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

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.

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 Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

Existence and multiplicity of solutions for a class of quasilinear elliptic field equation on ℝN

2018 ◽  
Vol 24 (3) ◽  
pp. 1231-1248
Author(s):  
Claudianor O. Alves ◽  
Alan C.B. dos Santos
Keyword(s):  
Field Equation ◽  
Continuous Function ◽  
Singular Function ◽  
Positive Continuous Function ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Quasilinear Elliptic

In this paper, we establish existence and multiplicity of solutions for the following class of quasilinear field equation    −Δu + V(x)u − Δpu + W′(u) = 0,  in  ℝN,    (P) where u = (u1, u2, … , uN+1), p > N ≥ 2, W is a singular function and V is a positive continuous function.

scholarly journals Solutions of Nonlocal -Laplacian Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Mustafa Avci ◽  
Rabil Ayazoglu (Mashiyev)
Keyword(s):  
Laplace Operator ◽  
Variational Approach ◽  
Nonlocal Problem ◽  
Type Equation ◽  
Multiplicity Of Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Equation ◽  
Laplacian Equations

In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation involving -Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.

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