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Generalized quasilinear Schrödinger equations with concave functions \( l(s^2)\). (English) Zbl 1408.35023

Summary: We establish the existence of nontrivial solutions for the following quasilinear Schrödinger equation with subcritical or critical growth:
\[ -\Delta u+W(x)u^{2\alpha-1}-ul'(u^2)\Delta l(u^2) = f(u) \text{ or }h(u)+u^{2^*-1},\quad x\in\mathbb{R}^N, \]
where \(W(x):\mathbb{R}^N \rightarrow \mathbb{R}\) is a given potential and \( l,h,f \) are real functions, \( u>0\), \( 2^* = 2N/(N-2) \), \( N\geq3 \). Our results cover physical models \( l(s) = s^{\frac{\alpha}{2}}\), \( \frac{1}{2}<\alpha<1 \).

MSC:

35J20 Variational methods for second-order elliptic equations
35J62 Quasilinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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