Positive bounded solutions for nonlinear polyharmonic problems in the unit ball

Abstract

In this paper, we study the existence of positive solutions for the following nonlinear polyharmonic equation (-∆)mu+λf(x, u) = 0 in B; subject to some boundary conditions, where m is a positive integer, λ is a nonnegative constant and B is the unit ball of ℝn (n ≥ 2). Under some appropriate assumptions on the nonnegative nonlinearity term f(x, u) and by using the Schäuder fixed point theorem, the existence of positive solutions is obtained. At last, examples are given for illustration.

Keywords

• green function
• kato class
• positive solution
• nonlinear polyharmonic equation
• schäuder fixed point theorem
• primary 35b09
• 35j40
• secondary 34b27