Normalized solutions for scalar field equation involving multiple critical nonlinearities

Abstract

Abstract This paper concerns the scalar field equation - Δ ⁢ u = λ ⁢ u + | u | q - 2 ⁢ u + a ⁢ | u | 4 ⁢ u + b ⁢ ( I 2 ∗ | u | 5 ) ⁢ | u | 3 ⁢ u   in  ⁢ ℝ 3 -\Delta u=\lambda u+|u|^{q-2}u+a|u|^{4}u+b(I_{2}\ast|u|^{5})|u|^{3}u\quad\text% {in }\mathbb{R}^{3} under the normalized constraint ∫ ℝ 3 u 2 ⁢ 𝑑 x = c 2 {\int_{\mathbb{R}^{3}}u^{2}\,dx=c^{2}} , where a , b , c > 0 {a,\,b,\,c>0} , 2 < q < 10 3 {2<q<\frac{10}{3}} and I 2 {I_{2}} is the Riesz potential. We prove that for small prescribed mass c the above equation has a positive ground state solution and an infinite sequence of normalized solutions with negative energies tending to zero. Asymptotic properties of ground state solutions as a → 0 + {a\to 0^{+}} and as b → 0 + {b\to 0^{+}} are also studied.