40 research outputs found
Concentrating solutions for a fractional Kirchhoff equation with critical growth
In this paper we consider the following class of fractional Kirchhoff
equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll}
\left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u
\quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0
&\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where
is a small parameter, are constants, , is the fractional critical
exponent, is the fractional Laplacian operator, is a
positive continuous potential and is a superlinear continuous function with
subcritical growth. Using penalization techniques and variational methods, we
prove the existence of a family of positive solutions which
concentrates around a local minimum of as .Comment: arXiv admin note: text overlap with arXiv:1810.0456
Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth
We investigate the existence, multiplicity and concentration of nontrivial
solutions for the following fractional magnetic Kirchhoff equation with
critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3}
[u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u
\quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where is a
small positive parameter, are fixed constants, , is the fractional critical exponent,
is the fractional magnetic Laplacian,
is a smooth magnetic potential,
is a positive continuous potential
verifying the global condition due to Rabinowitz \cite{Rab}, and
is a subcritical nonlinearity. Due
to the presence of the magnetic field and the critical growth of the
nonlinearity, several difficulties arise in the study of our problem and a
careful analysis will be needed. The main results presented here are
established by using minimax methods, concentration compactness principle of
Lions \cite{Lions}, a fractional Kato's type inequality and the
Ljusternik-Schnirelmann theory of critical points.Comment: arXiv admin note: text overlap with arXiv:1808.0929
Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent
Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent
Existence of positive solutions to Kirchhoff type problems with zero mass
The existence of positive solutions depending on a nonnegative parameter lambda to Kirchhoff type problems with zero mass is proved by using variational method, and the new result does not require usual compactness conditions. A priori estimate and a Pohozaev type identity are used to obtain the bounded Palais-Smale sequences for constant coefficient nonlinearity, while a cut-off functional and Pohozaev type identity are utilized to obtain the bounded Palais-Smale sequences for the variable-coefficient case. (C) 2013 Elsevier Inc. All rights reserved
Existence of positive ground state solutions of critical nonlinear Klein-Gordon-Maxwell systems
In this paper we study the following nonlinear Klein–Gordon–Maxwell system −∆u + [m2 0 − (ω + φ) 2 ]u = f(u) in R3 ∆φ = (ω + φ)u in R3 where 0 < ω < m0. Based on an abstract critical point theorem established by Jeanjean, the existence of positive ground state solutions is proved, when the nonlinear term f(u) exhibits linear near zero and a general critical growth near infinity. Compared with other recent literature, some different arguments have been introduced and some results are extended
A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator
In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained. 
Existence and multiplicity of nontrivial solutions to the modified Kirchhoff equation without the growth and Ambrosetti-Rabinowitz conditions
The paper focuses on the modified Kirchhoff equation a + b Z RN |∇u| 2 dx� ∆u − u∆(u 2 ) + V(x)u = λ f(u), x ∈ R N, where a, b > 0, V(x) ∈ C(RN, R) and λ < 1 is a positive parameter. We just assume that the nonlinearity f(t) is continuous and superlinear in a neighborhood of t = 0 and at infinity. By applying the perturbation method and using the cutoff function, we get existence and multiplicity of nontrivial solutions to the revised equation. Then we use the Moser iteration to obtain existence and multiplicity of nontrivial solutions to the above original Kirchhoff equation. Moreover, the nonlinearity f(t) may be supercritical