Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

Abstract

In this paper we prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growthvia variational methods, where λ ≥ 0, 0 < α < 1/2, 2 < q < 2 * . It is interesting that we do not need to add a weight function to control |u| q−2 u.

Existence of positive solutions for a class of quasilinear Schrödinger equation with critical exponent

“…in Li and Wu [17] via a perturbation method when 𝛼 ≥ 3 4 and in Wu [18] via a dual approach when 1 2 < 𝛼 ≤ 1. Li [19] considered the existence of positive solutions of problem (1.1) via variational methods where 0 < 𝛼 < 1 2 and the nonlinear term is at critical growth, more precisely, 𝜌(|u| 2 )u = |u| q−2 u + |u| 2 * −2 u, where 2 ≤ q < 2 * . For the case h(s) = (1 + s) 𝛼 2 , by using a change of variables and mountain pass theorem, Li [20] obtained the existence of positive solution for Equation (1.3) with 1 ≤ 𝛼 ≤ 2.…”

“…Specifically, Chen et al [27] studied the existence of positive solutions for two nonlinear terms by using the mountain pass theorem and Moser iterative method. Motivated by earlier studies [19,21], the main purpose of this paper is to consider the existence of positive solutions for (1.1) with 𝜅 > 0 and 3 4 < 𝛼 ≤ 4 3 . Our main results supplement the results in Alves et al [21] considering problem (1.1) with 𝛼 = 1 and subcritical nonlinearity and in Li [19] considering problem (1.1) with 𝜅 < 0 and 0 < 𝛼 < 1 2 .…”

“…Wu studied the existence of positive solutions, negative solutions, and sequence of high‐energy solutions for the following problem $$$$ -\Delta u&amp;#x0002B;V(x)u-\Delta \left({\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;{2\alpha}\right){\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;{2\alpha -2}u&amp;#x0003D;g\left(x,u\right),x\in {\mathrm{\mathbb{R}}}&amp;#x0005E;N $$$$ in Li and Wu [17] via a perturbation method when $$$ \alpha \ge \frac{3}{4} $$$ and in Wu [18] via a dual approach when $$$ \frac{1}{2}&lt;\alpha \le 1 $$$. Li [19] considered the existence of positive solutions of problem () via variational methods where $$$ 0&lt;\alpha &lt;\frac{1}{2} $$$ and the nonlinear term is at critical growth, more precisely, $$$ \rho \left({\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;2\right)u&amp;#x0003D;{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;{q-2}u&amp;#x0002B;{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;{2&amp;#x0005E;{\ast }-2}u $$$, where $$$ 2\le q&lt;{2}&amp;#x0005E;{\ast } $$$. For the case …”

Concentration behavior of ground states for a generalized quasilinear Choquard equation

“…The quasilinear Schrödinger Equation (1) is derived as models of several physical phenomenas, see, for example, previous studies. [1][2][3][4] To our best knowledge, the first work to study Equation (1) on mathematics can date back to de Bouard et al 5 Ever since then, there are many works focus on the existence, multiplicity, and concentration of (1), we can refer to previous studies [6][7][8][9][10][11][12][13][14][15] for the subcritical case and to previous studies [16][17][18][19][20][21] for the critical case, respectively. Equation (2) also can be used to describe many physical models.…”

Ground states for a class of quasilinear Schrödinger equations with vanishing potentials

“…On the other hand, problem with q at 2(2 * ) growth was studied by Moameni in [20] and the existence of a nonnegative solution was proved. We refer to [10,15,16,17,18,20,21,28] for the study of standing waves.…”