Abstract
We consider equations involving the one-dimensional p-Laplacian with the Dirichlet boundary conditions. By using time map methods, we show how changes of the sign of lead to multiple positive solutions of the problem for sufficiently large λ. MSC:34B10, 34B18.
Highlights
Let f : [, ∞) → R be continuous and change its sign
(and their finite difference analogues) have been extensively studied since early s. Several different approaches such as variational methods, bifurcation theory, lower and upper solutions method and quadrature arguments have been successfully applied to show the existence of multiple solutions
Where p is the p-Laplace operator for p ∈ (, ∞). They assumed that the nonlinearity f is a continuous function on R, f ( ) ≥, and there exist < a < b < a < b < · · · < bm– < am such that f ≤ on and f ≥ on for every k =, . . . , m
Summary
Let f : [ , ∞) → R be continuous and change its sign. Let be an open subset of RN with smooth boundary ∂. Loc and Schmitt [ ] considered the problem pu + λf (u) = in , u = on ∂ , where p is the p-Laplace operator for p ∈ ( , ∞) They assumed that the nonlinearity f is a continuous function on R, f ( ) ≥ , and there exist < a < b < a < b < · · · < bm– < am. We shall apply the time map method to show how changes of the sign of f (·) lead to multiple positive solutions of Let (λ, u) be a positive solution of the problem u (t) p– u (t) + λf u(t) = , t ∈ ( , ), u( ) = u( ) =
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