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Standing waves for a class of Schrödinger-Poisson equations in ${\mathbb{R}^3}$ involving critical Sobolev exponents (1412.4057v1)
Published 6 Dec 2014 in math.AP
Abstract: We are concerned with the following Schr\"odinger-Poisson equation with critical nonlinearity: [\left{\begin{gathered} - {\varepsilon 2}\Delta u + V(x)u + \psi u = \lambda |u{|{p - 2}}u + |u{|4}u{\text{in}}{\mathbb{R}3}, \hfill - {\varepsilon 2}\Delta \psi = {u2}{\text{in}}{\mathbb{R}3},{\text{}}u > 0,{\text{}}u \in {H1}({\mathbb{R}3}), \hfill \end{gathered} \right. ] where $\varepsilon > 0$ is a small positive parameter, $\lambda > 0$, $3 < p \le 4$. Under certain assumptions on the potential $V$, we construct a family of positive solutions ${u_\varepsilon} \in {H1}({\mathbb{R}3})$ which concentrates around a local minimum of $V$ as $\varepsilon \to 0$.