Existence and concentration of positive solutions for a p-fractional Choquard equation

<abstract><p>In this work, we study the existence, multiplicity and concentration behavior of positive solutions for the following problem involving the fractional $ p $-Laplacian</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p> <p>where $ 0 &lt; s &lt; 1 &lt; p &lt; \infty $, $ N &gt; ps $, $ 0 &lt; \mu &lt; ps $, $ p &lt; q &lt; \frac{p^{*}_{s}}{2}(2-\frac{\mu}{N}) $, $ (-\Delta)^{s}_{p} $ is the fractional $ p $-Laplacian and $ \varepsilon &gt; 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.</p></abstract>

Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.