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Existence and multiplicity of solutions for fractional SchrödingerKirchhoff equations with TrudingerMoser nonlinearity
Xiang, Mingqi
(
Author
),
Zhang, Binlin
(
Author
),
Repovš, Dušan
(
Author
)
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Abstract
We study the existence and multiplicity of solutions for a class of fractional SchrödingerKirchhoff type equations with the TrudingerMoser nonlinearity. More precisely, we consider ▫$$\begin{cases} M(\u\^{N/s}) \Big[ (\Delta)_{N/s}^s u + V(x)u^{\frac{N}{s}1}u \Big]= f(x,u) + \lambda h(x)u^{p2}u & \text{in} \quad \mathbb{R}^N \; , \\ \u\ = \Big( \iint_{\mathbb{R}^{2N}} \frac{u(x)u(y)^{N/s}}{xy^{2N}}dxdy + \int_{\mathbb{R}^N} V(x) u^{N/s}dx \Big)^{s/N} \; , \end{cases}$$▫ 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 SchrödingerKirchhoff equations
,
TrudingerMoser inequality
,
existence of solutions
Work type:
Article (dk_c)
Tipology:
1.01  Original Scientific Article
Organization:
PEF  Faculty of Education
Year:
2019
Number of pages:
Str. 7498
Numbering:
Vol. 186
UDC:
517.95
ISSN on article:
0362546X
DOI:
10.1016/j.na.2018.11.008
COBISS.SIID:
18500185
Views:
420
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304
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Record is a part of a journal
Title:
Nonlinear Analysis
Shortened title:
Nonlinear anal.
Publisher:
Pergamon Press
ISSN:
0362546X
COBISS.SIID:
26027520
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