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On existence and concentration behavior of ground state solutions for a class of problems with critical growth. (English) Zbl 1183.35121

Summary: We study the existence and the concentration behavior of ground state for the problem \[ -h^2\Delta u+V(z)u=\lambda u^{q-1}+u^{\frac{N+2}{N-2} -1}\text{ in } \mathbb R^N, \]
\[ u(z)>0\;\text{for all}\;z\in \mathbb R^N \] where \(h, \lambda >0\), \(1< q < \frac{N+2}{N-2}\), \(N\geq 3\) and \(V: \mathbb R^N\to \mathbb R\) is a positive function such that \(0< \inf_{z\in\mathbb R^N}V(z) < \lim\inf_{|z| \to \infty}V(z)=V_{\infty}.\)

MSC:

35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35J20 Variational methods for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
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