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23 pages, 364 KB

Open AccessArticle

Existence of the Nontrivial Solution for a p-Kirchhoff Problem with Critical Growth and Logarithmic Nonlinearity

by Lixiang Cai and Qing Miao

Axioms 2024, 13(8), 548; https://doi.org/10.3390/axioms13080548 - 13 Aug 2024

Cited by 3 | Viewed by 1336

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth: [...] Read more.

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth:

\{\begin{matrix} - M (\int_{Ω} {|\nabla u|}^{p} d x) Δ_{p} u = {|u|}^{p^{*} - 2} u + λ {|u|}^{p - 2} u - {|u|}^{p - 2} u \ln {|u|}^{2} x \in Ω, \\ u = 0 x \in \partial Ω, \end{matrix}

where

Ω \subset ℝ^{N}

is a bounded domain with a smooth boundary,

2 < p < p^{*} < N

, and both

p

and

N

are positive integers. By using the Nehari manifold and the Mountain Pass Theorem without the Palais-Smale compactness condition, it was proved that the equation had at least one nontrivial solution under appropriate conditions. It addresses the challenges posed by the critical term, the Kirchhoff nonlocal term and the logarithmic nonlinear term. Additionally, it extends partial results of the Brézis–Nirenberg problem with logarithmic perturbation from p = 2 to more general p-Kirchhoff type problems. Full article

20 pages, 384 KB

Open AccessArticle

(p(x),q(x))-Kirchhoff-Type Problems Involving Logarithmic Nonlinearity with Variable Exponent and Convection Term

by Weichun Bu, Tianqing An, Deliang Qian and Yingjie Li

Fractal Fract. 2022, 6(5), 255; https://doi.org/10.3390/fractalfract6050255 - 6 May 2022

Cited by 1 | Viewed by 2379

Abstract

In the present article, we study a class of Kirchhoff-type equations driven by the

(p (x), q (x))

-Laplacian. Due to the lack of a variational structure, ellipticity, and monotonicity, the well-known variational methods are not [...] Read more.

In the present article, we study a class of Kirchhoff-type equations driven by the

(p (x), q (x))

-Laplacian. Due to the lack of a variational structure, ellipticity, and monotonicity, the well-known variational methods are not applicable. With the help of the Galerkin method and Brezis theorem, we obtain the existence of finite-dimensional approximate solutions and weak solutions. One of the main difficulties and innovations of the present article is that we consider competing

(p (x), q (x))

-Laplacian, convective terms, and logarithmic nonlinearity with variable exponents, another one is the weaker assumptions on nonlocal term

M_{υ (x)}

and nonlinear term g. Full article

16 pages, 314 KB

Open AccessArticle

A New Proof of the Existence of Nonzero Weak Solutions of Impulsive Fractional Boundary Value Problems

by Asma Alharbi, Rafik Guefaifia and Salah Boulaaras

Mathematics 2020, 8(5), 856; https://doi.org/10.3390/math8050856 - 25 May 2020

Cited by 2 | Viewed by 2125

Abstract

The paper deals with the existence of at least two non zero weak solutions to a new class of impulsive fractional boundary value problems via Brezis and Nirenberg’s Linking Theorem. Finally, an example is presented to illustrate our results. Full article

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