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Zbl 0686.35045
Iannacci, R.; Nkashama, M.N.; Ward, J.R.jun.
Nonlinear second order elliptic partial differential equations at resonance.
(English)
[J] Trans. Am. Math. Soc. 311, No.2, 711-726 (1989). ISSN 0002-9947; ISSN 1088-6850/e

The authors consider nonlinear second order elliptic partial differential equations at resonance. More precisely, they study the solvability of selfadjoint boundary value problems of the form $$(1)\quad Lu+\lambda\sb 1u+g(x,u)=h\quad in\quad \Omega,\quad u/\partial \Omega =0,$$ and the corresponding nonselfadjoint problems $$(2)\quad Au+\lambda\sb 1u+g(x,u)=h\quad in\quad \Omega,\quad u/\partial \Omega =0,$$ where h is a given function on $\Omega$ and $\lambda\sb 1$ is the first (resp. principal) eigenvalue of a uniformly elliptic operator -L (resp. -A) on a bounded smooth domain $\Omega \subset {\bbfR}\sp N:$ $$Lu=\sum\sb{i,j}\partial /\partial x\sb i(a\sb{ij}(x)\partial u/\partial x\sb j)-a\sb 0(u)u,\quad Au=Lu+\sum\sb{i}b\sb i(x)\partial u/\partial x\sb i,$$ with the coefficients satisfying suitable regularity conditions. The objective is to show solvability of (1) or (2) for any h orthogonal to the first eigenfunction, in situations where the nonlinearity satisfies neither a monotonicity condition nor a Landesman- Lazer type condition. Instead, the nonlinearity is assumed to satisfy a sign condition g(x,u), $u\ge 0$, and a linear growth allowing interaction'' with the first and second eigenvalues. Moreover, some crossing of eigenvalues is allowed in certain cases.
[D.Costa]
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35J15 Second order elliptic equations, general
35J25 Second order elliptic equations, boundary value problems

Keywords: topological degree; Leray-Schauder method of continuity; first eigenfunction; monotonicity; sign condition

Cited in: Zbl 1113.35080 Zbl 0702.35090

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