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Sun, Jijiang; Li, Lin; Cencelj, Matija; Gabrovšek, Boštjan

doi:10.1016/j.na.2018.10.007

Cited by 46 publications

(13 citation statements)

References 34 publications

(70 reference statements)

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“…Precisely, they determine the existence of a first solution by using the method of sub-and super-solutions and then prove that this solution is the minimum of a suitable functional and apply the mountain pass theorem so ensuring the existence of a second solution. For other result of fourth-order problem and variational problem, we refer the reader to [1,5,8,[10][11][12][13][14][15][16] and references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

O’Regan

2021

NAMC

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

O’Regan

2021

NAMC

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“…Li and Ye [22] proved that problem (SK ε ) has a positive ground state solution for 3 < p < 6 when V satisfies some suitable conditions. For more related results, we refer the readers to [28,32] for the bounded domain, [13,35] for ground state solutions, [10,42] for nodal solutions, [33,36] for sign-changing solutions, [2,41] for steep well potential cases and [1,27,43] for periodic potential cases.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Semi-classical solutions for Kirchhoff type problem with a critical frequency

Xie¹,

Zhang²

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In the present paper, we consider the following Kirchhoff type problem $$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$ where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$ . As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

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“…Precisely, they proved the existence of a sign-changing solution, which changes signs exactly k times for any k ∈ N. Moreover, they investigated the energy property and the asymptotic behavior of the sign-changing solution. By using a combination of the invariant set method and the Ljusternik-Schnirelman type minimax method, Sun et al [39] obtained infinitely many sign-changing solutions for Kirchhoff problem (1.6) when f (x, u) = f (u) and f is odd in u. It is worth noticing that, in [39], the nonlinear term may not be 4-superlinear at infinity; in particular, it includes the power-type nonlinearity |u| p−2 u with p ∈ (2,4].…”

Section: Introductionmentioning

confidence: 99%

Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity

Liang

Rădulescu

2020

Anal.Math.Phys.

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In this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^p\right) (-\Delta )^s_pu = \lambda |u|^{q-2}u\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\quad \text {in } \Omega , \\ u=0 &{}\quad \text {in } {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$ a + b [ u ] s , p p ( - Δ ) p s u = λ | u | q - 2 u ln | u | 2 + | u | p s ∗ - 2 u in Ω , u = 0 in R N \ Ω , where $$N >sp$$ N > s p with $$s \in (0, 1)$$ s ∈ ( 0 , 1 ) , $$p>1$$ p > 1 , and $$\begin{aligned}{}[u]_{s,p}^p =\iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \end{aligned}$$ [ u ] s , p p = ∬ R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y , $$p_s^*=Np/(N-ps)$$ p s ∗ = N p / ( N - p s ) is the fractional critical Sobolev exponent, $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N $$(N\ge 3)$$ ( N ≥ 3 ) is a bounded domain with Lipschitz boundary and $$\lambda $$ λ is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution $$u_b$$ u b . Moreover, for any $$\lambda > 0$$ λ > 0 , we show that the energy of $$u_b$$ u b is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as $$b \rightarrow 0$$ b → 0 .

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Infinitely many sign-changing solutions for Kirchhoff type problems in R3

Cited by 46 publications

References 34 publications

Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

Semi-classical solutions for Kirchhoff type problem with a critical frequency

Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity

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