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-\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0,where {\lambda\in\mathbb{R}} is a parameter, {a\in L^{\infty}(0,1)} changes sign once in {(0,1)} and satisfies {\int_{0}^{1}a(x)\,dx<0}, and {f\in\mathcal{C}^{1}(\mathbb{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}_{\lambda_{0}}^{+}}, bifurcating from {(\lambda,0)} at some {\lambda_{0}>0} and from {(\lambda,\infty)} at some {\lambda_{\infty}>0}. It also establishes that {\mathscr{C}_{\lambda_{0}}^{+}} consists of regular solutions if and only if\int_{0}^{z}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty\quad\text% {or}\quad\int_{z}^{1}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty.Equivalently, the small positive regular solutions of {\mathscr{C}_{\lambda_{0}}^{+}} become singular as they are sufficiently large if and only if \Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(0,z)\quad\text{and}% \quad\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(z,1).This is achieved by providing a very sharp description of the asymptotic profile, as {\lambda\to\lambda_{\infty}}, of the solutions. According to the mutual positions of {\lambda_{0}} and {\lambda_{\infty}}, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.