\mathbb{R}^N$

Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition

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