## Positive solutions of a Schrödinger equation with critical nonlinearity.(English)Zbl 1060.35130

The authors study the nonlinear Schrödinger equation transformed into the problem $-\triangle u+\lambda a(x)u=\mu u+u^{2^{\ast }-1}, \quad u\in \mathbb R^N, \;\lambda >0, \;\mu \in \mathbb R$ with critical exponent $$2^{\ast }=2N/(N-2)$$, $$N\geq 4$$, where $$a\geq 0$$ has a potential well, and $$u>0$$, $$u\in H^1(\mathbb R^N)$$. Here $$a$$ is a nonnegative and continuous function, and the Lebesgue measure in $$\mathbb R^N$$ is defined by $$\mathcal L$$$$\{x\in \mathbb R^n:a(x)\leq M_0\}<\infty$$. 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.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35J65 Nonlinear boundary value problems for linear elliptic equations
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