Inbo Sim

Lyapunov Inequalities for One-Dimensional -Laplacian Problems with a Singular Weight Function

We estimate Lyapunov inequalities for a single equation, a cycled system and a coupled system of one-dimensional -Laplacian problems with weight functions having stronger singularities than .

Three positive solutions for one-dimensional p-Laplacian problem with sign-changing weight

Abstract We show that one-dimensional p -Laplacian with a sign-changing weight which is subject to Dirichlet boundary condition has three positive solutions suggesting suitable conditions on the weight function and nonlinearity. Proofs are mainly based on the directions of a bifurcation.

A-priori bounds and existence for solutions of weighted elliptic equations with a convection term

Abstract We investigate weighted elliptic equations containing a convection term with variable exponents that are subject to Dirichlet or Neumann boundary condition. By employing the De Giorgi iteration and a localization method, we give a-priori bounds for solutions to these problems. The existence of solutions is also established using Brezis’ theorem for pseudomonotone operators.

Existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators

We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniqueness and nonnegativeness of solutions when the principal operator is monotone and the nonlinearity is nonincreasing. Our operator is of the most general form containing all previous ones and we also weaken assumptions on the operator and the nonlinearity to get the above results. Moreover, we do not impose the restricted condition on p(x) and the uniform monotonicity of the operator to show the existence of three distinct solutions.

C1-regularity of solutions for p-Laplacian problems☆

Abstract In this work, we study C 1 -regularity of solutions for one-dimensional p -Laplacian problems and systems with a singular weight which may not be in L 1 . On the basis of the regularity result, we give an example to show the multiplicity of positive (or negative) solutions as well as sign-changing solutions especially when the nonlinear term is p -superlinear.

The exact multiplicity of positive solutions for a class of two-point boundary-value problems with the one-dimensional p -Laplacian

Employing the Kolodner–Coffman method, we show the exact multiplicity of positive solutions for the one-dimensional p -Laplacian that is subject to a Dirichlet boundary condition with a positive convex nonlinearity and an indefinite weight function.

Second-order initial value problems with a singular indefinite weight☆

Abstract We are concerned with the existence and uniqueness for the several types of second-order initial value problems with a singular indefinite weight which are related to singular p -Laplacian eigenvalue problems.

Multiplicity Results of Positive Radial Solutions for -Laplacian Problems in Exterior Domains

We find the second positive radial solution for the following -Laplacian problem: in Ω, as , where , , and . We also give some global existence results with respect to the parameter .

Symmetry-breaking bifurcation for the one-dimensional Hénon equation

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Henon equation u″ + |x|lup = 0,x ∈ (−1, 1),u(−1) = u(1) = 0, where l > 0 and p > 1. Moreover, employing a variant of Rabinowitz’s global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz’s global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz’s global bifurcation) for the problem u″ + |x|l(λ)up = 0,x ∈ (−1, 1),u(−1) = u(1) = 0, where l is a specified continuous parametrization function.