Elliptic Problems with Singular Potential and Double-Power Nonlinearity

Abstract.

We prove the existence of positive symmetric solutions to the semilinear elliptic problem

$$ - \Delta u + V(|y|)u = f(u),\quad u \in D^{1,2} (\mathbb{R}^N ), \quad \left( {y,z} \right) \in \mathbb{R}^k \times \mathbb{R}^{N - k} $$

in both the radial case N = k ≥ 3 and the cylindrical case N ≥ k + 3 ≥ 6. The potential V is measurable, positive and it is only required to satisfy a mild integrability condition. The nonlinearity is continuous and has a doublepower behavior, super-critical near the origin and sub-critical at infinity. If f is odd, we show that the radial problem has infinitely many solutions. In proving these results we exploit the compactness of suitable restrictions of the embedding $D^{{1,2}} (\mathbb{R}^{N} ) \hookrightarrow L^{p} {\left( {\mathbb{R}^{N} } \right)} + L^{q} {\left( {\mathbb{R}^{N} } \right)}{\text{ for }}2 < p < \frac{{2N}} {{N - 2}} < q.$