Abstract
We consider a class of generalized quasilinear Schrödinger equations −div(l2(u)∇u)+l(u)l′(u)|∇u|2+V(x)u=f(u),x∈RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -\\operatorname{div}\\bigl(l^{2}(u)\\nabla u\\bigr)+l(u)l'(u) \\vert \\nabla u \\vert ^{2}+V(x)u= f(u),\\quad x\\in \\mathbb{R}^{N}, $$\\end{document} where l(t): mathbb{R}tomathbb{R}^{+} is a nondecreasing function with respect to |t|, the potential function V is allowed to be sign-changing so that the Schrödinger operator -Delta+V possesses a finite-dimensional negative space. We obtain existence and multiplicity results for the problem via the Symmetric Mountain Pass Theorem and Morse theory.
Highlights
1 Introduction In our article, we study the generalized quasilinear Schrödinger problem as follows:
Our results extend and modify those obtained by S
Inspired by [11], we present our hypotheses on the potential V and the nonlinearity f : (V1) V ∈ C(RN, R) and infx∈RN V (x) > –∞; (V2) μ(V –1(–∞, M]) < ∞ for all M > 0, where μ is the Lebesgue measure; (f1) f ∈ C(R, R) and there exist C1, C2 > 0 such that for all t ∈ R, p ∈ (2, 2∗), f (t) ≤ C1l(t) L(t) + C2l(t) L(t) p–1; (f2) there exists μ > 2 such that for t = 0, 0 < μl(t)F(t) ≤ L(t)f (t); (f3) f (t) = o(t) as t → 0; (f4) f (t) = –f (–t)
Summary
– div(l2(u)∇u) + l(u)l (u)|∇u|2 + V(x)u = f (u), x ∈ RN, where l(t) : R → R+ is a nondecreasing function with respect to |t|, the potential function V is allowed to be sign-changing so that the Schrödinger operator – + V possesses a finite-dimensional negative space. We obtain existence and multiplicity results for the problem via the Symmetric Mountain Pass Theorem and Morse theory. MSC: 35J60; 35J20 Keywords: Quasilinear Schrödinger equations; Sign-changing potentials; Symmetric Mountain Pass Theorem; Morse theory
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