On the Existence of Positive Solutions of Slightly Supercritical Elliptic Equations

2003 ◽  
Vol 3 (3) ◽  
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Bounded Domains ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Blowing Up ◽  
Existence Of Positive Solutions

AbstractIn this paper we deal with existence and multiplicity of solutions for problem P(ε, Ω) below, in bounded domains Ω which do not satisfy some symmetry prop erties that play an important role in the proof of previous results on this subject. For ε > 0 small and k large enough, we construct solutions blowing up, as ε → 0, at exactly k points which approach the boundary of Ω as k tends to infinity.We also present some examples showing that solutions of this type also exist in some contractible domains which may be even very close to starshaped domains.

scholarly journals On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

2020 ◽  
Vol 9 (1) ◽  
pp. 1480-1503
Author(s):  
Mousomi Bhakta ◽  
Phuoc-Tai Nguyen
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Total Mass ◽  
Elliptic Systems ◽  
Linking Theorem ◽  
Bounded Domains ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Smallness Condition

Abstract We study positive solutions to the fractional Lane-Emden system $$\begin{array}{} \displaystyle \left\{ \begin{aligned} (-{\it\Delta})^s u &= v^p+\mu \quad &\text{in } {\it\Omega} \\(-{\it\Delta})^s v &= u^q+\nu \quad &\text{in } {\it\Omega}\\u = v &= 0 \quad &&\!\!\!\!\!\!\!\!\!\!\!\!\text{in } {\it\Omega}^c={\mathbb R}^N \setminus {\it\Omega}, \end{aligned} \right. \end{array}$$(S) where Ω is a C2 bounded domains in ℝN, s ∈ (0, 1), N > 2s, p > 0, q > 0 and μ, ν are positive measures in Ω. We prove the existence of the minimal positive solution of (S) under a smallness condition on the total mass of μ and ν. Furthermore, if p, q ∈ $\begin{array}{} (1,\frac{N+s}{N-s}) \end{array}$ and 0 ≤ μ, ν ∈ Lr(Ω), for some r > $\begin{array}{} \frac{N}{2s}, \end{array}$ we show the existence of at least two positive solutions of (S). The novelty lies at the construction of the second solution, which is based on a highly nontrivial adaptation of Linking theorem. We also discuss the regularity of the solutions.

scholarly journals Quasilinear Elliptic Equations with Hardy-Sobolev Critical Exponents: Existence and Multiplicity of Nontrivial Solutions

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Guanwei Chen
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Critical Exponents ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Existence Of Positive Solutions ◽  
Quasilinear Elliptic

We study the existence of positive solutions and multiplicity of nontrivial solutions for a class of quasilinear elliptic equations by using variational methods. Our obtained results extend some existing ones.

Fractional elliptic equations with critical exponential nonlinearity

2016 ◽  
Vol 5 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Jacques Giacomoni ◽  
Pawan Kumar Mishra ◽  
K. Sreenadh
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Exponential Nonlinearity ◽  
Existence Of Positive Solutions ◽  
Fractional Laplacian Operator

AbstractWe study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

scholarly journals Positive Solutions for Nonlinear Fractional Differential Equations with Boundary Conditions Involving Riemann-Stieltjes Integrals

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Jiqiang Jiang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Integral Boundary ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
Integral Boundary Value Problems

We consider the existence of positive solutions for a class of nonlinear integral boundary value problems for fractional differential equations. By using some fixed point theorems, the existence and multiplicity results of positive solutions are obtained. The results obtained in this paper improve and generalize some well-known results.

scholarly journals Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yong-Yi Lan ◽  
Xian Hu ◽  
Bi-Yun Tang
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Elliptic Problems ◽  
Semilinear Elliptic Equations ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

A Note on The Schrödinger-Poisson-Slater Equation on Bounded Domains

2008 ◽  
Vol 8 (1) ◽  
Author(s):  
David Ruiz ◽  
Gaetano Siciliano
Keyword(s):  
Bounded Domains ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Made In

AbstractIn this note we consider the problemwhere u, ϕ ∈ Hexistence and multiplicity of solutions depending on the parameters p and λ. Our results extend previous work made in ℝ

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