2 results
On the solvability of semilinear differential equations at resonance
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 43 / Issue 1 / February 2000
- Published online by Cambridge University Press:
- 20 January 2009, pp. 103-112
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In this paper we use the Leray–Schauder continuation method to study the existence of solutions for semilinear differential equations Lu + g(x, u) = h, in which the linear operator L on L2(Ω) may be non-self-adjoint, the L2(Ω)-function h belongs to N⊥(L), the nonlinear term g(x, u) ∈ O(|u|α) as |u| → ∞ for some 0 ≤ α < 1 and satisfies
for all v ∈ N(L) – {0}, where
and 
and 
On the Solvability of a Neumann Boundary Value Problem at Resonance
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- Journal:
- Canadian Mathematical Bulletin / Volume 40 / Issue 4 / 01 December 1997
- Published online by Cambridge University Press:
- 20 November 2018, pp. 464-470
- Print publication:
- 01 December 1997
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We study the existence of solutions of the semilinear equations (1)
in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, 
in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy 
are arbitrarily nonnegative constants, 
. The proofs are based upon degree theoretic arguments.
in which the non-linearity 
in which the nonlinear term g may grow superlinearly in 
are arbitrarily nonnegative constants, 
. The proofs are based upon degree theoretic arguments.