scholarly journals Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity

2003 ◽  
Vol 279 (2) ◽  
pp. 508-521 ◽  
Author(s):  
C.O. Alves ◽  
Y.H. Ding
Keyword(s):  
Positive Solutions ◽  
Critical Nonlinearity ◽  
Laplacian Equation ◽  
Multiplicity Of Positive Solutions

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 On Dirichlet problem for fractional p-Laplacian with singular non-linearity

2016 ◽  
Vol 8 (1) ◽  
pp. 52-72 ◽  
Author(s):  
Tuhina Mukherjee ◽  
Konijeti Sreenadh
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Smooth Boundary ◽  
Critical Growth ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

Abstract In this article, we study the following fractional p-Laplacian equation with critical growth and singular non-linearity: (-\Delta_{p})^{s}u=\lambda u^{-q}+u^{\alpha},\quad u>0\quad\text{in }\Omega,% \qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in {\mathbb{R}^{n}} with smooth boundary {\partial\Omega} , {n>sp} , {s\in(0,1)} , {\lambda>0} , {0<q\leq 1} and {1<p<\alpha+1\leq p^{*}_{s}} . We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ.

scholarly journals Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity

2020 ◽  
Vol 9 (1) ◽  
pp. 1420-1436
Author(s):  
Xiangdong Fang ◽  
Jianjun Zhang
Keyword(s):  
Positive Solutions ◽  
Category Theory ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Nehari Manifold ◽  
Quasilinear Elliptic Equations ◽  
Critical Nonlinearity ◽  
Multiplicity Of Positive Solutions ◽  
The Nehari Manifold ◽  
Quasilinear Elliptic

Abstract We are concerned with the following quasilinear elliptic equation $$\begin{array}{} \displaystyle -{\it\Delta} u-{\it\Delta}(u^{2})u=\mu |u|^{q-2}u+|u|^{2\cdot 2^*-2}u, u\in H_0^1({\it\Omega}), \end{array}$$(QSE) where Ω ⊂ ℝN is a bounded domain, N ≥ 3, qN < q < 2 ⋅ 2∗, 2∗ = 2N/(N – 2), qN = 4 for N ≥ 6 and qN = $\begin{array}{} \frac{2(N+2)}{N-2} \end{array}$ for N = 3, 4, 5, and μ is a positive constant. By employing the Nehari manifold and the Lusternik-Schnirelman category theory, we prove that there exists μ* > 0 such that (QSE) admits at least catΩ(Ω) positive solutions when μ ∈ (0, μ*).

scholarly journals Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 107
Author(s):  
Daliang Zhao ◽  
Juan Mao
Keyword(s):  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Integral Boundary ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

Sign in / Sign up

Export Citation Format

Share Document