Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Nehari manifold for non-local elliptic operator with concave-convex
  non-linearities and sign-changing weight function   In this article, we study the existence and multiplicity of non-negative
solutions of following $p$-fractional equation: $$ \quad
\left\{\begin{array}{lr}\ds
  \quad - 2\int_{\mb R^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\al}}
dxdy = \la h(x)|u|^{q-1}u+ b(x)|u|^{r-1} u \; \text{in}\; \Om
  \quad \quad \quad \quad u \geq 0 \; \mbox{in}\; \Om,\quad u\in W^{\al,p}(\mb
R^n),
  \quad \quad\quad \quad\quad u =0\quad\quad \text{on} \quad \mb R^n\setminus
\Om \end{array} \right. $$ where $\Om$ is a bounded domain in $\mb R^n$, $p\geq
2$, $n> p\al$, $\al\in(0,1)$, $0< q<p-1 <r < \frac{np}{n-ps}-1$, $\la>0$ and
$h$, $b$ are sign changing smooth functions. We show the existence of solutions
by minimization on the suitable subset of Nehari manifold using the fibering
maps. We find that there exists $\la_0$ such that for $\la\in (0,\la_0)$, it
has at least two solutions.

26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th