Critical Sobolev Hardy Exponents Research Articles

Multiple positive solutions for elliptic equations involving a concave term and critical Sobolev–Hardy exponent

In this paper, we establish the existence of multiple positive solutions for elliptic equations involving a concave term and critical Sobolev–Hardy exponent.

On Elliptic System Involving Critical Sobolev–Hardy Exponents

This paper deals with a class of nonlinear elliptic system involving Sobolev-Hardy exponents in bounded domain of \({\mathbb{R}}^{N}\). By using variational method, we show that the existence of solutions depend on certain parameters.

Positive solutions for Neumann elliptic problems involving critical Hardy–Sobolev exponent with boundary singularities

In this paper we consider the existence of positive solution for some semilinear elliptic equations with Neumann boundary condition involving a critical Hardy–Sobolev exponent and Hardy terms with boundary singularities. Using mountain pass lemma without (PS) condition and the strong maximum principle, we get the existence of a positive solution.

Multiplicity of solutions for singular semilinear elliptic equations with critical Hardy-Sobolev exponents

We consider the semilinear elliptic problem with critical Hardy-Sobolev exponents and Dirichlet boundary condition. By using variational methods we obtain the existence and multiplicity of nontrivial solutions and improve the former results.

Solutions of the quasilinear elliptic problem with a critical Sobolev–Hardy exponent and a Hardy-type term

In the present paper, the quasilinear elliptic problem with a critical Sobolev–Hardy exponent and a Hardy-type term is considered. By means of a variational method, the existence of nontrivial solutions to the problem is obtained.

On the quasilinear elliptic problem with a critical Hardy–Sobolev exponent and a Hardy term

In the present paper, a quasilinear elliptic problem with a critical Sobolev exponent and a Hardy-type term is considered. By means of a variational method, the existence of nontrivial solutions for the problem is obtained. The result depends crucially on the parameters p , t , s , λ and μ .

Existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving Hardy term and Hardy–Sobolev critical exponents

Existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with Hardy term, Hardy–Sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques.

Existence results for singular critical elliptic problems in [formula omitted

In the paper we consider the following semilinear elliptic problems with critical Sobolev–Hardy exponents: ( P μ , s ) { − Δ u − μ | x | 2 u + λ V ( x ) u = K ( x ) | u | 2 ∗ ( s ) − 2 | x | s u , in R N , lim | x | → ∞ u ( x ) = 0 , where N ⩾ 3 , 0 ⩽ μ < μ ¯ : = ( N − 2 2 ) 2 , λ > 0 , 0 ⩽ s < 2 , 2 ∗ ( s ) = 2 ( N − s ) N − 2 . Under suitable conditions, we prove that ( P μ , s ) has at least one nontrivial solution by means of Linking theorem. Also via the Pseudo-index theory, we consider the multiplicity of nontrivial solutions for ( P μ , s ) .

Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy–Sobolev critical exponents

Some existence and multiplicity results are obtained for solutions of semilinear elliptic equations with Hardy terms, Hardy–Sobolev critical exponents and superlinear nonlinearity by the variational methods and some analysis techniques.

On the quasilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy terms

Let Ω ⊂ R N be a smooth bounded domain such that 0 ∈ Ω , N ≥ 3 . In this paper, we study the critical quasilinear elliptic problems − Δ p u − μ | u | p − 2 u | x | p = | u | p ∗ ( t ) − 2 | x | t u + λ | u | q − 2 | x | s u , u ∈ W 0 1 , p ( Ω ) with Dirichlet boundary condition, where − Δ p u = − div ( | ∇ u | p − 2 ∇ u ) , 1 < p < N , 0 ≤ μ < μ ̄ ≔ ( N − p p ) p , λ > 0 , 0 ≤ s , t < p , p ≤ q < p ∗ ( s ) ≔ p ( N − s ) N − p , p ∗ ( t ) ≔ p ( N − t ) N − p , p ∗ ( s ) and p ∗ ( t ) are the critical Sobolev–Hardy exponents. Via variational methods, we deal with the conditions that ensure the existence of positive solutions for the equation. The results depend crucially on the parameters p , q , s , λ and μ .

RETRACTED: Multiple nodal solutions for quasilinear elliptic problems involving Hardy–Sobolev critical exponents in symmetric domain

This article has been retracted at the request of the Editor-in-Chief and author. Please see Elsevier Policy on Article Withdrawal ( http://www.elsevier.com/locate/withdrawalpolicy ). Reason: In the proof of Lemma 2.8 in this paper, the authors mistakenly claimed to have obtained u λ in M τ λ . However, u λ = v λ (·− y ) does not belong to u λ in M τ λ , making the statement u λ in M τ λ false. The same error occurs in the proof of Lemma 4.6. The failure of both Lemmas 2.8 and 4.6 makes Theorem 1.1 and 1.2 invalid. The authors sincerely apologize for the errors.

Singular elliptic problems in [formula omitted] with critical Sobolev–Hardy exponents

Let N ≥ 3 , λ > 0 , β ≤ 0 , N + β − 2 > 0 , N + α > 0 , N + σ > 0 , α + 2 > β , σ + 2 > β , β 2 ≥ α p ( β , α ) , β 2 ≥ σ p ( β , σ ) , 1 < q < min ( p ( β , α ) , p ( β , σ ) ) , p ( s , t ) ≔ 2 ( N + t ) N + s − 2 be the critical Sobolev–Hardy exponent. Via the variational methods, we prove the existence of a nontrivial solution to the singular semilinear problem − div ( | x | β ∇ u ) = | x | α | u | p ( β , α ) − 2 u + λ a ( x ) | u | q − 2 u , u ≥ 0 in R N for suitable parameters N , λ , q and some kinds of functions a ( x ) .

On symmetric solutions of a singular elliptic equation with critical Sobolev–Hardy exponent

This paper is concerned with the existence of the nontrivial solutions of the following problem: { − Δ u = μ u | x | 2 + K ( x ) u 2 ∗ ( s ) − 1 | x | s , x ∈ R n , u ∈ D G 1 , 2 ( R n ) , where n > 2 , K ( x ) is a bounded, continuous function satisfying some conditions. D G 1 , 2 ( R n ) is an appropriate Sobolev space of G-symmetric functions. 2 ∗ ( s ) = 2 ( n − s ) ( n − 2 ) is the critical Sobolev–Hardy exponent, and 0 ⩽ s < 2 , 0 < μ < μ ¯ = ( n − 2 2 ) 2 .

SOLUTIONS FOR A NONHOMOGENEOUS ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT IN RN

For the following elliptic problem { - Δ u - μ u | x | 2 = | u | 2 * ( s ) - 2 u | x | s + h ( x ) , on R N u ∈ D 1 , 2 ( R N ) , N ≥ 3 , 0 ≤ μ < μ ¯ = ( N - 2 ) 2 4 , 0 ≤ s < 2 , where 2 * ( s ) = 2 ( N - s ) N - 2 is the critical Sobolev-Hardy exponent, h ( x ) ∈ ( D 1 , 2 ( R N ) * , the dual space of ( D 1 , 2 ( R N ) ) , with h( x)≥(≢)0. By Ekeland's variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ∥ h ∥ * < C N , s A s N - s 4 - 2 s ( 1 - μ μ ) 1 2 , C N , s = 4 - 2 s N - 2 ( N - 2 N + 2 - 2 s ) N + 2 - 2 s 4 - 2 s and A s = inf u ∈ D 1 , 2 ( R N ) \\ { 0 } ∫ R N ( | ∇ u | 2 - μ u 2 | x | 2 ) d x ( ∫ R N | u | 2 * ( s ) | x | s d x ) 2 2 * ( s ) .

On the elliptic problems involving multi-singular inverse square potentials and multi-critical Sobolev–Hardy exponents

Let Ω ⊂ R N ( N ≥ 3 ) be a smooth bounded domain such that the different points a i ∈ Ω , i = 1 , 2 , … , k , k ≥ 2 , 0 ≤ μ i < μ ̄ = ( N − 2 2 ) 2 , 0 < s i < 2 , p i = 2 ( N − s i ) N − 2 , are the critical Sobolev–Hardy exponents. We deal with the conditions that ensure the existence of positive solutions for the multi-singular and multi-critical elliptic problem − Δ u − ∑ i = 1 k μ i | x − a i | 2 u = ∑ i = 1 k u p i − 1 | x − a i | s i with the Dirichlet boundary condition, involving the Hardy inequality and the Sobolev–Hardy inequality. The results depend crucially on the parameters s i and μ i , i = 1 , 2 , … , k .

Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy–Sobolev exponent

In this work we improve some known results for a singular operator and also for a wide class of lower-order terms by proving a multiplicity result. The proof is made by applying the generalized mountain-pass theorem due to Ambrosetti and Rabinowitz. To do this, we show that the minimax levels are in a convenient range by combining a special class of approximating functions, due to Gazzola and Ruf, with the concentrating functions of the best Sobolev constant.

Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents in [formula omitted

Let N ≥ 3 , 0 ≤ s < 2 , 0 ≤ μ < ( N − 2 2 ) 2 and 2 ∗ ( s ) ≔ 2 ( N − s ) N − 2 be the critical Sobolev–Hardy exponents. Via variational methods and the analytic technique, we prove the existence of a nontrivial solution to the singular semilinear problem − Δ u − μ u | x | 2 + u = | u | 2 ∗ ( s ) − 2 | x | s u + f ( u ) , u ∈ H r 1 ( R N ) , for N ≥ 4 , 0 ≤ μ ≤ μ ̄ − 1 and suitable functions f ( u ) .

Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy potential

Let Ω ⊂ R N be a smooth bounded domain such that 0 ∈ Ω , N ≥ 5 , 0 ≤ s < 2 , 2 ∗ ( s ) = 2 ( N − s ) N − 2 . We prove the existence of nontrivial solutions for the singular critical problem − Δ u − μ u | x | 2 = | u | 2 ∗ ( s ) − 2 | x | s u + λ u with Dirichlet boundary condition on Ω for all λ > 0 and 0 ≤ μ < ( N − 2 2 ) 2 − ( N + 2 N ) 2 .

Existence of solutions for elliptic problems with critical Sobolev-Hardy exponents

Let Ω ⊂ ℝ N be a smooth bounded domain such that 0 ∈ Ω,N≥3, 0≤s<2,2* (s)=2(N−s)/(N−2). We prove the existence of nontrival solutions for the singular critical problem $$ - \Delta u - \mu \frac{u}{{\left| x \right|^2 }} = \frac{{\left| u \right|^{2^* \left( s \right) - 2} }}{{\left| x \right|^s }}u + \lambda u$$ with Dirichlet boundary condition on Ω for suitable positive parameters λ and μ.

Multiple solutions for inhomogeneous elliptic problems involving critical Sobolev–Hardy exponents

Let Ω ⊂ R N be a smooth bounded domain such that 0 ∈ Ω , N ⩾ 3 , 0 ⩽ s < 2 , 2 * ( s ) := 2 ( N - s ) / N - 2 is the critical Sobolev–Hardy exponent, f ( x ) is a given function. By using the Ekeland's variational principle and the mountain pass lemma, we prove the existence of multiple solutions for the singular critical inhomogeneous problem - Δ u - μ u | x | 2 = | u | 2 * ( s ) - 2 | x | s u + λ u + f ( x ) with Dirichlet boundary condition on ∂ Ω under some assumptions on f ( x ) , λ and μ .