Regular Versus Singular Solutions in a Quasilinear Indefinite Problem with an Asymptotically Linear Potential

The aim of this paper is analyzing the positive solutions of the quasilinear problem -(u'/root 1+(u')(2))' = lambda a(x)f(u) in (0, 1), u'(0) = 0, u'(1) = 0, where lambda is an element of R is a parameter, a is an element of L-infinity(0, 1) changes sign once in (0, 1) and satisfies integral(1)(0) a(x) dx < 0, f is an element of C-1 (R) is positive and increasing in (0, +infinity) with a potential, F(s) = integral(s)(0) f(t) dt, quadratic at zero and linear at +infinity. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions, C-lambda 0(+), bifurcating from (lambda, 0) at some lambda(0) > 0 and from (lambda, infinity) at some lambda(infinity )> 0. It also establishes that C-lambda 0(+), consists of regular solutions if and only if integral(z )(0)(integral(z)(x) a(t) dt)(-1/2) dx = +infinity or integral(1)(z) (integral(z)(x) a(t) dt )(-1/2) dx = +infinity. Equivalently, the small positive regular solutions of C-lambda 0(+), become singular as they are sufficiently large if and only if (integral(z)(x) a(t) dt)(-1/2) is an element of L-1(0, z) and (integral(z)(x) a(t) dc)(-1/2) is an element of L-1(z, 1). This is achieved by providing a very sharp description of the asymptotic profile, as lambda -> lambda(infinity), of the solutions. According to the mutual positions of lambda(0) and lambda(infinity), as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.