Soliton solutions for quasilinear Schrödinger equations with critical growth. (English) Zbl 1182.35205
Summary: We establish the existence of standing wave solutions for quasilinear Schrödinger equations involving critical growth. By using a change of variables, the quasilinear equations are reduced to semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution \(v\). In the proof that \(v\) is nontrivial, the main tool is the concentration-compactness principle due to P.L. Lions together with some classical arguments used by H. Brézis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437–477 (1983; Zbl 0541.35029)].
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35J60 | Nonlinear elliptic equations |
| 35B33 | Critical exponents in context of PDEs |
| 35C08 | Soliton solutions |
Keywords:
Schrödinger equations; standing wave solutions; variational methods; minimax methods; critical exponentCitations:
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