Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials

Zhang, Xia (author), Zhang, Binlin (author), Repovš, Dušan (author)

This paper is concerned with the following fractional Schrödinger equations involving critical exponents: ▫$$(-\Delta)^\alpha u + V(x)u = k(x)f(u) + \lambda|u|^{2_\alpha^\ast-2}u \quad \text{in} \; \mathbb{R}^N,$$▫ where ▫$(-\Delta)^\alpha$▫ is the fractional Laplacian operator with ▫$\alpha \in (0,1)$▫, ▫$N \ge 2$▫, ▫$\lambda$▫ is a positive real parameter and ▫$2_\alpha^\ast = 2N/(N-2\alpha)$▫ is the critical Sobolev exponent, ▫$V(x)$▫ and ▫$k(x)$▫ are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti-Rabinowitz condition on the subcritical nonlinearity.

• Language: English
• Keywords: fractional Schrödinger equations, critical Sobolev exponent, Ambrosetti-Rabinowitz condition, concentration compactness principle
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2016
• Number of pages: Str. 48-68
• Numbering: Vol. 142
• PID: 20.500.12556/RUL-111174
• UDC: 517.95
• ISSN on article: 0362-546X
• DOI: http://dx.doi.org/10.1016/j.na.2016.04.012
• COBISS.SI-ID: 17674585
• Publication date in RUL: 25.09.2019

Record is a part of a journal

• Title: Nonlinear Analysis
• Shortened title: Nonlinear anal.
• Publisher: Pergamon Press
• ISSN: 0362-546X
• COBISS.SI-ID: 26027520