Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compact Sobolev embeddings and positive solutions to a quasilinear
  equation with indefinite nonlinearities   In this paper, we study the existence and multiplicity results of nontrivial
positive solutions to the following quasilinear elliptic equation on $\RN$,
when $N\geq2$, \begin{equation} \Lp
u=\lambda\hspace{0.2mm}K(x)u^{r-1}-V(x)u^{q-1}.\nonumber \end{equation} Here,
$K(x),V(x)>0$ are suitable potentials, $1<p<r<q<\infty$, and $\lambda>0$ is a
parameter. To study this problem, some compact embedding results regarding
$\MVR\hookrightarrow\LKR$ are proved that unify and extend some recent results
of the author \cite{Ha1,Ha2,Ha3}.

17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th