id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-164	Wang, Lixia; Xiong, Chunlian; Zhao, Pingping	Two solutions for nonhomogeneous Klein-Gordon equations coupled with Born-Infeld type equations	2022	11	.pdf	application/pdf	4711	258	81	[(2ω + φun )φun un − (2ω + φu)φuu](un − u) dx = 2ω ∫ R3 [(φunun − φuu)(un − u) dx+ ∫ R3 [(φ2 un un − φ2 uu)(un − u) dx→ 0 as n→ +∞. Indeed, by the Hölder inequality, the Sobolev inequality, and Propo- sition 2.1, we have∣∣ ∫ R3 (φun − φu)(un − u)un dx ∣∣ ≤ |(φun − φu)(un − u)|2|un|2 ≤ |φun − φu|6|un − u|3|un|2 ≤ C‖φun − φu‖D|un − u|3|un|2, where C is a positive constant. − C7|vn|22 − 2ω2|vn|22 + (1− µ) ∫ R3 h(x)un ‖un‖2 dx ≥ (µ 2 − 1 ) − (C7 + 2ω2)|vn|22 + (1− µ) ∫ R3 h(x)un ‖un‖2 dx → µ 2 − 1 as n→∞. Then we have 0 ≥ 1 2 − 1 µ , which contradicts with µ > 2.	cache/ejde-164.pdf	txt/ejde-164.txt