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

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

Abstract

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 ▫$M(0) = 0$▫.

Keywords

fractional Schrödinger-Kirchhoff equations;Trudinger-Moser inequality;existence of solutions;

Data

Language:	English
Year of publishing:	2019
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.95
COBISS:	18500185
ISSN:	0362-546X
Views:	655
Downloads:	429
Average score:	0 (0 votes)
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Other data

Type (COBISS):	Article
Pages:	str. 74-98
Issue:	ǂVol. ǂ186
Chronology:	Sep. 2019
DOI:	10.1016/j.na.2018.11.008
ID:	11193525