Dirichlet problems involving the Hardy-Leray operators with multiple polars

Our aim of this article is to study qualitative properties of Dirichlet problems involving the Hardy-Leray operator ℒV≔−Δ+V{{\mathcal{ {\mathcal L} }}}_{V}:= -\Delta +V, where V(x)=∑i=1mμi∣x−Ai∣2V\left(x)={\sum }_{i=1}^{m}\frac{{\mu }_{i}}{{| x-{A}_{i}| }^{2}}, with μi≥−(N−2)24{\mu }_{i}\ge -\frac{{\left(N-2)}^{2}}{4} being the Hardy-Leray potential containing the polars’ set Am={Ai:i=1,…,m}{{\mathcal{A}}}_{m}=\left\{{A}_{i}:i=1,\ldots ,m\right\} in RN{{\mathbb{R}}}^{N} (N≥2N\ge 2). Since the inverse-square potentials are critical with respect to the Laplacian operator, the coefficients {μi}i=1m{\left\{{\mu }_{i}\right\}}_{i=1}^{m} and the locations of polars {Ai}\left\{{A}_{i}\right\} play an important role in the properties of solutions to the related Poisson problems subject to zero Dirichlet boundary conditions. Let Ω\Omega be a bounded domain containing Am{{\mathcal{A}}}_{m}. First, we obtain increasing Dirichlet eigenvalues: ℒVu=λuinΩ,u=0on∂Ω,{{\mathcal{ {\mathcal L} }}}_{V}u=\lambda u\hspace{1.0em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1.0em}{\rm{on}}\hspace{0.33em}\partial \Omega , and the positivity of the principle eigenvalue depends on the strength μi{\mu }_{i} and polars’ setting. When the spectral does not contain the origin, we then consider the weak solutions of the Poisson problem (E)ℒVu=νinΩ,u=0on∂Ω,\left(E)\hspace{1.0em}\hspace{1.0em}{{\mathcal{ {\mathcal L} }}}_{V}u=\nu \hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1em}{\rm{on}}\hspace{0.33em}\partial \Omega , when ν\nu belongs to Lp(Ω){L}^{p}\left(\Omega ), with p>2NN+2p\gt \frac{2N}{N+2} in the variational framework, and we obtain a global weighted L∞{L}^{\infty } estimate when p>N2p\gt \frac{N}{2}. When the principle eigenvalue is positive and ν\nu is a Radon measure, we build a weighted distributional framework to show the existence of weak solutions of problem (E)\left(E). Moreover, via this weighted distributional framework, we can obtain a sharp assumption of ν∈Cγ(Ω¯\Am)\nu \in {{\mathcal{C}}}^{\gamma }\left(\bar{\Omega }\setminus {{\mathcal{A}}}_{m}) for the existence of isolated singular solutions for problem (E)\left(E).