| Abstract: |
| A vast literature addresses problems involving the Laplacian operator of the kind
\[
-\Delta u=\lambda u^s+u^{p}\quad \mbox{in }\Omega\Subset R^N,
\]
the main focus being the existence of positive solutions (vanishing at the boundary) when the nonlinearity hits the critical Sobolev exponent $2^\star=\frac{2N}{N-2}$.
In both the singular case with $s\in (-1,0)$ and the concave-convex case with $s\in(0,1)$, there exists a threshold $\Lambda>0$ such that the problem admits at least two solutions for $\lambda\in(0,\Lambda)$ and admits no solutions for $\lambda>\Lambda$. For $\lambda=\Lambda$ at least one solution exists.
In two recent works (joint with Biagi-Vecchi and Vecchi), we consider the same problem but in a Carnot group $G$,
where the operator is the sub-Laplacian $-\Delta_G$, and prove an analogous behaviour to the Euclidean case. In this sub-Riemannian setting, the critical case is $2^\star_G=\frac{2Q}{Q-2}$, where $Q$ is the homogeneous dimension of $G$.
In the singular case, we obtain the first solution via Perron`s method and the second by adapting an argument by Tarantello, with new estimates for the group convolution and for substitutes of the Aubin--Talenti functions. In the concave-convex case, the minimality argument for the first solution relies on regularity results available in the Euclidean setting but not in the sub-Riemannian one; to overcome this, we develop an ad hoc variational approach. |
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