Nontrivial solutions for a class of critical elliptic equations of Caffarelli–Kohn–Nirenberg type

Liu, Haidong; Zhao, Leiga

doi:10.1016/j.jmaa.2013.03.016

Cited by 4 publications

(1 citation statement)

References 32 publications

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“…The mathematical interest lies in the fact that these problems like (2) are doubly critical due to the presence of the Sobolev embedding and the singularities. For this reason, many existence, nonexistence, and multiplicity results of nontrivial solutions for the single equations like (2) have been established with different assumptions on the potentials ( ), ℎ( ) and the parameters , , and ; we refer to [2][3][4][5][6][7][8] and the references therein. In a recent paper, Deng and Jin [9] considered the following single semilinear elliptic problem:…”

Section: Introductionmentioning

confidence: 99%

Multiple Symmetric Results for Quasilinear Elliptic Systems Involving Singular Potentials and Critical Sobolev Exponents inRN

Deng

Huang

2014

Abstract and Applied Analysis

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Section: Introductionmentioning

confidence: 99%

Multiple Symmetric Results for Quasilinear Elliptic Systems Involving Singular Potentials and Critical Sobolev Exponents inRN

Deng

Huang

2014

Abstract and Applied Analysis

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Several logarithmic Caffarelli–Kohn–Nirenberg inequalities and applications

Feng

Niu

Qiao

2018

Journal of Mathematical Analysis and Applications

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Solutions for a Kirchhoff Equation with Weight and Nonlinearity with Subcritical and Critical Caffarelli–Kohn–Nirenberg Growth

Figueiredo¹,

Guimarães²,

Rodrigues³

2016

Proceedings of the Edinburgh Mathematical Society

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In this paper we study the multiplicity of non-trivial solutions to a class of nonlinear boundaryvalue problems of Kirchhoff type. We prove existence results when the problem has nonlinearities with subcritical and with critical Caffarelli-Kohn-Nirenberg exponent.Problem (1.1) with a = b = δ = β = 0 and p = 2, that is,is called non-local because of the presence of the term M ( Ω |∇u| 2 dx), which implies that (1.2) is no longer a pointwise identity. This phenomenon causes some mathematical difficulties, which makes the study of such a class of problem particularly interesting. Besides this, this class of problem has a physical motivation. Indeed, the operator M ( Ω |∇u| 2 dx)Δu appears in the Kirchhoff equation, which arises in nonlinear vibrations, namely,Proof . Fix k ∈ N and let X k be a k-dimensional subspace of D 1,p a . Thus, there exists C(k) > 0 such that C(k) u r C 1 u r L r (Ω,|x| −δ ) for all u ∈ X k . Consideringρ > 0 such that 0 < u =ρ and 0 < u p < t 1 , we obtain that J λ (u) = I λ (u). Arguing as in the proof of Theorem 1.1, we can take R > 0 such thatfor all u ∈ S = {u ∈ X k : u = s}, with 0 < s < min{R,ρ}. Hence, S ⊂ J −ε and, since J −ε is symmetric and closed, from Corollary 2.3,We define now, for each k ∈ N, the sets Γ k = {C ⊂ D 1,p a \{0} : C is closed, C = −C and γ(C) k}, K c = {u ∈ D 1,p a : J λ (u) = 0 and J λ (u) = c} and the number c k = inf C∈Γ k sup u∈C J λ (u).Lemma 4.6. Given k ∈ N, the number c k is negative.Proof . From Lemma 4.5, for each k ∈ N there exists ε > 0 such that γ(J −ε ) k. Moreover, 0 / ∈ J −ε and J −ε ∈ Γ k . On the other hand,The next lemma allows us to prove the existence of critical points of J.Lemma 4.7. If c = c k = c k+1 = · · · = c k+r for some r ∈ N, then γ(K c ) r + 1 for all λ ∈ (0, λ * ).

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Nontrivial solutions for a class of critical elliptic equations of Caffarelli–Kohn–Nirenberg type

Cited by 4 publications

References 32 publications

Multiple Symmetric Results for Quasilinear Elliptic Systems Involving Singular Potentials and Critical Sobolev Exponents inRN

Multiple Symmetric Results for Quasilinear Elliptic Systems Involving Singular Potentials and Critical Sobolev Exponents inRN

Several logarithmic Caffarelli–Kohn–Nirenberg inequalities and applications

Solutions for a Kirchhoff Equation with Weight and Nonlinearity with Subcritical and Critical Caffarelli–Kohn–Nirenberg Growth

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