Nonlinear pseudo-differential equations defined by elliptic symbols on $\text{lp}$ and the fractional Laplacian

We develop an $L^p({\mathbb{R}}^n)$-functional calculus appropriated for interpreting “non-classical symbols” of the form $a(-\mathrm{\Delta })$, and for proving existence in $L^q({\mathbb{R}}^n)$, some $q>p$, of solutions to nonlinear pseudo-differential equations of the form $[1 + a(-\Delta)]^{s/2} (u) = V(\cdot, u)$. More precisely, we use the theory of Fourier multipliers for constructing suitable domains on which the formal operator appearing in the above equation can be rigorously defined, and we prove existence of solutions belonging to these domains. We also include applications of the theory to equations of physical interest involving the fractional Laplace operator such as the Allen-Cahn equation.