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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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