A priori bounds for positive solutions of Kirchhoff type equations

作者：Dai, Qiuyi; Lan, Enhao; Shi, Feilin^*

来源：Computers & Mathematics with Applications, 2018, 76(6): 1525-1534.

DOI：10.1016/j.camwa.2018.07.004

• A priori estimates
• Kirchhoff type equations
• Positive solutions

Let Omega be a bounded smooth domain in RN. Assume that 0 < alpha < 2*-1/2, a > 0, and b > 0. We consider the following Dirichlet problem of Kirchhoff type equation
{-(a + b parallel to del u parallel to(2 alpha)(2))Delta u = vertical bar u vertical bar(p-1)u + h(x, u, del u) in Omega,
u = 0 on partial derivative Omega (0.1)
with p is an element of (0, 2*) \ {1}. Where 2* = +infinity for N = 2, and 2* = N+2/N-2 for N >= 3. Under suitable conditions of h(x, u, del u) (see (A), (H-1) and (H-2) in Section 3), we get a priori estimates for positive solutions to problem (0.1). By making use of these estimates and the continuous method, we further get some existence results for positive solutions to problem (0.1) when 0 < p < 1, or 2 alpha + 1 < p < 2*. Effects of the term a + b parallel to del u parallel to(2 alpha)(2) on the solution set of problem (0.1) can be seen in an example given in Section 2.

• 出版日期2018-9-15
• 单位湖南师范大学