scholarly journals Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity

2019 ◽  
pp. 1 ◽  
Author(s):  
Jinguo Zhang ◽  
Xiaochun Liu ◽  
Hongying Jiao
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Critical Nonlinearity ◽  
Multiplicity Of Positive Solutions ◽  
Laplacian Equations

Existence and Multiplicity of Positive Solutions for Singular Semipositone p-Laplacian Equations

2006 ◽  
Vol 58 (3) ◽  
pp. 449-475 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Daomin Cao ◽  
Haishen Lü ◽  
Donal O'Regan
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Solution Method ◽  
Degree Theory ◽  
Lower Solution ◽  
Existence And Multiplicity ◽  
Lower Solution Method ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Laplacian Equations

AbstractPositive solutions are obtained for the boundary value problemHere f (t, u) ≥ –M, (M is a positive constant) for (t, u) ∈ [0, 1]×(0, ∞). We will show the existence of two positive solutions by using degree theory together with the upper–lower solution method.

Monotonicity and symmetry of positive solutions to fractional p-Laplacian equation

2021 ◽  
pp. 2150005
Author(s):  
Wei Dai ◽  
Zhao Liu ◽  
Pengyan Wang
Keyword(s):  
Dirichlet Problem ◽  
Nonlinear Equations ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Fractional Laplacian ◽  
Unbounded Domains ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Laplacian Equation ◽  
Laplacian Equations

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional [Formula: see text]-Laplacian: [Formula: see text] where [Formula: see text] is a bounded or an unbounded domain which is convex in [Formula: see text]-direction, and [Formula: see text] is the fractional [Formula: see text]-Laplacian operator defined by [Formula: see text] Under some mild assumptions on the nonlinearity [Formula: see text], we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional [Formula: see text]-Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017) 1750018].

The maximum principles for fractional Laplacian equations and their applications

2017 ◽  
Vol 19 (06) ◽  
pp. 1750018 ◽  
Author(s):  
Tingzhi Cheng ◽  
Genggeng Huang ◽  
Congming Li
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Dirichlet Condition ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Laplacian Equation ◽  
Left Half ◽  
Monotonicity Properties ◽  
Fractional Laplacian Operator ◽  
Laplacian Equations

This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: [Formula: see text] Here [Formula: see text] is a domain (bounded or unbounded) in [Formula: see text] which is convex in [Formula: see text]-direction. [Formula: see text] is the nonlocal fractional Laplacian operator which is defined as [Formula: see text] Under various conditions on [Formula: see text] and on a solution [Formula: see text] it is shown that [Formula: see text] is strictly increasing in [Formula: see text] in the left half of [Formula: see text], or in [Formula: see text]. Symmetry (in [Formula: see text]) of some solutions is proved.

scholarly journals Multiplicity of Positive Solutions for ap-q-Laplacian Type Equation with Critical Nonlinearities

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Positive Solutions ◽  
Type Equation ◽  
Critical Sobolev Exponent ◽  
Critical Nonlinearity ◽  
Bounded Smooth Domain ◽  
Laplacian Equation ◽  
Critical Nonlinearities ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions

We study the effect of the coefficientf(x)of the critical nonlinearity on the number of positive solutions for ap-q-Laplacian equation. Under suitable assumptions forf(x)andg(x), we should prove that for sufficiently smallλ>0, there exist at leastkpositive solutions of the followingp-q-Laplacian equation,-Δpu-Δqu=fxu|p*-2u+λgxu|r-2u  in  Ω,u=0  on  ∂Ω,whereΩ⊂RNis a bounded smooth domain,N>p,1<q<N(p-1)/(N-1)<p≤max⁡{p,p^*-q/(p-1)}<r<p^*,p^*=Np/(N-p)is the critical Sobolev exponent, andΔsu=div(|∇u|s-2∇uis thes-Laplacian ofu.

scholarly journals Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

2021 ◽  
Vol 11 (1) ◽  
pp. 432-453
Author(s):  
Qi Han
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Fractional Laplacian ◽  
Linear Growth ◽  
Critical Sobolev Exponent ◽  
Compact Embedding ◽  
Multiple Positive Solutions ◽  
Measurable Functions ◽  
Sobolev Exponent ◽  
Laplacian Equations

Abstract In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (−Δ) s in ℝ n , for n ≥ 2, such as (0.1) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = K ( x ) f ( u ) + u 2 s ⋆ − 1 . $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$ Here, s ∈ (0, 1), q ∈ 2 , 2 s ⋆ $q \in\left[2,2_{s}^{\star}\right)$ with 2 s ⋆ := 2 n n − 2 s $2_{s}^{\star}:=\frac{2 n}{n-2 s}$ being the fractional critical Sobolev exponent, E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, and f (u) is a continuous function on ℝ of quasi-critical, super-q-linear growth with f (u) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝ n such as (0.2) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = λ K ( x ) u r − 1 , $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$ where 2 < r < q < ∞(both possibly (super-)critical), E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces Ms ;q,p (ℝ n ) as well as their associated compact embedding results.

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