168 research outputs found

    Equations involving fractional Laplacian operator: Compactness and application

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    In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \, \partial\Omega, \end{equation} where Ω\Omega is a smooth bounded domain in RN\mathbb{R}^N, ε∈[0,2α∗−2)\varepsilon\in [0, 2^*_\alpha-2), 0<α<1, 2α∗=2NN−2α0<\alpha<1,\, 2^*_\alpha = \frac {2N}{N-2\alpha}. We show that for any sequence of solutions unu_n of \eqref{eq:0.1} corresponding to εn∈[0,2α∗−2)\varepsilon_n\in [0, 2^*_\alpha-2), satisfying ∥un∥H≤C\|u_n\|_{H}\le C in the Sobolev space HH defined in \eqref{eq:1.1a}, unu_n converges strongly in HH provided that N>6αN>6\alpha and λ>0\lambda>0. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions.Comment: 34 page

    Multiple nodal solutions of nonlinear Choquard equations

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    In this paper, we consider the existence of multiple nodal solutions of the nonlinear Choquard equation \begin{equation*} \ \ \ \ (P)\ \ \ \ \begin{cases} -\Delta u+u=(|x|^{-1}\ast|u|^p)|u|^{p-2}u \ \ \ \text{in}\ \mathbb{R}^3, \ \ \ \ \\ u\in H^1(\mathbb{R}^3),\\ \end{cases} \end{equation*} where p∈(52,5)p\in (\frac{5}{2},5). We show that for any positive integer kk, problem (P)(P) has at least a radially symmetrical solution changing sign exactly kk-times

    Multiple solutions of semilinear elliptic systems

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    summary:We obtain in this paper a multiplicity result for strongly indefinite semilinear elliptic systems in bounded domains as well as in RN\Bbb R^N

    Asymptotics of ground states for fractional H\'enon systems

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    We investigate the asymptotic behavior of positive ground states for H\'enon type systems involving a fractional Laplacian on a bounded domain, when the powers of the nonlinearity approach the Sobolev critical exponent.Comment: 18 page
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