1,105 research outputs found

    Concentrating solutions for a fractional Kirchhoff equation with critical growth

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    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 ε>0\varepsilon>0 is a small parameter, a,b>0a, b>0 are constants, s∈(34,1)s\in (\frac{3}{4}, 1), 2s∗=63−2s2^{*}_{s}=\frac{6}{3-2s} is the fractional critical exponent, (−Δ)s(-\Delta)^{s} is the fractional Laplacian operator, VV is a positive continuous potential and ff is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions uεu_{\varepsilon} which concentrates around a local minimum of VV as ε→0\varepsilon\rightarrow 0.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

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    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 ε\varepsilon is a small positive parameter, a,b>0a, b>0 are fixed constants, s∈(34,1)s\in (\frac{3}{4}, 1), 2s∗=63−2s2^{*}_{s}=\frac{6}{3-2s} is the fractional critical exponent, (−Δ)As(-\Delta)^{s}_{A} is the fractional magnetic Laplacian, A:R3→R3A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3} is a smooth magnetic potential, V:R3→RV:\mathbb{R}^{3}\rightarrow \mathbb{R} is a positive continuous potential verifying the global condition due to Rabinowitz \cite{Rab}, and f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} is a C1C^{1} 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

    A multiplicity result for a fractional Kirchhoff equation in RN\mathbb{R}^{N} with a general nonlinearity

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    In this paper we deal with the following fractional Kirchhoff equation \begin{equation*} \left(p+q(1-s) \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^{2}}{|x-y|^{N+2s}} \, dx\,dy \right)(-\Delta)^{s}u = g(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where s∈(0,1)s\in (0,1), N≥2N\geq 2, p>0p>0, qq is a small positive parameter and g:R→Rg: \mathbb{R}\rightarrow \mathbb{R} is an odd function satisfying Berestycki-Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that qq is sufficiently small

    Infinitely many periodic solutions for a class of fractional Kirchhoff problems

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    We prove the existence of infinitely many nontrivial weak periodic solutions for a class of fractional Kirchhoff problems driven by a relativistic Schr\"odinger operator with periodic boundary conditions and involving different types of nonlinearities

    On the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

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    We consider the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity ε2sM([u]s,Aε2)(−Δ)Aεsu+V(x)u=\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-\Delta)_{A_\varepsilon}^su + V(x)u = ∣u∣2s∗−2u+h(x,∣u∣2)u,|u|^{2_s^\ast-2}u + h(x,|u|^2)u,   x∈RN,\ \ x\in \mathbb{R}^N, where u(x)→0 u(x) \rightarrow 0 as ∣x∣→∞,|x| \rightarrow \infty, and (−Δ)Aεs(-\Delta)_{A_\varepsilon}^s is the fractional magnetic operator with 0<s<10<s<1, 2s∗=2N/(N−2s),2_s^\ast = 2N/(N-2s), M:R0+→R+M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+} is a continuous nondecreasing function, V:RN→R0+,V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0, and A:RN→RNA: \mathbb{R}^N \rightarrow \mathbb{R}^N are the electric and the magnetic potential, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that ε<E\varepsilon < \mathcal {E}; and (ii) for any m∗∈Nm^\ast \in \mathbb{N}, has m∗m^\ast pairs of solutions if ε<Em∗\varepsilon < \mathcal {E}_{m^\ast}, where E\mathcal {E} and Em∗\mathcal {E}_{m^\ast} are sufficiently small positive numbers. Moreover, these solutions uε→0u_\varepsilon \rightarrow 0 as ε→0\varepsilon \rightarrow 0
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