Quasilinear asymptotically periodic Schrödinger equations with critical growth

Abstract

It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in R N with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H 1 (R N ) and satisfy the geometric hypotheses of the Mountain Pass Theorem. The Concentration-Compactness Principle and a comparison argument allow to verify that the problem possesses a nontrivial solution.

Bound states to critical quasilinear Schrödinger equations

“…associated to the problem Under the conditions (g 1 ) − (g 4 ), similar to the arguments of [41], (3.1) possesses a nontrivial solution ω at the level…”

Positive solutions of quasilinear Schrödinger equations with critical growth

“…In [5], by applying a change of variables introduced in [8,9], the existence of positive solutions of (1.3) with 3 < p < 22 * − 1 was proved. By the same change of variables as in [5], critical quasilinear equations like (1.3) with periodic potential were studied by Silva and Vieira in [10]. Under the assumption p > max{ N+6 N−2 , 3}, Liu, Liu and Wang [11] established a existence result of solutions of (1.3) with a bounded potential via the Nehari method.…”

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

“…The same method was also used in [3], but the usual Sobolev space H 1 (R N ) framework was used as the working space. We refer the reader to [5,6,9,20,21] for more results. Usually, the authors consider the case that the function g(x, t) is sublinear at the origin and superlinear at infinity.…”