A mixed local and nonlocal supercritical Dirichlet problems

In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem \begin{equation} Lu=|u|^{p-2}u+\mu|u|^{q-2}u~~\text{in}~~\Omega,~~~~~ u=0~~\text{in}~~\mathbb{R}^N\setminus\Omega,~~~ (0.1) \end{equation} where $L=-\Delta+(-\Delta)^s$ for $s\in(0,1)$ and $\Omega\subset\mathbb{R}^N$ is a bounded domain. Precisely, we show that problem (0.1) for $1<q<2<p$ has a positive solution as well as a sequence of sign-changing solutions with a negative energy for small values of $\mu$. Here $u$ can be either a scalar function, or a vector valued function so that (0.1) turns into a system with supercritical nonlinearity. Moreover, whenever the domain is symmetric, we also prove the existence of symmetric solutions enjoying the same symmetry properties. We shall also prove an existence result for the supercritical Hamiltonian system \begin{equation} Lu=|v|^{p-2}v,~~~~~~~ Lv=|u|^{d-2}u+\mu |u|^{q-2}u \end{equation} with the Dirichlet boundary condition on $\Omega$ where $1<q<2<p, d$. Our method is variational, and in both problems the lack of compactness for the supercritical problem is recovered by working on a closed convex subset of an appropriate function space.