Asymptotic behavior of solution curves of nonlocal one-dimensional elliptic equations

Abstract

AbstractWe study the one-dimensional nonlocal elliptic equation $$\begin{aligned}& -A\bigl( \bigl\Vert u' \bigr\Vert _{p}^{p} \bigr) u''(x) = \lambda B\bigl( \bigl\Vert u' \bigr\Vert _{q}^{q} \bigr)u(x)^{r} , \quad x \in I:= (0,1), u(x) > 0, x \in I, \\& u(0) = u(1) = 0, \end{aligned}$$ − A ( ∥ u ′ ∥ p p ) u ″ ( x ) = λ B ( ∥ u ′ ∥ q q ) u ( x ) r , x ∈ I : = ( 0 , 1 ) , u ( x ) > 0 , x ∈ I , u ( 0 ) = u ( 1 ) = 0 , where $A = A(y)$ A = A ( y ) and $B = B(y)$ B = B ( y ) are continuous functions, satisfying $A(y) > 0$ A ( y ) > 0 , $B(y) > 0$ B ( y ) > 0 for $y > 0$ y > 0 , $p \ge 1$ p ≥ 1 , $q \ge 1$ q ≥ 1 , and $r > 1$ r > 1 are given constants, and $\lambda > 0$ λ > 0 is a bifurcation parameter. We establish the global behavior of solution curves and precise asymptotic formulas for $u_{\lambda}(x)$ u λ ( x ) as $\lambda \to \infty $ λ → ∞ .