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Các mô hình sắc nét của các nghiệm dương cho một số bài toán elliptic bán tuyến tính có trọng số
Tóm tắt
Bài báo này đề cập đến bài toán elliptic bán tuyến tính
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda m(x)u-[a(x)+\varepsilon b(x)]u^p &{}\text { trong } \Omega ,\\ Bu=0 &{}\text { trên } \partial \Omega , \end{array}\right. } \end{aligned}$$
trong đó
$$p>1$$
,
$$\lambda >0$$
,
$$m,a,b\in C({\bar{\Omega }})$$
, với
$$a \gneq 0$$
,
$$b\gneq 0$$
,
$$\Omega $$
là một miền hữu hạn
$$C^{2}$$
trong
$${\mathbb {R}}^N$$
(
$$N\ge 1$$
), B là một toán tử biên hỗn hợp cổ điển tổng quát, và
$$\varepsilon \ge 0$$
. Do đó, a(x) và b(x) có thể biến mất trên một số miền con của
$$\Omega $$
và hàm trọng số m(x) có thể đổi dấu trong
$$\Omega $$
. Qua bài báo này, chúng tôi luôn xem xét các nghiệm cổ điển. Đầu tiên, chúng tôi định hình sự tồn tại của các nghiệm dương cho bài toán này trong trường hợp đặc biệt khi
$$\varepsilon =0$$
. Sau đó, chúng tôi điều tra các mô hình sắc nét của các nghiệm dương khi
$$\varepsilon \downarrow 0$$
và
$$\varepsilon \uparrow \infty $$
. Nghiên cứu của chúng tôi tiết lộ cách mà sự tồn tại của các profile sắc nét được xác định bởi hành vi của b(x).
Từ khóa
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