Bifurcation and Nodal Solutions for the Half-Linear Problems with Nonasymptotic Nonlinearity at 0 and ∞

Abstract EN

We study the existence of nodal solutions for the following problem: - x ″ = α x + + β x - + r a ( t ) f ( x ) , 0 < t < 1 , x ( 0 ) = x ( 1 ) = 0 , where r ≠ 0 is a parameter, a ( t ) ∈ C ( [ 0,1 ] , ( 0 , ∞ ) ) with a ( t ) ≢ 0 on any subinterval of [ 0,1 ] , x + = m a x { x , 0 } , x - = - m i n { x , 0 } , and α , β ∈ C [ 0,1 ] ; f ∈ C ( R , R ) , s f ( s ) > 0 for s ≠ 0 , and f 0 , f ∞ ∉ ( 0 , ∞ ) , where f 0 = l i m | s | → 0 f ( s ) / s and f ∞ = l i m | s | → + ∞ f ( s ) / s .

We use bifurcation techniques to prove our main results.