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 egin{equation*} label{P} -(u'/sqrt{1+(u')^2})' = lambda a(x) f(u) ; ; ext{in } (0,1), u'(0)=0,;u'(1)=0, end{equation*} where $lambdain R$ is a parameter, $ain L^infty(0,1)$ changes sign once in $(0,1)$ and satisfies $int_0^1a(x),dx<0$, and $f in mc{C}^1(R)$ is positive and increasing in $(0,+infty)$ with a potential, $F(s)=int_0^{s}f(t),dt$, quadratic at zero and linear at $+infty$. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions, $mathscr{C}_{l_0}^+$, bifurcating from $(l,0)$ at some $l_0>0$ and from $(l,infty)$ at some $l_infty>0$. It also establishes that $mathscr{C}_{l_0}^+$ consists of regular solutions, if, and only if, centerline{ $ int_0^z left( int_x^z a(t),dt ight)^{-rac{1}{2}}dx =+infty, quad hbox{or}quad int_z^1 left( int_x^z a(t),dt ight)^{-rac{1}{2}}dx =+infty. $} Equivalently, the small positive regular solutions of $mathscr{C}_{l_0}^+$ become singular as they are sufficiently large if, and only if, centerline{ $ left( int_x^z a(t),dt ight)^{-rac{1}{2}}in L^1(0,z) quad ext{and} quad left( int_x^z a(t),dt ight)^{-rac{1}{2}}in L^1(z,1). $} This is achieved by providing a very sharp description of the asymptotic profile, as $l ol_infty$, of the solutions. According to the mutual positions of $l_0$ and $l_infty$, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.