AbstractThe aim of this paper is analyzing the positive solutions of the quasilinear problem-(u′/1+(u′)2)′=λa(x)f(u) in (0,1),u′(0)=0,u′(1)=0,-\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,whereλ∈ℝ{\lambda\in\mathbb{R}}is a parameter,a∈L∞(0,1){a\in L^{\infty}(0,1)}changes sign once in(0,1){(0,1)}and satisfies∫01a(x)𝑑x<0{\int_{0}^{1}a(x)\,dx<0}, andf∈𝒞1(ℝ){f\in\mathcal{C}^{1}(\mathbb{R})}is positive and increasing in(0,+∞){(0,+\infty)}with a potential,F(s)=∫0sf(t)𝑑t{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,𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}, bifurcating from(λ,0){(\lambda,0)}at someλ0>0{\lambda_{0}>0}and from(λ,∞){(\lambda,\infty)}at someλ∞>0{\lambda_{\infty}>0}. It also establishes that𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}consists of regular solutions if and only if∫0z(∫xza(t)dt)-1/2dx=+∞ or ∫z1(∫xza(t)dt)-1/2dx=+∞.\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 of𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}become singular as they are sufficiently large if and only if(∫xza(t)dt)-1/2∈L1(0,z) and (∫xza(t)dt)-1/2∈L1(z,1).\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, asλ→λ∞{\lambda\to\lambda_{\infty}}, of the solutions. According to the mutual positions ofλ0{\lambda_{0}}andλ∞{\lambda_{\infty}}, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.