The authors prove existence of a nontrivial weak solution to a double critical fractional Laplacian equation with a Hardy term, using a new improved Sobolev inequality in weighted Morrey spaces.
Nonlocal Pertubations of Fractional Choquard Equation
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abstract
We study the equation \begin{equation} (-\Delta)^{s}u+V(x)u= (I_{\alpha}*|u|^{p})|u|^{p-2}u+\lambda(I_{\beta}*|u|^{q})|u|^{q-2}u \quad\mbox{ in } \R^{N}, \end{equation} where $I_\gamma(x)=|x|^{-\gamma}$ for any $\gamma\in (0,N)$, $p, q >0$, $\alpha,\beta\in (0,N)$, $N\geq 3$ and $ \lambda \in R$. First, the existence of a groundstate solutions using minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.
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2019 1verdicts
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The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${\R}^n$ with a Hardy term
The authors prove existence of a nontrivial weak solution to a double critical fractional Laplacian equation with a Hardy term, using a new improved Sobolev inequality in weighted Morrey spaces.