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

Abstract

Abstract The aim of this paper is analyzing the positive solutions of the quasilinear problem - ( u ′ / 1 + ( u ′ ) 2 ) ′ = λ ⁢ a ⁢ ( 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, where λ ∈ 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 ∫ 0 1 a ⁢ ( x ) ⁢ 𝑑 x < 0 {\int_{0}^{1}a(x)\,dx<0} , and f ∈ 𝒞 1 ⁢ ( ℝ ) {f\in\mathcal{C}^{1}(\mathbb{R})} is positive and increasing in ( 0 , + ∞ ) {(0,+\infty)} with a potential, F ⁢ ( s ) = ∫ 0 s f ⁢ ( t ) ⁢ 𝑑 t {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, 𝒞 λ 0 + {\mathscr{C}_{\lambda_{0}}^{+}} , bifurcating from ( λ , 0 ) {(\lambda,0)} at some λ 0 > 0 {\lambda_{0}>0} and from ( λ , ∞ ) {(\lambda,\infty)} at some λ ∞ > 0 {\lambda_{\infty}>0} . It also establishes that 𝒞 λ 0 + {\mathscr{C}_{\lambda_{0}}^{+}} consists of regular solutions if and only if ∫ 0 z ( ∫ x z a ( t ) d t ) - 1 / 2 d x = + ∞   or   ∫ z 1 ( ∫ x z a ( t ) d t ) - 1 / 2 d x = + ∞ . \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 𝒞 λ 0 + {\mathscr{C}_{\lambda_{0}}^{+}} become singular as they are sufficiently large if and only if ( ∫ x z a ( t ) d t ) - 1 / 2 ∈ L 1 ( 0 , z )   and   ( ∫ x z a ( t ) d t ) - 1 / 2 ∈ L 1 ( 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, as λ → λ ∞ {\lambda\to\lambda_{\infty}} , of the solutions. According to the mutual positions of λ 0 {\lambda_{0}} and λ ∞ {\lambda_{\infty}} , as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.