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Positive solutions of a Schrödinger equation with critical nonlinearity. (English) Zbl 1060.35130
The authors study the nonlinear Schrödinger equation transformed into the problem $$ -\triangle u+\lambda a(x)u=\mu u+u^{2^{\ast }-1}, \quad u\in \Bbb R^N, \ \lambda >0, \ \mu \in \Bbb R$$ with critical exponent $2^{\ast }=2N/(N-2)$, $N\ge 4$, where $a\ge 0$ has a potential well, and $u>0$, $u\in H^1(\Bbb R^N)$. Here $a$ is a nonnegative and continuous function, and the Lebesgue measure in $\Bbb R^N$ is defined by $\cal L$$\{x\in \Bbb R^n:a(x)\le M_0\}<\infty $. Having in mind the above stated conditions the authors prove existence and multiplicity of positive solutions which localize near the potential well for $\mu $ small and $\lambda $ large.

35Q55NLS-like (nonlinear Schrödinger) equations
35J65Nonlinear boundary value problems for linear elliptic equations
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