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23 pages, 437 KiB

Open AccessArticle

Monotone Positive Radial Solution of Double Index Logarithm Parabolic Equations

by Mengru Liu and Lihong Zhang

Fractal Fract. 2024, 8(3), 173; https://doi.org/10.3390/fractalfract8030173 - 16 Mar 2024

Cited by 2 | Viewed by 1464

This article mainly studies the double index logarithmic nonlinear fractional

g -

Laplacian parabolic equations with the Marchaud fractional time derivatives

\partial_{t}^{α}

. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double [...] Read more.

This article mainly studies the double index logarithmic nonlinear fractional

g -

Laplacian parabolic equations with the Marchaud fractional time derivatives

\partial_{t}^{α}

. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional

g -

Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional

g -

Laplacian parabolic equations is studied. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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14 pages, 313 KiB

Open AccessArticle

Infinitely Many Solutions for the Fractional p&q-Laplacian Problems in R^N

by Liyan Wang, Kun Chi, Jihong Shen and Bin Ge

Symmetry 2022, 14(12), 2486; https://doi.org/10.3390/sym14122486 - 23 Nov 2022

Cited by 1 | Viewed by 1396

Abstract

In this paper, we consider the following class of the fractional

p & q

-Laplacian problem: [...] Read more.

In this paper, we consider the following class of the fractional

p & q

-Laplacian problem:

{(- Δ)}_{p}^{s} u + {(- Δ)}_{q}^{s} u + {V (x) (| u |}^{p - 2} u + {| u |}^{q - 2} {u) + g (x) | u |}^{r - 2} u = K (x) f (x, u) + h (u), x \in R^{N},

V : R^{N} \to R^{+}

is a potential function, and

h : R \to R

is a perturbation term. We studied two cases: if

f (x, u)

is sublinear, by means of Clark’s theorem, which considers the symmetric condition about the functional, we get infinitely many solutions; if

f (x, u)

is superlinear, using the symmetric mountain-pass theorem, infinitely many solutions can be obtained. Full article

(This article belongs to the Section Mathematics)

13 pages, 313 KiB

Open AccessArticle

On Non Local p-Laplacian with Right Hand Side Radon Measure

by Mohammed Kbiri Alaoui

Fractal Fract. 2022, 6(9), 464; https://doi.org/10.3390/fractalfract6090464 - 25 Aug 2022

Viewed by 1518

Abstract

The aim of this paper is to investigate the following non local p-Laplacian problem with data a bounded Radon measure

ϑ \in M_{b} (Ω)

{(- Δ)}_{p}^{s} u = ϑ in Ω,

with vanishing [...] Read more.

The aim of this paper is to investigate the following non local p-Laplacian problem with data a bounded Radon measure

ϑ \in M_{b} (Ω)

{(- Δ)}_{p}^{s} u = ϑ in Ω,

with vanishing conditions outside

Ω,

and where

s \in (0, 1), 2 - \frac{s}{N} < p \leq N .

An existence result is provided, and some sharp regularity has been investigated. More precisely, we prove by using some fractional isoperimetric inequalities the existence of weak solution u such that: 1. If

ϑ \in M_{b} (Ω),

then

u \in W_{0}^{s_{1}, q} (Ω)

for all

s_{1} < s

and

q < \frac{N (p - 1)}{N - s} .

2. If

ϑ

belongs to the Zygmund space

L L o g^{α} L (Ω), α > \frac{N - s}{N},

then the limiting regularity

u \in W_{0}^{s_{1}, \frac{N (p - 1)}{N - s}} (Ω)

(for all

s_{1} < s

). 3. If

ϑ \in L L o g^{α} L (Ω),

and

α = \frac{N - s}{N}

with

p = N,

then we reach the maximal regularity with respect to s and

N, u \in W_{0}^{s, N} (Ω) .

Full article

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17 pages, 341 KiB

Open AccessArticle

Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝ^N

by Yun-Ho Kim

Mathematics 2020, 8(10), 1792; https://doi.org/10.3390/math8101792 - 15 Oct 2020

Cited by 5 | Viewed by 2221

Abstract

We are concerned with the following elliptic equations: [...] Read more.

We are concerned with the following elliptic equations:

{(- Δ)}_{p}^{s} v + {V (x) | v |}^{p - 2} v = λ a (x) {| v |}^{r - 2} v + g (x, v) i n R^{N},

where

{(- Δ)}_{p}^{s}

is the fractional p-Laplacian operator with

0 < s < 1 < r < p < + \infty

s p < N

, the potential function

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

is a continuous potential function, and

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

satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

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