Abstract
We herein discuss the following elliptic equations: M ∫ R N ∫ R N | u ( x ) − u ( y ) | p | x − y | N + p s d x d y ( − Δ ) p s u + V ( x ) | u | p − 2 u = λ f ( x , u ) in R N , where ( − Δ ) p s is the fractional p-Laplacian defined by ( − Δ ) p s u ( x ) = 2 lim ε ↘ 0 ∫ R N \ B ε ( x ) | u ( x ) − u ( y ) | p − 2 ( u ( x ) − u ( y ) ) | x − y | N + p s d y , x ∈ R N . Here, B ε ( x ) : = { y ∈ R N : | x − y | < ε } , V : R N → ( 0 , ∞ ) is a continuous function and f : R N × R → R is the Carathéodory function. Furthermore, M : R 0 + → R + is a Kirchhoff-type function. This study has two aims. One is to study the existence of infinitely many large energy solutions for the above problem via the variational methods. In addition, a major point is to obtain the multiplicity results of the weak solutions for our problem under various assumptions on the Kirchhoff function M and the nonlinear term f. The other is to prove the existence of small energy solutions for our problem, in that the sequence of solutions converges to 0 in the L ∞ -norm.
Highlights
It is well known that the studies in [14,17,19,21,22,26,29,32,33] as well as our first primary result essentially demand some global conditions on f ( x, t) for t, such as oddness and behavior at infinity, for applying the fountain theorem to allow an infinite number of solutions
We show that the functional Eλ satisfies the Cerami condition ((C )c -condition, briefly), i.e., for c ∈ R, any sequence {un } ⊂ W (R N ) such that Eλ → c and ||Eλ0||W ∗ (RN )
This paper is devoted to the study of weak solutions for Kirchhoff–Schrödinger-type equations involving the fractional p-Laplacian
Summary
Significant attention has been focused on the study of fractional-type operators in view of the mathematical theory to some phenomena: the social sciences, quantum mechanics, continuum mechanics, phase transition phenomena, game theory, and Levy processes [1,2,3,4,5]. It is well known that the studies in [14,17,19,21,22,26,29,32,33] as well as our first primary result essentially demand some global conditions on f ( x, t) for t, such as oddness and behavior at infinity, for applying the fountain theorem to allow an infinite number of solutions In contrast to these studies that yield large solutions in that they form an unbounded sequence, by modifying and extending the function f ( x, t) to a adequate function f( x, t), the authors in [27,28,29] investigated the existence of small energy solutions to equations of the elliptic type. L∞ -bound of a weak solution and the modified functional method
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