Infinitely many solutions for a class of elliptic problems involving the fractional Laplacian
| Autor: | Ying-Xin Cui, Liang-Liang Sun, Massimiliano Ferrara, Bin Ge, Ting-Ting Zhao |
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| Rok vydání: | 2018 |
| Předmět: |
Class (set theory)
Algebra and Number Theory Sublinear function Applied Mathematics 010102 general mathematics 01 natural sciences 010101 applied mathematics Combinatorics Computational Mathematics Mountain pass theorem Geometry and Topology 0101 mathematics Fractional Laplacian Analysis Nonlinear operators Mathematics |
| Zdroj: | Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. 113:657-673 |
| ISSN: | 1579-1505 1578-7303 |
| DOI: | 10.1007/s13398-018-0498-8 |
| Popis: | In this paper, we study the existence of nontrivial solution to a quasi-linear problem $$\begin{aligned} (-\Delta )_{p}^{s} u+ V(x)|u|^{p-2}u=Q(x)f(x,u),\quad x\in {\mathbb {R}}^{N}, \end{aligned}$$ where $$\begin{aligned} (-\Delta )_{p}^{s} u(x)=2\lim \limits _{\epsilon \rightarrow 0}\int _{{\mathbb {R}}^N \backslash B_{\varepsilon }(x)} \frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))}{| x-y | ^{N+sp}}dy,\quad x\in {\mathbb {R}}^N \end{aligned}$$ is a nonlocal and nonlinear operator and $$ p\in (1,\infty )$$ , $$ s \in (0,1) $$ . We study two cases: if f(x, u) is sublinear, then we get infinitely many solutions for (P) by Clark’s theorem; if f(x, u) is superlinear, we obtain infinitely many solutions of the problem (P) by symmetric mountain pass theorem. |
| Databáze: | OpenAIRE |
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