Abstract
We study the following generalized quasilinear Schrödinger equations with critical growth -divg2u∇u+gug′u∇u|2+Vxu=λfx,u+guGu|2⁎-2Gu,x∈RN, where λ>0, N≥3, g(s):R→R+ is a C1 even function, g(0)=1, and g′(s)≥0 for all s≥0, where G(u)≔∫0ug(t)dt. Under some suitable conditions, we prove that the equation has a nontrivial solution by variational method.
Highlights
We study the following generalized quasilinear Schrodinger equations with critical growth −div(g2(u)∇u) + g(u)g(u)|∇u|2 + V(x)u = λf(x, u) + g(u)|G(u)|2∗−2G(u), x ∈ RN, where λ > 0, N ≥ 3, g(s) : R → R+ is a C1 even function, g(0) = 1, and g(s) ≥ 0 for all s ≥ 0, where G(u) fl ∫0u g(t)dt
We prove that the equation has a nontrivial solution by variational method
The equations are related to the existence of solitary wave solutions for quasilinear Schrodinger equations izt = −Δz + W (x) z − k (x, |z|) z (2)
Summary
We study the following generalized quasilinear Schrodinger equations with critical growth −div(g2(u)∇u) + g(u)g(u)|∇u|2 + V(x)u = λf(x, u) + g(u)|G(u)|2∗−2G(u), x ∈ RN, where λ > 0, N ≥ 3, g(s) : R → R+ is a C1 even function, g(0) = 1, and g(s) ≥ 0 for all s ≥ 0, where G(u) fl ∫0u g(t)dt. Consider the following generalized quasilinear Schrodinger equations with critical growth: The research on the existence of solitary wave solutions of Schrodinger equations (2) is for some given special function l(s).
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