Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonlinear equations involving the square root of the Laplacian   In this paper we discuss the existence and non-existence of weak solutions to
parametric fractional equations involving the square root of the Laplacian
$A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$)
and with zero Dirichlet boundary conditions. Namely, our simple model is the
following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda
f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega.
\end{array}\right. \end{equation*} The existence of at least two non-trivial
$L^{\infty}$-bounded weak solutions is established for large value of the
parameter $\lambda$ requiring that the nonlinear term $f$ is continuous,
superlinear at zero and sublinear at infinity. Our approach is based on
variational arguments and a suitable variant of the Caffarelli-Silvestre
extension method.

37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg