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Concentrating ground state solutions for quasilinear Schrödinger equations with steep potential well. (English) Zbl 1479.35275

Summary: We are concerned with the following quasilinear Schrödinger equations \[ \begin{cases} -\Delta u+\lambda V(x)u+\displaystyle\frac{\kappa}{2} [\Delta (u^2)]u=q(x)f(u), x\in \mathbb{R}^N,\\ u \in H^1 (\mathbb{R}^N), \end{cases} \quad (1) \] where \(N\geq 3\), \(\lambda, \kappa >0\) are parameters, \(V\) and \(f\) are nonnegative continuous functions, \(q(x)\) is a positive bounded function. By using variational methods, we study the existence of positive ground state solutions to problem (1) when \(V,q\) and \(f\) satisfy some suitable conditions. Furthermore, the concentrating behavior of ground state solutions to problem (1) is proved. We mainly extend the results in Severo, Gloss and da Silva [U. B. Severo et al., J. Differ. Equations 263, No. 6, 3550–3580 (2017; Zbl 1378.35097)], which considered quasilinear Schrödinger equations with positive potential function, to quasilinear Schrödinger equations with steep potential well.

MSC:

35J10 Schrödinger operator, Schrödinger equation
35Q55 NLS equations (nonlinear Schrödinger equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J20 Variational methods for second-order elliptic equations

Citations:

Zbl 1378.35097
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Full Text: DOI

References:

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