Elliptic Equations with Hardy Potential and Gradient-Dependent Nonlinearity

Abstract

Let Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} (N≥3{N\geq 3}) be a C2{C^{2}} bounded domain, and let δ be the distance to ∂⁡Ω{\partial\Omega}. We study equations (E±){(E_{\pm})}, -Lμ⁢u±g⁢(u,|∇⁡u|)=0{-L_{\mu}u\pm g(u,\lvert\nabla u\rvert)=0} in Ω, where Lμ=Δ+μδ2{L_{\mu}=\Delta+\frac{\mu}{\delta^{2}}}, μ∈(0,14]{\mu\in(0,\frac{1}{4}]} and g:ℝ×ℝ+→ℝ+{g\colon\mathbb{R}\times\mathbb{R}_{+}\to\mathbb{R}_{+}} is nondecreasing and locally Lipschitz in its two variables with g⁢(0,0)=0{g(0,0)=0}. We prove that, under some subcritical growth assumption on g, equation (E+){(E_{+})} with boundary condition u=ν{u=\nu} admits a solution for any nonnegative bounded measure on ∂⁡Ω{\partial\Omega}, while equation (E-){(E_{-})} with boundary condition u=ν{u=\nu} admits a solution provided that the total mass of ν is small. Then we analyze the model case g⁢(s,t)=|s|p⁢tq{g(s,t)=\lvert s\rvert^{p}t^{q}} and obtain a uniqueness result, which is even new with μ=0{\mu=0}. We also describe isolated singularities of positive solutions to (E+){(E_{+})} and establish a removability result in terms of Bessel capacities. Various existence results are obtained for (E-){(E_{-})}. Finally, we discuss existence, uniqueness and removability results for (E±){(E_{\pm})} in the case g⁢(s,t)=|s|p+tq{g(s,t)=\lvert s\rvert^{p}+t^{q}}.

Keywords

• hardy potential
• singular solutions
• boundary trace
• uniqueness
• critical exponent
• gradient term
• isolated singularities
• 35j10
• 35j25
• 35j60
• 35j66
• 35j75