Minimal Energy Solutions and Infinitely Many Bifurcating Branches for a Class of Saturated Nonlinear Schrödinger Systems

Abstract

Abstract We prove a conjecture recently formulated by Maia, Montefusco and Pellacci saying that minimal energy solutions of the saturated nonlinear Schrödinger system { - Δ ⁢ u + λ 1 ⁢ u = α ⁢ u ⁢ ( α ⁢ u 2 + β ⁢ v 2 ) 1 + s ⁢ ( α ⁢ u 2 + β ⁢ v 2 ) in ⁢ ℝ n , - Δ ⁢ v + λ 2 ⁢ v = β ⁢ v ⁢ ( α ⁢ u 2 + β ⁢ v 2 ) 1 + s ⁢ ( α ⁢ u 2 + β ⁢ v 2 ) in ⁢ ℝ n $\left\{\begin{aligned} \displaystyle-\Delta u+\lambda_{1}u&\displaystyle=\frac% {\alpha u(\alpha u^{2}+\beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&% \displaystyle\text{in }\mathbb{R}^{n},\\ \displaystyle-\Delta v+\lambda_{2}v&\displaystyle=\frac{\beta v(\alpha u^{2}+% \beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&\displaystyle\text{in }\mathbb{R% }^{n}\end{aligned}\right.$ are necessarily semitrivial whenever α , β , λ 1 , λ 2 > 0 ${\alpha,\hskip 0.5pt\beta,\hskip 0.5pt\lambda_{1},\hskip 0.5pt\lambda_{2}>0}$ and 0 < s < max ⁡ { α / λ 1 , β / λ 2 } ${0<s<\max\{\alpha/\lambda_{1},\hskip 0.5pt\beta/\lambda_{2}\}}$ except for the symmetric case λ 1 = λ 2 ${\lambda_{1}=\lambda_{2}}$ , α = β ${\alpha=\beta}$ . Moreover, it is shown that for most parameter samples α , β , λ 1 , λ 2 ${\alpha,\beta,\lambda_{1},\lambda_{2}}$ , there are infinitely many branches containing seminodal solutions which bifurcate from a semitrivial solution curve parametrized by s.