Abstract
In this paper, we deal with the fractional Laplacian equations $$(\mathrm{P})\quad \left \{ \textstyle\begin{array}{@{}l@{\quad}l} (-\Delta)^{s} u = f(x,u), & x\in \Omega, u(x)= 0, & x\in \mathbb{R}^{N}\backslash\Omega, \end{array}\displaystyle \right . $$ where $0< s<1<p<+\infty$ , $N\in\mathbb{N}, N>2s$ , $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary. Under local growth conditions of $f(x,t)$ , infinitely many solutions for problem (P) are obtained via variational methods.
Highlights
1 Introduction and main results In this article we are concerned with the multiplicity of solutions for the following fractional Laplacian equations: (– )su = f (x, u), x ∈, (P) u(x) =, x ∈ RN \, where < s < < p < +∞, N > s, ⊂ RN is an open bounded domain with smooth boundary, f (x, t) is a Carathéodory function defined on × (–δ, δ) for some δ >, and (– )s is known as the fractional Laplacian operator, which may be defined as u(x + y) + u(x – y) – u(x)
The topic of fractional Laplacian operators (– )s and more generally non-local operators is a classical one in harmonic analysis and partial differential equations
Li and Wei Advances in Difference Equations (2016) 2016:244 attention has been focused on the study of them, both for the pure mathematical research and in view of concrete applications
Summary
Let E be a Banach space, we say that a functional ∈ C (E, R) satisfies Palais-Smale condition at the level c ∈ R ((PS)c in short) if any sequence {un} ⊂ E satisfying (un) → c, (un) → as n → ∞, has a convergent subsequence. ( ) There exists a sequence of critical points {uk} satisfying (uk) < for all k and uk → as k → ∞. It is easy to see that f(x, t) = f (x, t) for (x, t) ∈ × [–τ , τ ] and a critical point u ofis a solution of the original problem (P) if and only if |u|∞ ≤ τ. In order to get the weak solutions of the original problem (P), we will prove that the above sequence of critical points {um} forenjoys the following property.
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