Existence and multiplicity of solutions for fractional Schrödinger-Kirchhoff equations with Trudinger-Moser nonlinearity

Xiang, Mingqi (author), Zhang, Binlin (author), Repovš, Dušan (author)

We study the existence and multiplicity of solutions for a class of fractional Schrödinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider ▫$$\begin{cases} M(\|u\|^{N/s}) \Big[ (-\Delta)_{N/s}^s u + V(x)|u|^{\frac{N}{s}-1}u \Big]= f(x,u) + \lambda h(x)|u|^{p-2}u & \text{in} \quad \mathbb{R}^N \; , \\ \|u\| = \Big( \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy + \int_{\mathbb{R}^N} V(x) |u|^{N/s}dx \Big)^{s/N} \; , \end{cases}$$▫ where ▫$M \colon [0, \infty] \to [0, \infty)$▫ is a continuous function, ▫$s \to (0,1)$▫, ▫$N \ge 2$▫, ▫$\lambda > 0$▫ is a parameter, ▫$1 < p < \infty$▫, ▫$(-\Delta)_{N/s}^s$▫ is the fractional ▫$N/s$▫-Laplacian, ▫$V \colon \mathbb{R} \to (0, \infty)$▫ is a continuous function, ▫$f \colon \mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$▫ is a continuous function, and ▫$h \colon \mathbb{R} \to [0, \infty)$▫ is a measurable function. First, using the mountain pass theorem, a nonnegative solution is obtained when ▫$f$▫ satisfies exponential growth conditions and ▫$\lambda$▫ is large enough, and we prove that the solution converges to zero in ▫$W_V^{s, N/s} (\mathbb{R}^N)$▫ as ▫$\lambda \to \infty$▫. Then, using the Ekeland variational principle, a nonnegative nontrivial solution is obtained when ▫$\lambda$▫ is small enough, and we show that the solution converges to zero in ▫$W_V^{s, N/s} (\mathbb{R}^N)$▫ as ▫$\lambda \to 0$▫. Furthermore, using the genus theory, infinitely many solutions are obtained when ▫$M$▫ is a special function and ▫$\lambda$▫ is small enough. We note that our paper covers a novel feature of Kirchhoff problems, that is, the Kirchhoff function ▫$M(0) = 0$▫.

• Language: English
• Keywords: fractional Schrödinger-Kirchhoff equations, Trudinger-Moser inequality, existence of solutions
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2019
• Number of pages: Str. 74-98
• Numbering: Vol. 186
• PID: 20.500.12556/RUL-108783
• UDC: 517.95
• ISSN on article: 0362-546X
• DOI: 10.1016/j.na.2018.11.008
• COBISS.SI-ID: 18500185
• Publication date in RUL: 24.07.2019

Record is a part of a journal

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