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Nonlinear second order elliptic partial differential equations at resonance. (English) Zbl 0686.35045
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 $(1)\quad Lu+\lambda_ 1u+g(x,u)=h\quad in\quad \Omega,\quad u/\partial \Omega =0,$ and the corresponding nonselfadjoint problems $(2)\quad Au+\lambda_ 1u+g(x,u)=h\quad in\quad \Omega,\quad u/\partial \Omega =0,$ where h is a given function on $$\Omega$$ and $$\lambda_ 1$$ is the first (resp. principal) eigenvalue of a uniformly elliptic operator -L (resp. -A) on a bounded smooth domain $$\Omega \subset {\mathbb{R}}^ N:$$ $Lu=\sum_{i,j}\partial /\partial x_ i(a_{ij}(x)\partial u/\partial x_ j)-a_ 0(u)u,\quad Au=Lu+\sum_{i}b_ i(x)\partial u/\partial x_ i,$ 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), $$u\geq 0$$, and a linear growth allowing “interaction” with the first and second eigenvalues. Moreover, some crossing of eigenvalues is allowed in certain cases.
Reviewer: D.Costa

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J15 Second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations
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##### References:
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