Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Infinitely many sign-changing solutions for Kirchhoff type problems in
  $\mathbb{R}^3$   In this paper, we consider the following nonlinear Kirchhoff type problem: \[
\left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla
u|^2\right)\Delta u+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in
H^1(\mathbb{R}^3), \end{array}\right. \] where $a,b>0$ are constants, the
nonlinearity $f$ is superlinear at infinity with subcritical growth and $V$ is
continuous and coercive. For the case when $f$ is odd in $u$ we obtain
infinitely many sign-changing solutions for the above problem by using a
combination of invariant sets method and the Ljusternik-Schnirelman type
minimax method. To the best of our knowledge, there are only few existence
results for this problem. It is worth mentioning that the nonlinear term may
not be 4-superlinear at infinity, in particular, it includes the power-type
nonlinearity $|u|^{p-2}u$ with $p\in(2,4]$.

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The categories of this paper are: gr-qc
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: hep-ph
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: q-fin