scholarly journals Multiple positive solutions for a semilinear elliptic equation with critical Sobolev exponent

2009 ◽  
Vol 354 (2) ◽  
pp. 451-458 ◽  
Author(s):  
Haidong Liu
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Critical Sobolev Exponent ◽  
Semilinear Elliptic Equation ◽  
Multiple Positive Solutions ◽  
Semilinear Elliptic ◽  
Sobolev Exponent

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.

scholarly journals Multiple Positive Solutions for Semilinear Elliptic Equations in Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Weight Functions ◽  
Multiple Positive Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions

We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation in , , where , if , if ), , satisfy suitable conditions, and may change sign in .

Multiple positive solutions for an unbounded Dirichlet boundary problem involving sign-changing weight

2009 ◽  
Vol 139 (6) ◽  
pp. 1297-1325
Author(s):  
Tsung-fang Wu
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Infinite Strip ◽  
Multiple Positive Solutions ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Dirichlet Boundary Problem

We study the multiplicity of positive solutions for the following semilinear elliptic equation:where 1 < q < 2 < p < 2* (2* = 2N/(N − 2) if N ≥ 3, 2* = ∞ if N = 2), the parameters λ, μ ≥ 0, is an infinite strip in ℝN and Θ is a bounded domain in ℝN−1 We assume that fλ(x) = λf+(x) + f−(x) and gμ(x) = a(x) + μb(x), where the functions f±, a and b satisfy suitable conditions.

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