Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties

Abstract

We consider the following nonlinear nonlocal elliptic problem: −a+b∫R3∣∇ψ∣2dxΔψ+λψ=∫R3G(ψ(y))∣x−y∣αdyG′(ψ),x∈R3,-\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla \psi | }^{2}{\rm{d}}x\right)\Delta \psi +\lambda \psi =\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}\frac{G\left(\psi (y))}{{| x-y| }^{\alpha }}{\rm{d}}y\right)G^{\prime} \left(\psi ),\hspace{1em}x\in {{\mathbb{R}}}^{3}, where a,b>0a,b\gt 0 are constants, λ>0\lambda \gt 0 is a parameter, α∈(0,3)\alpha \in \left(0,3), and G∈C1(R,R)G\in {{\mathcal{C}}}^{1}\left({\mathbb{R}},{\mathbb{R}}). By using variational methods, we establish the existence of least energy solutions for the above equation under conditions on the nonlinearity GG we believe to be almost necessary. Some qualitative properties of the least energy solutions are also obtained

Keywords

• nonlocal elliptic problem
• variational method
• least energy solution
• qualitative property
• 35j60
• 35j20
• 35q55