A note on a nonlinear elliptic problem with a nonlocal coefficient

作者：Naimen Daisuke^*

来源：Journal of Mathematical Analysis and Applications, 2016, 435(2): 1710-1737.

DOI：10.1016/j.jmaa.2015.11.030

• Nonlocal
• Elliptic
• Bifurcation
• Variational method
• Kirchhoff type

In this paper, we investigate a nonlocal and nonlinear elliptic problem,
{-a (integral(Omega) vertical bar del(u)vertical bar(2)dx) Delta u = lambda u + u(P) in Omega,
u = 0 on partial derivative Omega, (P)
where N <= 3, Omega subset of R-N is a bounded domain with smooth boundary partial derivative Omega, a is a nondegenerate continuous function, p > 1 and lambda is an element of R. We show several effects of the nonlocal coefficient a on the structure of the solution set of (P). We first introduce a scaling observation and describe the solution set by using that of an associated semilinear problem. This allows us to get unbounded continua of solutions (lambda, u) of (P). A rich variety of new bifurcation and multiplicity results are observed. We also prove that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way. Lastly, we give some remarks from the variational point of view.

• 出版日期2016-3-15