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26 pages, 387 KB

Open AccessFeature PaperArticle

Multiplicity Results of Solutions to the Fractional p-Laplacian Problems of the Kirchhoff–Schrödinger–Hardy Type

by Yun-Ho Kim

Mathematics 2025, 13(1), 47; https://doi.org/10.3390/math13010047 - 26 Dec 2024

Viewed by 1020

This paper is devoted to establishing multiplicity results of nontrivial weak solutions to the fractional p-Laplacian equations of the Kirchhoff–Schrödinger type with Hardy potentials. The main features of the paper are the appearance of the Hardy potential and nonlocal Kirchhoff coefficients, and the absence of the compactness condition of the Palais–Smale type. To demonstrate the multiplicity results, we exploit the fountain theorem and the dual fountain theorem as the main tools, respectively. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

10 pages, 298 KB

Open AccessArticle

Uniqueness Results and Asymptotic Behaviour of Nonlinear Schrödinger–Kirchhoff Equations

by Dongdong Sun

Symmetry 2023, 15(10), 1856; https://doi.org/10.3390/sym15101856 - 2 Oct 2023

Cited by 1 | Viewed by 1250

Abstract

In this paper, we first study the uniqueness and symmetry of solution of nonlinear Schrödinger–Kirchhoff equations with constant coefficients. Then, we show the uniqueness of the solution of nonlinear Schrödinger–Kirchhoff equations with the polynomial potential. In the end, we investigate the asymptotic behaviour of the positive least energy solutions to nonlinear Schrödinger–Kirchhoff equations with vanishing potentials. The vanishing potential means that the zero set of the potential is non-empty. The uniqueness results of Schrödinger equations and the scaling technique are used in our proof. The elliptic estimates and energy analysis are applied in the proof of the asymptotic behaviour of the above Schrödinger–Kirchhoff-type equations. Full article

(This article belongs to the Special Issue Symmetry in Differential Equations and Integral Operators)

16 pages, 361 KB

Open AccessArticle

Multiple Solutions to a Non-Local Problem of Schrödinger–Kirchhoff Type in ℝ^N

by In Hyoun Kim, Yun-Ho Kim and Kisoeb Park

Fractal Fract. 2023, 7(8), 627; https://doi.org/10.3390/fractalfract7080627 - 17 Aug 2023

Cited by 5 | Viewed by 1521

Abstract

The main purpose of this paper is to show the existence of a sequence of infinitely many small energy solutions to the nonlinear elliptic equations of Kirchhoff–Schrödinger type involving the fractional p-Laplacian by employing the dual fountain theorem as a key tool. Because of the presence of a non-local Kirchhoff coefficient, under conditions on the nonlinear term given in the present paper, we cannot obtain the same results concerning the existence of solutions in similar ways as in the previous related works. For this reason, we consider a class of Kirchhoff coefficients that are different from before to provide our multiplicity result. In addition, the behavior of nonlinear terms near zero is slightly different from previous studies. Full article

(This article belongs to the Special Issue Variational Problems and Fractional Differential Equations)

16 pages, 384 KB

Open AccessArticle

Infinitely Many Small Energy Solutions to Schrödinger-Kirchhoff Type Problems Involving the Fractional

r (\cdot)

-Laplacian in

R^{N}

by Yun-Ho Kim

Fractal Fract. 2023, 7(3), 207; https://doi.org/10.3390/fractalfract7030207 - 21 Feb 2023

Cited by 2 | Viewed by 1914

Abstract

This paper is concerned with the existence result of a sequence of infinitely many small energy solutions to the fractional

r (\cdot)

-Laplacian equations of Kirchhoff–Schrödinger type with concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The [...] Read more.

This paper is concerned with the existence result of a sequence of infinitely many small energy solutions to the fractional

r (\cdot)

-Laplacian equations of Kirchhoff–Schrödinger type with concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, under suitable assumptions on a nonlinear term, is to discuss the multiplicity result of non-trivial solutions by using the dual fountain theorem as the main tool. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Laplacian Problems)

14 pages, 330 KB

Open AccessArticle

Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in

R^{N}

by Kisoeb Park

Axioms 2022, 11(2), 38; https://doi.org/10.3390/axioms11020038 - 19 Jan 2022

Cited by 2 | Viewed by 2662

Abstract

In this paper, we establish the existence of a nontrivial weak solution to Schrödinger-kirchhoff type equations with the fractional magnetic field without Ambrosetti and Rabinowitz condition using mountain pass theorem under a suitable assumption of the external force. Furthermore, we prove the existence of infinitely many large- or small-energy solutions to this problem with Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem by applying the variational methods, that is, the fountain and the dual fountain theorem with Cerami condition. Full article

(This article belongs to the Special Issue Recent Advances in Nonlinear Differential Equations: Theory, Methods and Applications)

22 pages, 369 KB

Open AccessArticle

Infinitely Many Solutions for Fractional p-Laplacian Schrödinger–Kirchhoff Type Equations with Symmetric Variable-Order

by Weichun Bu, Tianqing An, José Vanteler da C. Sousa and Yongzhen Yun

Symmetry 2021, 13(8), 1393; https://doi.org/10.3390/sym13081393 - 31 Jul 2021

Cited by 1 | Viewed by 2029

Abstract

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations [...] Read more.

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations

{(a + b \int_{Ω \times Ω} \frac{{| ξ (x) - ξ (y) |}^{^{p}}}{{| x - y |}^{N + p s (x, y)}} d x d y)}^{p - 1} {(- Δ)}_{p}^{s (\cdot)} ξ + λ V (x) {| ξ |}^{p - 2} ξ = f (x, ξ), x \in Ω, ξ = 0, x \in \partial Ω,

where

Ω

is a bounded Lipschitz domain in

R^{N}

1 < p < + \infty

a, b > 0

are constants,

s (\cdot) : R^{N} \times R^{N} \to (0, 1)

is a continuous and symmetric function with

N > s (x, y) p

for all

(x, y) \in Ω \times Ω

λ > 0

is a parameter,

{(- Δ)}_{p}^{s (\cdot)}

is a fractional p-Laplace operator with variable-order,

V (x) : Ω \to R^{+}

is a potential function, and

f (x, ξ) : Ω \times R^{N} \to R

is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition. Full article

21 pages, 390 KB

Open AccessFeature PaperArticle

Multiplicity of Small or Large Energy Solutions for Kirchhoff–Schrödinger-Type Equations Involving the Fractional p-Laplacian in ℝ^N

by Jae-Myoung Kim, Yun-Ho Kim and Jongrak Lee

Symmetry 2018, 10(10), 436; https://doi.org/10.3390/sym10100436 - 26 Sep 2018

Cited by 4 | Viewed by 3056

Abstract

We herein discuss the following elliptic equations: [...] Read more.

We herein discuss the following elliptic equations:

M (\int_{R^{N}} \int_{R^{N}} \frac{{| u (x) - u (y) |}^{p}}{{| x - y |}^{N + p s}} d x d y) {(- Δ)}_{p}^{s} u + V (x) {| u |}^{p - 2} u = λ f (x, u) in R^{N},

where

{(- Δ)}_{p}^{s}

is the fractional p-Laplacian defined by

{(- Δ)}_{p}^{s} u (x) = 2 {lim}_{ε ↘ 0} \int_{R^{N} B_{ε} (x)} \frac{{| u (x) - u (y) |}^{p - 2} (u (x) - u (y))}{{| x - y |}^{N + p s}} d y, x \in R^{N} .

Here,

B_{ε} (x) : = {y \in R^{N} : | x - y | < ε}

V : R^{N} \to (0, \infty)

is a continuous function and

f : R^{N} \times R \to R

is the Carathéodory function. Furthermore,

M : R_{0}^{+} \to R^{+}

is a Kirchhoff-type function. This study has two aims. One is to study the existence of infinitely many large energy solutions for the above problem via the variational methods. In addition, a major point is to obtain the multiplicity results of the weak solutions for our problem under various assumptions on the Kirchhoff function

M

and the nonlinear term f. The other is to prove the existence of small energy solutions for our problem, in that the sequence of solutions converges to 0 in the

L^{\infty}

-norm. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)

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