Nonlinear biharmonic problems with singular potentials. (English) Zbl 1304.35256
Summary: We deal with the problem
\[
\Delta^2 u +V(|x|)u = f(u), u\in D^{2,2}(\mathbb R^N)
\]
where \(\Delta^2\) is a biharmonic operator and the potential \(V > 0\) is measurable, singular at the origin and may also have a continuous set of singularities. The nonlinearity is continuous and has a super-linear power-like behaviour; both sub-critical and super-critical cases are considered. We prove the existence of nontrivial radial solutions. If \(f\) is odd, we show that the problem has infinitely many radial solutions.
MSC:
| 35J35 | Variational methods for higher-order elliptic equations |
| 35J70 | Degenerate elliptic equations |
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
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