Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity

Abstract

For the following quasilinear Choquard-type equation: −Δu−Δ(u2)u+V(x)u=(Iμ*∣u∣p)∣u∣p−2u,x∈RN,-\Delta u-\Delta \left({u}^{2})u+V\left(x)u=\left({I}_{\mu }* {| u| }^{p}){| u| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N≥3,0<μ<NN\ge 3,0\lt \mu \lt N, V(x)=a−b1+∣x∣2V\left(x)=a-\frac{b}{1+{| x| }^{2}}, 1<a<+∞1\lt a\lt +\infty , 0<b<120\lt b\lt \frac{1}{2}, 2(N+μ)N<p<2(N+μ)N−2\frac{2\left(N+\mu )}{N}\lt p\lt \frac{2\left(N+\mu )}{N-2}, and Iμ{{I}}_{\mu } is the Riesz potential. Our work is finding the positive solutions and the ground-state solutions. Using a change of variables method, we overcome the difficulties which the quasilinear term may bring us and consider the corresponding functional with variational arguments. Then, we establish the nonexistence results via the Pohožaev identity.

Keywords

• quasilinear choquard equation
• variational method
• positive solutions
• ground-state solutions
• pohožaev identity
• 35b09
• 35b38
• 35j20
• 35j62