Abstract
We consider the following nonlinear biharmonic equations: \t\t\tΔ2u−Δu+Vλ(x)u=f(x,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}$$ \\Delta^{2} u-\\Delta u+ V_{\\lambda }(x)u=f(x,u),\\quad \\text{in } \\mathbb{R}^{N}, $$\\end{document} where V_{lambda }(x) is allowed to be sign-changing and f is an indefinite function. Under some suitable assumptions, the existence of nontrivial solutions and the high energy solutions are obtained by using variational methods. Moreover, the phenomenon of concentration of solutions is explored. The results extend the main conclusions in recent literature.
Highlights
Motivated by [16], Ye and Tang [24] studied problem (1.1) under the following more general case imposed on the potential V (x): (V1 ) V (x) ≥ 0 for all x ∈ RN ; (V2∗) There exists b > 0 such that the set {x ∈ RN : V (x) ≤ b} has finite measure
Introduction and main resultsThis paper concerns the existence results and the phenomenon of concentration of solutions for the following biharmonic equation: 2u – u + Vλ(x)u = f (x, u), in RN, (1.1)where 2 = ( ) is the biharmonic operator, f is an indefinite function and the potential Vλ(x) = λV +(x) – V –(x) with V + having a bounded potential well whose depth is controlled by λ and V –(x) ≥ 0 for all x ∈ RN
In the last two decades, the existence of bound states, ground states, semi-classical states, and infinitely many nontrivial solutions of biharmonic equations have been widely discussed under various conditions no matter on a bounded domain or on the whole space
Summary
Motivated by [16], Ye and Tang [24] studied problem (1.1) under the following more general case imposed on the potential V (x): (V1 ) V (x) ≥ 0 for all x ∈ RN ; (V2∗) There exists b > 0 such that the set {x ∈ RN : V (x) ≤ b} has finite measure. This is an interesting question, and we mainly consider the following two problems in the present paper: (i) The existence results of problem (1.1) when f is indefinite and satisfies the superquadratic linear conditions; (ii) The phenomenon of concentration of nontrivial solutions.
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