Dirichlet problems involving the Hardy-Leray operators with multiple polars

Abstract

Abstract Our aim of this article is to study qualitative properties of Dirichlet problems involving the Hardy-Leray operator ℒ V ≔ − Δ + V {{\mathcal{ {\mathcal L} }}}_{V}:= -\Delta +V , where V ( x ) = ∑ i = 1 m μ i ∣ x − A i ∣ 2 V\left(x)={\sum }_{i=1}^{m}\frac{{\mu }_{i}}{{| x-{A}_{i}| }^{2}} , with μ i ≥ − ( N − 2 ) 2 4 {\mu }_{i}\ge -\frac{{\left(N-2)}^{2}}{4} being the Hardy-Leray potential containing the polars’ set A m = { A i : i = 1 , … , m } {{\mathcal{A}}}_{m}=\left\{{A}_{i}:i=1,\ldots ,m\right\} in R N {{\mathbb{R}}}^{N} ( N ≥ 2 N\ge 2 ). Since the inverse-square potentials are critical with respect to the Laplacian operator, the coefficients { μ i } i = 1 m {\left\{{\mu }_{i}\right\}}_{i=1}^{m} and the locations of polars { A i } \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 A m {{\mathcal{A}}}_{m} . First, we obtain increasing Dirichlet eigenvalues: ℒ V u = λ u in Ω , u = 0 on ∂ 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 μ i {\mu }_{i} and polars’ setting. When the spectral does not contain the origin, we then consider the weak solutions of the Poisson problem ( E ) ℒ V u = ν in Ω , u = 0 on ∂ 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 L p ( Ω ) {L}^{p}\left(\Omega ) , with p > 2 N N + 2 p\gt \frac{2N}{N+2} in the variational framework, and we obtain a global weighted L ∞ {L}^{\infty } estimate when p > N 2 p\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 ∈ C γ ( Ω ¯ \ A m ) \nu \in {{\mathcal{C}}}^{\gamma }\left(\bar{\Omega }\setminus {{\mathcal{A}}}_{m}) for the existence of isolated singular solutions for problem ( E ) \left(E) .