Quasilinear asymptotically periodic elliptic equations with critical growth

Quasilinear asymptotically periodic Schrödinger equations with critical growth

“…To overcome this difficulty, we shall use two versions of the Mountain Pass Theorem. Next, we state the first version of this theorem (see also [11,15,25]). …”

“…We will also need to establish a local version of Theorem 3, which has been proved in [15] (see also [14]). …”

“…We also use a change of variable to reformulate the problem obtaining a semilinear problem which has an associated functional well-defined in the Sobolev space H 1 (R N ) and satisfies the geometric hypotheses of the Mountain Pass Theorem (see [2]). After that, we adapt the argument employed in [15], supposing by contradiction that the only possible solution for the Problem 1.3 is the trivial solution. Considering the functional associated with the modified problem, we use a version of the Mountain Pass Theorem without compactness condition [12] to get a Cerami sequence associated with the minimax level.…”

Existence of solutions for fractional Schrodinger equation with asymptotica- lly periodic terms

“…Motivated by above results, in this paper we study non-trivial solution and ground state solution to problem (1.1) under asymptotically periodic case of V and f at infinity. In the context about asymptotic periodic, we refer the reader to [1,12,23,24,31,32].…”

Solutions for quasilinear Schrödinger systems with critical exponents

“…Due to its significant applications in mathematical physics (see [1][2][3] and the references therein), this type of quasilinear equations have been widely studied in the literature and lots of results are achieved, see for example, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein. On the other hand, the quasilinear Schrödinger system like (1.1) is also a challenging physical problem that has been studied in recent years, which describes the interaction between two electrons and includes the MNLS.…”