Normalized solutions for Schrödinger-Poisson equation with prescribed mass: the Sobolev subcritical case and the Sobolev critical case with mixed dispersion. (English) Zbl 1540.35196
Summary: In this paper, we study the following Schrödinger Poisson equation
\[
\begin{cases}
-\Delta u -\lambda u -\gamma (|x|^{-1} \ast |u|^2) u = f(u), & x \in \mathbb{R}^3; \\
\displaystyle\int\limits_{\mathbb{R}^3} u^2 dx = c,
\end{cases}
\]
where \(c>0\), \(\lambda \in \mathbb{R}\) and \(f \in \mathcal{C}(\mathbb{R}, \mathbb{R})\). When \(\gamma <0\) and \(f\) satisfies some weaker \(L^2\)-supercritical conditions in Sobolev subcritical case, we show the existence of normalized solutions. When \(f(u) = \mu |u|^{q-2} u + |u|^4 u\) with \(\mu >0\), \(\gamma >0\) and \(2 < q < \frac{10}{3}\), a more complicated situation, we obtain the existence of multiple solutions. Our main tools are some technically simpler than the N. Ghoussoub minimax principle [Duality and perturbation methods in critical point theory. Cambridge: Cambridge University Press (1993; Zbl 0790.58002)], which not only allow us to construct a bounded (PS) sequence under some weaker conditions on \(f\) than before, but help to yield richer results. In particular, our results generalize and improve some ones in [L. Jeanjean and T. T. Le, J. Differ. Equations 303, 277–325 (2021; Zbl 1475.35163)] and some other related literatures.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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