Elliptic problems with mixed nonlinearities and potentials singular at the origin and at the boundary of the domain

Abstract

AbstractWe are interested in the following Dirichlet problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\textrm{dist}(x,\mathbb {R}^N \setminus \Omega )^2} = f(x,u) &{} \quad \text{ in } \Omega \\ u = 0 &{} \quad \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$ - Δ u + λ u - μ u | x | 2 - ν u dist ( x , R N \ Ω ) 2 = f ( x , u ) in Ω u = 0 on ∂ Ω , on a bounded domain $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N with $$0 \in \Omega $$ 0 ∈ Ω . We assume that the nonlinear part is superlinear on some closed subset $$K \subset \Omega $$ K ⊂ Ω and asymptotically linear on $$\Omega \setminus K$$ Ω \ K . We find a solution with the energy bounded by a certain min–max level, and infinitely, many solutions provided that f is odd in u. Moreover, we study also the multiplicity of solutions to the associated normalized problem.