Abstract
The purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation: 0.1Δ2u−(a+b∫RN|∇u|2dx)Δu+V(x)u=K(x)f(u)in 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}$$ \\Delta ^{2}u- \\biggl(a+ b \\int _{\\mathbb{R}^{N}} \\vert \\nabla u \\vert ^{2}\\,dx \\biggr) \\Delta u+V(x)u=K(x)f(u) \\quad\\text{in } \\mathbb{R}^{N}, $$\\end{document} where 5leq Nleq 7, Delta ^{2} denotes the biharmonic operator, K(x), V(x) are positive continuous functions which vanish at infinity, and f(u) is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally, f is an odd function, we obtain that equation has infinitely many nontrivial solutions.
Highlights
Guan and Zhang Journal of Inequalities and ApplicationsAs pointed out in the article, since the nonlocal term ( RN |∇u|2 dx) u is involved, there is an essential difference between problem (1.1) and problem (1.2) when we discussed the existence of sign-changing solutions, see [6,7,8, 10, 25, 37, 61]
Introduction and main resultsThis article is concerned with the following fourth-order Kirchhoff-type equation: 2u – a + b |∇u|2 dx u + V (x)u = K (x)f (u), x ∈ RN, (1.1)where 5 ≤ N ≤ 7, 2 denotes the biharmonic operator, and a, b are positive constants
As pointed out in the article, since the nonlocal term ( RN |∇u|2 dx) u is involved, there is an essential difference between problem (1.1) and problem (1.2) when we discussed the existence of sign-changing solutions, see [6,7,8, 10, 25, 37, 61]
Summary
As pointed out in the article, since the nonlocal term ( RN |∇u|2 dx) u is involved, there is an essential difference between problem (1.1) and problem (1.2) when we discussed the existence of sign-changing solutions, see [6,7,8, 10, 25, 37, 61]. It is noticed that there are some interesting results, for example, [11, 14, 23, 24, 30, 31, 34, 35, 38,39,40,41,42,43, 47, 56, 60], which considered sign-changing solutions for other nonlocal problems.
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