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Ground state solutions for the quasilinear Schrödinger equation. (English) Zbl 1234.35246

Summary: This paper is concerned with the quasilinear Schrödinger equation \[ -\Delta u + (\lambda a(x) +1)u - \frac{1}{2}(\Delta |u|^2)u = u^p, \qquad u>0, x \in {\mathbb R}^N \] with \(a(x)\geq 0\) having a potential well and \(4<p+1<2\cdot 2^{\ast }\). By using the variational method and the concentration compactness method in Orlicz space, we prove the existence of a ground state solution which localizes near the potential well \(\text{int}\{ a^{-1}(0)\}\) for \(\lambda\) large enough.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J62 Quasilinear elliptic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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