Let $\Omega\subset \mathbb{R}^3$ be a Lipschitz domain and let $S_\mathrm{curl}(\Omega)$ the largest constant such that $$ \int_{\mathbb{R}^3}|\nabla\times u|^2\, dx\geq S_{\mathrm{curl}}(\Omega) \inf_{\substack{w\in W_0^6(\mathrm{curl};\mathbb{R}^3)\\ \nabla\times w=0}}\Big(\int_{\mathbb{R}^3}|u+w|^6\,dx\Big)^{\frac13} for any $u$ in $W_0^6(\mathrm{curl};\Omega)\subset W_0^6(\mathrm{curl};\mat...

We prove the existence of a positive SOLA (Solutions Obtained as Limits Approximations) to following PDE involving fractional power Laplacian ( ? ? ) s u = 1 ? + ? 2 ? ? in ? , > 0 ? N ? . Here, is bounded domain ?N, s?(0,1), 2s<N, ?,??(0,1), 2s?=2N?N?2s critical Sobolev exponent and nonnegative Radon measure ?.

We study a nonlocal elliptic equation of $p$-Kirchhoff type involving the critical Sobolev exponent. First we give sufficient conditions for $(\text{PS})$ condition to hold. Then prove some existence and multiplicity results using tools from Morse theory, in particular, notion cohomological local splitting eigenvalues based on Fadell-Rabinowitz index.