Multiple positive solutions for a class of quasi-linear elliptic equations involving critical sobolev exponent

2014 ◽  
Vol 34 (4) ◽  
pp. 1111-1126 ◽  
Author(s):  
Haining FAN ◽  
Xiaochun LIU
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Critical Sobolev Exponent ◽  
Multiple Positive Solutions ◽  
Sobolev Exponent

Multiple positive solutions for an elliptic problem involving a critical Sobolev exponent

2021 ◽  
pp. 1-24
Author(s):  
Tiantian Zheng ◽  
Zhiyong Wang ◽  
Pei Ma ◽  
Jihui Zhang
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Critical Sobolev Exponent ◽  
Multiple Positive Solutions ◽  
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.

Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent

2014 ◽  
Vol 144 (4) ◽  
pp. 691-709 ◽  
Author(s):  
Ching-yu Chen ◽  
Tsung-fang Wu
Keyword(s):  
Positive Solutions ◽  
Critical Sobolev Exponent ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problems ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions ◽  
The Nehari Manifold ◽  
Indefinite Elliptic Problem

In this paper, we study the decomposition of the Nehari manifold by exploiting the combination of concave and convex nonlinearities. The result is subsequently used, in conjunction with the Ljusternik–Schnirelmann category and variational methods, to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical Sobolev exponent.

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