On Nodal Solutions of the Nonlinear Choquard Equation

Abstract

Abstract This paper deals with the general Choquard equation - Δ ⁢ u + V ⁢ ( | x | ) ⁢ u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u   in ⁢ ℝ N , -\Delta u+V(|x|)u=(I_{\alpha}*|u|^{p})|u|^{p-2}u\quad\text{in }\mathbb{R}^{N}, where V∈C⁢([0,∞),ℝ+){V\in C([0,\infty),\mathbb{R}^{+})} is bounded below by a positive constant, and Iα{I_{\alpha}} denotes the Riesz potential of order α∈(0,N){\alpha\in(0,N)}. In view of the convolution term, the nonlocal property makes the variational functional completely different from the one for local pure power-type nonlinearity. By combining the Brouwer degree and developing some new techniques, a family of radial solutions with a prescribed number of zeros is constructed for p∈[2,N+αN-2){p\in[2,\frac{N+\alpha}{N-2})}, while the degeneracy happens for p∈(N+αN,2){p\in(\frac{N+\alpha}{N},2)}. This result complements and improves the ones in the literature in the aspect of the range of p.