Ground state solutions for Kirchhoff type equations with asymptotically 4-linear nonlinearity. (English) Zbl 1443.35040
Summary: This paper is concerned with the following Kirchhoff type equation:
\[
\begin{cases}
-\left(a+b\int_\Omega |\nabla v|^2\right)\Delta v=f(x,v)\quad & \text{in }\Omega,\\
v=0 & \text{on }\partial\Omega.
\end{cases}
\]
Assuming that the primitive of \(f\) is asymptotically 4-linear as \(|v|\to\infty\), a homeomorphism between the Nehari manifold and a subset of unit sphere is constructed based on some observations and new techniques. Inspired by recent work of A. Szulkin and T. Weth [in: Handbook of nonconvex analysis and applications. Somerville, MA: International Press. 597–632 (2010; Zbl 1218.58010)], ground state solutions for the equation above are obtained, as well as infinitely many pairs of nontrivial solutions for case \(f(x,v)\) is odd with respect to \(v\).
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
Keywords:
Kirchhoff type equations; variational method; asymptotically 4-linearity; Nehari manifold; ground stateCitations:
Zbl 1218.58010References:
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