Elliptic equations with decaying cylindrical potentials and power-type nonlinearities. (English) Zbl 1158.35032

The authors study the following cylindrical nonlinear elliptic equation:
\[ -\Delta u+\frac{A}{| y| ^\alpha}u=f(u), \] where \(x=(y,z)\in \mathbb{R}^k\times\mathbb{R}^{N-k}\) (\(2\leq k<N\)), \(A, \alpha >0\), and \(f\) is a nonlinearity of power growth \(p\). Main results concern existence, nonexistence and asymptotic behavior of solutions, and rely on certain compatibility conditions between the exponents \(\alpha\) and \(p\). The proof of existence uses critical point theory.


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs