Explicit solutions and multiplicity results for some equations with the $p$-Laplacian

We derive explicit ground state solutions for several equations with the $p$-Laplacian in $R^n$, including (here $\varphi (z)=z|z|^{p-2}$, with $p>1$) \[ \varphi \left (u’(r)\right )’ +\frac {n-1}{r} \varphi \left (u’(r)\right )+u^M+u^Q=0 . \] The constant $M>0$ is assumed to be below the critical power, while $Q=\frac {M p-p+1}{p-1}$ is above the critical power. This explicit solution is used to give a multiplicity result, similarly to C. S. Lin and W.-M. Ni (1998). We also give the $p$-Laplace version of G. Bratu’s solution, connected to combustion theory.

In another direction, we present a change of variables which removes the non-autonomous term $r^{\alpha }$ in \[ \varphi \left (u’(r)\right )’ +\frac {n-1}{r} \varphi \left (u’(r)\right )+r^{\alpha } f(u)=0 , \] while preserving the form of this equation. In particular, we study singular equations, when $\alpha <0$, that occur often in applications. The Coulomb case $\alpha =-1$ turned out to give the critical power.