Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 [11]. We also give the $p$-Laplace version of G. Bratu's
solution [3].
  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 $\alpha <0$. The Coulomb case $\alpha=-1$ turned out to give
the critical power.

27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ