Abstract
We are concerned with the following elliptic equations: ( − Δ ) p , K s u + V ( x ) | u | p − 2 u = λ f ( x , u ) in R N , where ( − Δ ) p , K s is the nonlocal integrodifferential equation with 0 < s < 1 < p < + ∞ , s p < N the potential function V : R N → ( 0 , ∞ ) is continuous, and f : R N × R → R satisfies a Carathéodory condition. The present paper is devoted to the study of the L ∞ -bound of solutions to the above problem by employing De Giorgi’s iteration method and the localization method. Using this, we provide a sequence of infinitely many small-energy solutions whose L ∞ -norms converge to zero. The main tools were the modified functional method and the dual version of the fountain theorem, which is a generalization of the symmetric mountain-pass theorem.
Highlights
In recent years, the study of fractional and nonlocal problems of the elliptic type has received enormous attention because the interest in such operators has sustainedly increased within the framework of mathematical theory to confirm some phenomena such as fractional quantum mechanics, material sciences, continuum mechanics, phase-transition phenomena, image processes, the thin-obstacle problem, game theory, and Lévy processes.The fractional Schrödinger equation, which was introduced by Laskin [5], has especially received considerable attention in recent years.Stimulated by the large interest in the current literature and taking advantage of variational methods, we investigate the existence of weak solutions for nonlocal equations involving the fractional p-Laplacian
We indicate that u ∈ X (R N ) is a weak solution to the problem in Equation (1) if
As mentioned in the Introduction, even if the dual-fountain theorem plays a decisive role in obtaining the existence of multiple small-energy solutions to elliptic equations of the variational type, the boundedness of these solutions is not ensured by this theorem
Summary
The study of fractional and nonlocal problems of the elliptic type has received enormous attention because the interest in such operators has sustainedly increased within the framework of mathematical theory to confirm some phenomena such as fractional quantum mechanics, material sciences, continuum mechanics, phase-transition phenomena, image processes, the thin-obstacle problem, game theory, and Lévy processes (see [1,2,3,4,5,6,7] and the references therein). Many researchers applied this theorem to the various problems involving p- or p( x )-Laplacian (see [8,9,28,29,30,31,32,33,34,35,36] and the references therein) Such a multiplicity result that utilizes the dual-fountain theorem to derive the existence of infinitely many small-energy solutions for nonlinear elliptic equations of the variational type can be found in [9,29] (and the references therein). The authors of [31] investigated the L∞ bound of small-energy solutions to Kirchhoff–Schrödinger-type equations involving the fractional p-Laplacian by applying the modified functional method and a regularity-type result inspired by the work of P. We provide a sequence of infinitely many small-energy solutions whose L∞ -norms converge to zero
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