Soliton solutions for quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$

Abstract

We study the existence of positive solutions to the quasilinear elliptic problemwhere g has superlinear growth at infinity without any restriction from above on its growth.Mountain pass in a suitable Orlicz space is employed to establish this result. These equations contain strongly singular nonlinearities which include derivatives of the second order which make the situation more complicated. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schrödinger equations. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

Bound states to critical quasilinear Schrödinger equations

“…It seems that the existence of solutions for critical case was first studied by Moameni in [30] when the potential function V satisfies some geometry conditions, and it was established the existence of multiple solutions in [31] by the fibering method. Recently, doÓ et al [20] …”

Quasilinear asymptotically periodic Schrödinger equations with critical growth

“…A problem of type (1.2) was studied by Moameni [20], for N = 2, with V and g being two continuous 1-periodic functions and g with critical exponential growth. For N ≥ 3, Moameni [19] established the existence of a nonnegative solution for the critical exponent case, when the potential function V is radially symmetrical and satisfies some geometric condition other than periodic one. Furthermore, an Orlicz space framework was used.…”

A class of quasilinear Schrödinger equations with critical or supercritical exponents

“…Compared with the previous work [21,22], we are allowed p ≥ 2(2 * ). Moreover, we will investigate the influence of the signs of parameters κ on the existence of nontrivial solutions.…”

“…Later, B. Silva and G.F. Vieira in [22] established the existence of solutions for asymptotically periodic quasilinear Schrödinger equations (1.1) with the nonlinearity l(u) = |u| p−2 u replaced by K (x)u 2(2 * )−1 + g(x, u). We remark that in [21], the conditions on V (x) imply that (1.5) does not involve critical Sobolev exponent any more since they work on the radially Sobolev space to avoid the loss of compactness, while in [22], this difficulty can be overcome by adding a lower order perturbation term g(x, u). We also refer to [23,24] for more results.…”