A three solution theorem for singular nonlinear elliptic boundary value problems

作者：Dhanya R; Ko Eunkyung; Shivaji R^*

来源：Journal of Mathematical Analysis and Applications, 2015, 424(1): 598-612.

DOI：10.1016/j.jmaa.2014.11.012

• Three solutions
• Singular nonlinear elliptic BVPs

We establish a three solution theorem for singular elliptic boundary value problems of the form -Delta u = f(u)/u(beta) in Omega, u = 0 on partial derivative Omega where Omega is a bounded domain in R-N, N >= 1 with a smooth boundary partial derivative Omega. Here f : [0, infinity) -> [0, infinity) is a C-1 function in [0, infinity) with f(0) > 0 and beta is an element of(0,1). In particular if there exist two pairs of sub-supersolutions (psi(1), phi(1)) and (psi(2), phi(2)) where psi(1) <= psi(2) <= phi(1,) psi(1) <= phi(2) <= phi(1) with psi(2) not less than or equal to phi(2), and psi(2), phi(2) are strict sub and supersolutions, then we establish existence of three solutions u(1), u(2) and u(3) for the above boundary value problem such that u(1) is an element of[psi(1),phi(2)], u(2) is an element of[psi(2), phi(1).] and u(3) is an element of[psi(1), phi(2)]\([psi(1), phi(2)]boolean OR[psi(2), phi(2)]). Our results extend the work in [1] (and also in [17]) where such multiplicity results were discussed in the non-singular case beta = 0. Further our results strengthen the multiplicity results in [12] and [13] for such singular problems (beta not equal 0), namely, establish an additional third solution. We also establish a sub-super solution theorem for infinite semipositone problems (i.e. the case when f(0) < 0).

• 出版日期2015-4-1