Mathematics and System Engineering Faculty Publications

Document Type

Article

Publication Title

Open Mathematics

Abstract

We consider quasilinear elliptic problems of the form - div (φ (| ∇ u |) ∇ u) + V (x) φ (| u |) u = f (u), u ∈ W 1, φ (R N), where φ and f satisfy suitable conditions. The positive potential V ∈ C (R N) exhibits a finite or infinite potential well in the sense that V (x) V\left(x) tends to its supremum V ∞ ≤ + ∞ as | x | → ∞. Nontrivial solutions are obtained by variational methods. When V ∞ = + ∞, a compact embedding from a suitable subspace of W 1, φ (R N)) into L φ (R N)) is established, which enables us to get infinitely many solutions for the case that f f is odd. For the case that V (x) = λ a (x) + 1 exhibits a steep potential well controlled by a positive parameter λ \lambda, we get nontrivial solutions for large λ \lambda. © 2021 Shibo Liu, published by De Gruyter 2021.

First Page

971

DOI

10.1515/math-2021-0053

Publication Date

2021

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