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Article Contents

# Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents

• We are concerned with standing waves for the following Schrödinger-Poisson equation with critical nonlinearity: \begin{eqnarray} && - {\varepsilon ^2}\Delta u + V(x)u + \psi (x)u = \lambda W(x){\left| u \right|^{p - 2}}u + {\left| u \right|^4}u\;\;{\text{ in }}\mathbb{R}^3, \\ && - {\varepsilon ^2}\Delta \psi = {u^2}\;\;{\text{ in }}\mathbb{R}^3, u>0, u \in {H^1}(\mathbb{R}^3), \end{eqnarray} where $\varepsilon$ is a small positive parameter, $\lambda > 0$, $3 < p \le 4$, $V$ and $W$ are two potentials. Under proper assumptions, we prove that for $\varepsilon > 0$ sufficiently small, the above problem has a positive ground-state solution ${u_\varepsilon }$ by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to [C. Gui, Commun. Partial Differential Equations 21 (1996) 787-820] to show that ${u_\varepsilon }$ concentrates around a set which is related to the set where the potential $V(x)$ attains its global minima or the set where the potential $W(x)$ attains its global maxima as $\varepsilon \to 0$. As far as we know, the existence and concentration behavior of the positive solutions to the Schrödinger-Poisson equation with critical nonlinearity $g(u): = \lambda W(x)|u{|^{p - 2}}u + |u{|^4}u$ $(3 Mathematics Subject Classification: Primary: 35J20, 35J60, 35J92.  Citation: •  [1] A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.doi: 10.1007/s00032-008-0094-z. [2] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 14 (1973), 349-381. [3] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson equation, Commun. Contemp. Math., 10 (2008), 1-14.doi: 10.1142/S021919970800282X. [4] A. 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