## Bound state solutions for a class of nonlinear Schrödinger equations.(English)Zbl 1156.35084

Summary: We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form
$-\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N,$
where $$V, K$$ are positive continuous functions and $$p > 1$$ is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential $$V$$ is allowed to vanish at infinity and the competing function $$K$$ does not have to be bounded. In the semi-classical limit, i.e. for $$\varepsilon\sim 0$$, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function $$\mathcal{A} = V^\theta K^{-\frac{2}{p-1}}$$, where $$\theta=(p+1)/(p-1)-N/2$$. A special attention is devoted to the qualitative properties of these solutions as $$\varepsilon$$ goes to zero.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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### References:

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