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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 $$ (1)\quad Lu+\lambda\sb 1u+g(x,u)=h\quad in\quad \Omega,\quad u/\partial \Omega =0, $$ 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.
Reviewer: D.Costa

35J65Nonlinear boundary value problems for linear elliptic equations
35J15Second order elliptic equations, general
35J25Second order elliptic equations, boundary value problems
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