Abstract
On the solvability of a boundary value problem for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:math>-Laplacian differential equations
Highlights
This paper is devoted to the solvability of the boundary value problem (BVP)(φp(x )) = f (t, x, x ), t ∈ [0, 1], (1.1)x(0) = A, x (1) = B. (1.2)Here φp(s) = s|s|p−2, p > 2, the scalar function f (t, x, y) is defined for (t, x, y) ∈ [0, 1] × Dx × Dy, where the sets Dx, Dy ⊆ R may be bounded, and B ≥ 1
We study the existence of C2[0, 1]-solutions of the boundary value problem (φp(x )) = f (t, x, x ), x(0) = A, x (1) = B, where φp(s) = s|s|p−2, p > 2
The solvability of various singular and nonsingular BVPs with p-Laplacian has been studied, for example, in [1,2,3,4,5, 7,8,9,10,11,12, 14]. Conditions used in these works or do not allow the main nonlinearity to change sign, [2, 11], or impose a growth restriction on it, [3, 9, 11], or require the existence of upper and lower solutions, [1, 3, 5, 8, 9, 12]; other type conditions have been used in [7], where the main nonlinearity may changes its sign
Summary
The obtained results guarantee the existence of positive solutions. The existence of at least one positive and monotone C2[0, 1]-solution is established therein under the following barrier condition: Corresponding author. Let H and (1.5) hold, and f (t, x, y) be continuous on the set [0, 1] × [A − σ,L + σ] ×[F1 − σ, L1 + σ].
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