Nodal solutions for the Schrödinger–Poisson system with an asymptotically cubic term. (30th April 2022) - Guo, Hui; Tang, Ronghua; Wang, Tao
Abstract : This paper deals with the following Schrödinger–Poisson system 0.1 − Δ u + u + λ ϕ u = f ( u ) in ℝ 3, − Δ ϕ = u 2 in ℝ 3, $$ \left\{\begin{array}{cc}\hfill & -\Delta u+u+\lambda \phi u=f(u)\kern0.30em \mathrm{in}\kern0.4em {\mathbb{R}}^3, \hfill \\ {}\hfill & -\Delta \phi ={u}^2\kern0.30em \mathrm{in}\kern0.4em {\mathbb{R}}^3, \hfill \end{array}\right. $$ where λ > 0 $$ \lambda >0 $$ and f ( u ) $$ f(u) $$ is a nonlinear term asymptotically cubic at the infinity. Taking advantage of the Miranda's theorem and deformation lemma, we combine some new analytic techniques to prove that for each positive integer k $$ k $$, system (0.1 ) admits a radial nodal solution U k λ $$ {U}_k^{\lambda } $$, which has exactly k + 1 $$ k+1 $$ nodal domains and the corresponding energy is strictly increasing in k $$ k $$ . Moreover, for any sequence { λ n } → 0 + $$ \left\{{\lambda}_n\right\}\to {0}_{+} $$ as n → ∞ $$ n\to \infty $$, up to a subsequence, U k λ n $$ {U}_k^{\lambda_n} $$ converges to some U k 0 ∈ H r 1 ( ℝ 3 ) $$ {U}_k^0\in {H}_r^1\left({\mathbb{R}}^3\right) $$, which is a radial nodal solution with exactly k + 1 $$ k+1 $$ nodal domains of (0.1 ) for λ = 0 $$ \lambda =0 $$ . These results give an affirmative answer to the open problem proposed in Kim and Seok (2012) about the existence of nodal solutions with prescribed number of nodal domains for the Schrödinger–Poisson system with an asymptotically cubic term.