Positive solution for quasilinear Schrödinger equations with a parameter

Abstract

In this paper, we study the following quasilinear Schrödinger equations of the formand g ∈ C(R N × R, R). By using a change of variables, we get new equations, whose respective associated functionals are well defined in H 1 (R N ) and satisfy the geometric hypotheses of the mountain pass theorem. Using the special techniques, the existence of positive solutions is studied.

Generalized quasilinear Schrödinger equations with concave functions <inline-formula><tex-math id="M1">\begin{document}$ l(s^2) $\end{document}</tex-math></inline-formula>

“…Here, the condition (q 2 ) is generalized from the positive mass condition −∞ < lim inf u < −γ < 0, which was introduced in [5] when the authors investigated the semilinear equation − u = s(u). We remark that by the positive mass condition, if g(s) is decreasing and g(0) = ∞ such as the corresponding l(s) = s α 2 with α < 1, there must exists the term W u 2α−1 in the equation (16). If not, the condition (q 2 ) will be invalid.…”

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

“…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.…”

“…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 $$$ h(s)&amp;#x0003D;{\left(1&amp;#x0002B;s\right)}&amp;#x0005E;{\frac{\alpha }{2}} $$$, by using a change of variables and mountain pass theorem, Li [20] obtained the existence of positive solution for Equation () with $$$ 1\le \alpha \le 2 $$$.…”

Existence of ground state solutions for generalized quasilinear Schr\”{o}dinger equations with asymptotically periodic potential

“…It is worth pointing out that there is no result for equation (1.5) when the potential is asymptotically periodic. For the periodic potential, there are references [7,8], they discussed the following equation…”

“…Jalilian [7] considered equation (1.10) with 1.36 < α ≤ 2 and proved that (1.10) had infinitely many geometrically distinct solutions. Li [8] proved the existence of a ground state solution for equation (1.10) with 1 ≤ α ≤ 2 if g satisfies some conditions and…”