In this work, we study the existence, multiplicity and concentration behavior of positive solutions for the following problem involving the fractional $ p $-Laplacian
$ \begin{eqnarray*} \varepsilon^{ps}(-\Delta )^{s}_{p}u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}(\frac{1}{|x|^{\mu}}\ast K|u|^{q})K(x)|u|^{q-2}u \hskip0.2cm\text{in}\hskip0.1cm \mathbb{R}^{N}, \end{eqnarray*} $
where $ 0 < s < 1 < p < \infty $, $ N > ps $, $ 0 < \mu < ps $, $ p < q < \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N}) $, $ (-\Delta)^{s}_{p} $ is the fractional $ p $-Laplacian and $ \varepsilon > 0 $ is a small parameter. Under certain conditions on $ V $ and $ K $, we prove the existence of a positive ground state solution and express the location of concentration in terms of the potential functions $ V $ and $ K $. In particular, we relate the number of solutions with the topology of the set where $ V $ attains its global minimum and $ K $ attains its global maximum.
Citation: Xudong Shang. Existence and concentration of positive solutions for a p-fractional Choquard equation[J]. AIMS Mathematics, 2021, 6(11): 12929-12951. doi: 10.3934/math.2021748
In this work, we study the existence, multiplicity and concentration behavior of positive solutions for the following problem involving the fractional $ p $-Laplacian
$ \begin{eqnarray*} \varepsilon^{ps}(-\Delta )^{s}_{p}u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}(\frac{1}{|x|^{\mu}}\ast K|u|^{q})K(x)|u|^{q-2}u \hskip0.2cm\text{in}\hskip0.1cm \mathbb{R}^{N}, \end{eqnarray*} $
where $ 0 < s < 1 < p < \infty $, $ N > ps $, $ 0 < \mu < ps $, $ p < q < \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N}) $, $ (-\Delta)^{s}_{p} $ is the fractional $ p $-Laplacian and $ \varepsilon > 0 $ is a small parameter. Under certain conditions on $ V $ and $ K $, we prove the existence of a positive ground state solution and express the location of concentration in terms of the potential functions $ V $ and $ K $. In particular, we relate the number of solutions with the topology of the set where $ V $ attains its global minimum and $ K $ attains its global maximum.
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Xudong Shang. Existence and concentration of positive solutions for a p-fractional Choquard equation[J]. AIMS Mathematics, 2021, 6(11): 12929-12951. doi: 10.3934/math.2021748
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