Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Biharmonic nonlinear scalar field equations   We prove a Brezis-Kato-type regularity result for weak solutions to the
biharmonic nonlinear equation $$ \Delta^2 u = g(x,u)\qquad\text{in
}\mathbb{R}^N$$ with a Carath\'eodory function $g:\mathbb{R}^N\times
\mathbb{R}\to \mathbb{R}$, $N\geq 5$. The regularity results give rise to the
existence of ground state solutions provided that $g$ has a general subcritical
growth at infinity. We also conceive a new biharmonic logarithmic Sobolev
inequality $$ \int_{\mathbb{R}^N}|u|^2\log |u|\,dx\leq\frac{N}{8}\log
\Big(C\int_{\mathbb{R}^N}|\Delta u|^2\,dx \Big), \quad\text{for } u \in
H^2(\mathbb{R}^N), \; \int_{\mathbb{R}^N}u^2\,dx = 1, $$ for a constant $0<C<
\big(\frac{2}{\pi e N}\big)^2$ and we characterize its minimizers.

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The categories of this paper are: q-bio
19
The categories of this paper are: eess
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int