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12 pages, 245 KiB

Open AccessArticle

Multiple Solutions for Nonlocal Fourth-Order Equation with Concave–Convex Nonlinearities

by Ruiting Jiang and Chengbo Zhai

Mathematics 2025, 13(12), 1985; https://doi.org/10.3390/math13121985 - 16 Jun 2025

Viewed by 253

This paper is devoted to a class of general nonlocal fourth-order elliptic equation with concave–convex nonlinearities. First, using the

Z_{2}

-mountain pass theorem in critical point theory, we obtain the existence of infinitely many large energy solutions. Then, using the dual fountain [...] Read more.

This paper is devoted to a class of general nonlocal fourth-order elliptic equation with concave–convex nonlinearities. First, using the

Z_{2}

-mountain pass theorem in critical point theory, we obtain the existence of infinitely many large energy solutions. Then, using the dual fountain theorem, we prove that the equation has infinitely many negative energy solutions, whose energy converges at 0. Our results extend and complement existing findings in the literature. Full article

26 pages, 387 KiB

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 714

Abstract

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 [...] Read more.

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)

31 pages, 459 KiB

Open AccessArticle

Multiple Solutions to the Fractional p-Laplacian Equations of Schrödinger–Hardy-Type Involving Concave–Convex Nonlinearities

by Yun-Ho Kim

Fractal Fract. 2024, 8(7), 426; https://doi.org/10.3390/fractalfract8070426 - 20 Jul 2024

Cited by 1 | Viewed by 925

Abstract

This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the

L^{\infty}

-bound for any possible weak solution to our problem. As far as we know, the global a priori bound for [...] Read more.

This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the

L^{\infty}

-bound for any possible weak solution to our problem. As far as we know, the global a priori bound for weak solutions to nonlinear elliptic problems involving a singular nonlinear term such as Hardy potentials has not been studied extensively. To overcome this, we utilize a truncated energy technique and the De Giorgi iteration method. As its application, we demonstrate that the problem above has at least two distinct nontrivial solutions by exploiting a variant of Ekeland’s variational principle and the classical mountain pass theorem as the key tools. Furthermore, we prove the existence of a sequence of infinitely many weak solutions that converges to zero in the

L^{\infty}

-norm. To derive this result, we employ the modified functional method and the dual fountain theorem. Full article

(This article belongs to the Special Issue Fractional Elliptic and Parabolic Equations: Analysis and Related Topics)

35 pages, 444 KiB

Open AccessArticle

Multiplicity Results of Solutions to the Double Phase Problems of Schrödinger–Kirchhoff Type with Concave–Convex Nonlinearities

by Yun-Ho Kim and Taek-Jun Jeong

Mathematics 2024, 12(1), 60; https://doi.org/10.3390/math12010060 - 24 Dec 2023

Cited by 3 | Viewed by 1376

Abstract

The present paper is devoted to establishing several existence results for infinitely many solutions to Schrödinger–Kirchhoff-type double phase problems with concave–convex nonlinearities. The first aim is to demonstrate the existence of a sequence of infinitely many large-energy solutions by applying the fountain theorem [...] Read more.

The present paper is devoted to establishing several existence results for infinitely many solutions to Schrödinger–Kirchhoff-type double phase problems with concave–convex nonlinearities. The first aim is to demonstrate the existence of a sequence of infinitely many large-energy solutions by applying the fountain theorem as the main tool. The second aim is to obtain that our problem admits a sequence of infinitely many small-energy solutions. To obtain these results, we utilize the dual fountain theorem. In addition, we prove the existence of a sequence of infinitely many weak solutions converging to 0 in

L^{\infty}

-space. To derive this result, we exploit the dual fountain theorem and the modified functional method. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

16 pages, 361 KiB

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 4 | Viewed by 1292

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. [...] Read more.

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)

20 pages, 361 KiB

Open AccessArticle

Infinitely Many Small Energy Solutions to the Double Phase Anisotropic Variational Problems Involving Variable Exponent

by Jun-Hyuk Ahn and Yun-Ho Kim

Axioms 2023, 12(3), 259; https://doi.org/10.3390/axioms12030259 - 2 Mar 2023

Viewed by 1676

Abstract

This paper is devoted to double phase anisotropic variational problems for the case of a combined effect of concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, on a class of superlinear term which [...] Read more.

This paper is devoted to double phase anisotropic variational problems for the case of a combined effect of concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, on a class of superlinear term which is different from the previous related works, is to discuss the multiplicity result of non-trivial solutions by applying the dual fountain theorem as the main tool. In particular, our main result is obtained without assuming the conditions on the nonlinear term at infinity. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics)

16 pages, 384 KiB

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 1730

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 KiB

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 2509

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 [...] Read more.

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)

13 pages, 280 KiB

Open AccessArticle

Multiple Solutions for a Class of New p(x)-Kirchhoff Problem without the Ambrosetti-Rabinowitz Conditions

by Bei-Lei Zhang, Bin Ge and Xiao-Feng Cao

Mathematics 2020, 8(11), 2068; https://doi.org/10.3390/math8112068 - 19 Nov 2020

Cited by 11 | Viewed by 2369

Abstract

In this paper, we consider a nonlocal

p (x)

-Kirchhoff problem with a

p^{+}

-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Under some certain assumptions, we prove the existence of nontrivial solutions and many solutions. [...] Read more.

In this paper, we consider a nonlocal

p (x)

-Kirchhoff problem with a

p^{+}

-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Under some certain assumptions, we prove the existence of nontrivial solutions and many solutions. Our results are an improvement and generalization of the corresponding results obtained by Hamdani et al. (2020). Full article

(This article belongs to the Special Issue New Trends in Variational Methods in Nonlinear Analysis)

15 pages, 309 KiB

Open AccessArticle

Multiplicity of Radially Symmetric Small Energy Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators

by Jun Ik Lee and Yun-Ho Kim

Mathematics 2020, 8(1), 128; https://doi.org/10.3390/math8010128 - 15 Jan 2020

Cited by 5 | Viewed by 2123

Abstract

We investigate the multiplicity of radially symmetric solutions for the quasilinear elliptic equation of Kirchhoff type. This paper is devoted to the study of the

L^{\infty}

-bound of solutions to the problem above by applying De Giorgi’s iteration method and the localization [...] Read more.

We investigate the multiplicity of radially symmetric solutions for the quasilinear elliptic equation of Kirchhoff type. This paper is devoted to the study of the

L^{\infty}

-bound of solutions to the problem above by applying De Giorgi’s iteration method and the localization method. Employing this, we provide the existence of multiple small energy radially symmetric solutions whose

L^{\infty}

-norms converge to zero. We utilize the modified functional method and the dual fountain theorem as the main tools. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

15 pages, 316 KiB

Open AccessFeature PaperArticle

Existence of Small-Energy Solutions to Nonlocal Schrödinger-Type Equations for Integrodifferential Operators in ℝ^N

by Jun Ik Lee, Yun-Ho Kim and Jongrak Lee

Symmetry 2020, 12(1), 5; https://doi.org/10.3390/sym12010005 - 18 Dec 2019

Cited by 3 | Viewed by 1812

Abstract

We are concerned with the following elliptic equations:

{(- Δ)}_{p, K}^{s} u + V (x) {| u |}^{p - 2} u = λ f (x, u) in R^{N}

, where [...] Read more.

We are concerned with the following elliptic equations:

{(- Δ)}_{p, K}^{s} u + V (x) {| u |}^{p - 2} u = λ f (x, u) in R^{N}

, where

{(- Δ)}_{p, K}^{s}

is the nonlocal integrodifferential equation with

0 < s < 1 < p < + \infty

,

s p < N

the potential function

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

is continuous, and

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

satisfies a Carathéodory condition. The present paper is devoted to the study of the

L^{\infty}

-bound of solutions to the above problem by employing De Giorgi’s iteration method and the localization method. Using this, we provide a sequence of infinitely many small-energy solutions whose

L^{\infty}

-norms converge to zero. The main tools were the modified functional method and the dual version of the fountain theorem, which is a generalization of the symmetric mountain-pass theorem. Full article

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