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Quasilinear elliptic problems under asymptotically linear conditions at infinity and at the origin

2014; Birkhäuser; Volume: 66; Issue: 2 Linguagem: Inglês

10.1007/s00033-014-0406-9

1420-9039

Marcelo F. Furtado, Edcarlos D. Silva, Maxwell L. Silva,

Advanced Mathematical Physics Problems

We obtain existence and multiplicity of solutions for the quasilinear Schrödinger equation $$-\Delta u + V(x)u - \Delta(u^2)u = g(x,u), \,\, x \in \mathbb{R}^N,$$ where V is a positive potential and the nonlinearity g(x, t) behaves like t at the origin and like t 3 at infinity. In the proof, we apply a changing of variables besides variational methods. The obtained solutions belong to $${W^{1,2}(\mathbb{R}^N)}$$ .