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Existence and multiplicity of solutions for fractional Schrödinger-Kirchhoff equations with Trudinger-Moser nonlinearity
ID
Xiang, Mingqi
(
Avtor
),
ID
Zhang, Binlin
(
Avtor
),
ID
Repovš, Dušan
(
Avtor
)
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MD5: C37CB4E3006542F134227A8BC3022F30
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Izvleček
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 ▫$M(0) = 0$▫.
Jezik:
Angleški jezik
Ključne besede:
fractional Schrödinger-Kirchhoff equations
,
Trudinger-Moser inequality
,
existence of solutions
Vrsta gradiva:
Članek v reviji
Tipologija:
1.01 - Izvirni znanstveni članek
Organizacija:
PEF - Pedagoška fakulteta
FMF - Fakulteta za matematiko in fiziko
Leto izida:
2019
Št. strani:
Str. 74-98
Številčenje:
Vol. 186
PID:
20.500.12556/RUL-108783
UDK:
517.95
ISSN pri članku:
0362-546X
DOI:
10.1016/j.na.2018.11.008
COBISS.SI-ID:
18500185
Datum objave v RUL:
24.07.2019
Število ogledov:
1285
Število prenosov:
598
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Objavi na:
Gradivo je del revije
Naslov:
Nonlinear Analysis
Skrajšan naslov:
Nonlinear anal.
Založnik:
Pergamon Press
ISSN:
0362-546X
COBISS.SI-ID:
26027520
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