Compact embedding results of Sobolev spaces and positive solutions to an elliptic equation

2016 ◽  
Vol 146 (4) ◽  
pp. 693-721 ◽  
Author(s):  
Qi Han
Keyword(s):  
Elliptic Equation ◽  
Sobolev Space ◽  
Positive Solutions ◽  
Borel Measure ◽  
Nonlinear Term ◽  
Compact Embedding ◽  
Proper Subspace ◽  
Superlinear Growth ◽  
Regular Borel Measure ◽  
The Laplace Operator

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.

Positive solutions for parametric semilinear Robin problems with indefinite and unbounded potential

2017 ◽  
Vol 121 (2) ◽  
pp. 263 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Positive Solution ◽  
Laplace Operator ◽  
Positive Solutions ◽  
Type Theorem ◽  
Robin Problem ◽  
Continuity Properties ◽  
Superlinear Growth ◽  
Unbounded Potential ◽  
Carathéodory Function ◽  
The Laplace Operator

We consider a parametric Robin problem driven by the Laplace operator plus an indefinite and unbounded potential. The reaction term is a Carathéodory function which exhibits superlinear growth near $+\infty $ without satisfying the Ambrosetti-Rabinowitz condition. We are looking for positive solutions and prove a bifurcation-type theorem describing the dependence of the set of positive solutions on the parameter. We also establish the existence of the minimal positive solution $u^*_{\lambda }$ and investigate the monotonicity and continuity properties of the map $\lambda \mapsto u^*_{\lambda }$.

Liouville type theorem for nonlinear elliptic equation involving Grushin operators

2015 ◽  
Vol 17 (05) ◽  
pp. 1450050 ◽  
Author(s):  
Xiaohui Yu
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Type Theorem ◽  
Integral Form ◽  
Liouville Type ◽  
Nonlinear Elliptic ◽  
Main Technique ◽  
Grushin Operators ◽  
Moving Plane

In this paper, we study the nonexistence of positive solutions for the following elliptic equation [Formula: see text] where Lαu = Δxu + (α + 1)2|x|2αΔyu, α > 0, (x, y) ∈ ℝm × ℝk. We will prove that this problem possesses no positive solutions under some assumptions on the nonlinear term f. The main technique we use is the moving plane method in an integral form.

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 Positive solutions of a discrete second-order boundary value problems with fully nonlinear term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Liyun Jin ◽  
Hua Luo
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Index Theory ◽  
Second Order ◽  
Fully Nonlinear ◽  
Fixed Point Index Theory ◽  
Weaker Conditions ◽  
Lower Limits

Abstract In this paper, we mainly consider a kind of discrete second-order boundary value problem with fully nonlinear term. By using the fixed-point index theory, we obtain some existence results of positive solutions of this kind of problems. Instead of the upper and lower limits condition on f, we may only impose some weaker conditions on f.

Positive Solutions for the Generalized Nonlinear Logistic Equations

2016 ◽  
Vol 59 (01) ◽  
pp. 73-86 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Elliptic Equation ◽  
Positive Solutions ◽  
Type Result ◽  
Logistic Equations ◽  
Nonhomogeneous Differential Operator

AbstractWe consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiòusive type. Using variationalmethods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies.

scholarly journals On the existence and uniqueness of solution to Volterra equation on a time scale

2019 ◽  
Vol 27 (3) ◽  
pp. 177-194
Author(s):  
Bartłomiej Kluczyński
Keyword(s):  
Integral Equation ◽  
Sobolev Space ◽  
Time Scale ◽  
Existence And Uniqueness ◽  
Nonlinear Term ◽  
Volterra Equation ◽  
Uniqueness Of Solution ◽  
Inversion Theorem ◽  
T Tau

AbstractUsing a global inversion theorem we investigate properties of the following operator\matrix{\matrix{ V(x)( \cdot ): = {x^\Delta }( \cdot ) + \int_0^ \cdot {v\left( { \cdot ,\tau ,x,\left( \tau \right)} \right)} \Delta \tau , \hfill \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x(0) = 0, \hfill \cr}\cr {} \cr }in a time scale setting. Under some assumptions on the nonlinear term v we then show that there exists exactly one solution {x_y} \in W_{\Delta ,0}^{1,p}\left( {{{[0,1]}_\mathbb{T}},{\mathbb{R}^N}} \right) to the associated integral equation\left\{ {\matrix{{{x^\Delta }(t) + \int_0^t {v\left( {t,\tau ,x\left( \tau \right)} \right)} \Delta \tau = y(t)\,\,\,for\,\Delta - a.e.\,\,\,t \in {{[0.1]}_\mathbb{T}},} \cr {x(0) = 0,} \cr } } \right.which is considered on a suitable Sobolev space.

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