Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: − a − b ∫ Ω | ∇ H u | 2 d ξ Δ H u + μ ϕ u = λ | u | q − 2 u + | u | 2 u , in Ω , − Δ H ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 $ \mathbb{H}^1 $ , and Ω ⊂ H 1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a, b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ ∈ R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient (a − b∫Ω∣∇ H u∣2 dx) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

Existence of multiple positive solutions for singular p-q-Laplacian problems with critical nonlinearities

In this article, we consider the following p-q-Laplacian system with singular and critical nonlinearity \begin{equation*} \left \{ \begin{array}{lllll} -\Delta_{p}u-\Delta_{q}u=\frac{h_{1}(x)}{u^{r}}+\lambda\frac{\alpha}{\alpha+\beta}u^{\alpha-1}v^{\beta} \ \ in\ \Omega ,\\ -\Delta_{p}v-\Delta_{q}v=\frac{h_{2}(x)}{v^{r}}+\lambda\frac{\beta}{\alpha+\beta}u^{\alpha}v^{\beta-1} \ \ in\ \Omega, \\ u,v>0 \ \ \ \ \ \ in \ \Omega, \ \ \ \ \ u=v=0 \ \ \ \ \ \ \ on \ \partial\Omega, \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb {R}^{n}$ with smooth boundary $\partial\Omega$. $11,\lambda\in(0,\Lambda_{*})$ is parameter with $\Lambda _{*}$ is a positive constant and $h_{1}(x),h_{2}(x)\in L^{\infty},h_{1}(x),h_{2}(x)>0$. We show the existence and multiplicity of weak solution of equation above for suitable range of $\lambda$.