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A variational approach to multiplicity in elliptic problems near resonance. (English) Zbl 0869.35041
Summary: We consider the nonlinear elliptic problem \[ \pm(\Delta u+\lambda u)+ f(x,u)= h(x)\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(\lambda\) is near the first eigenvalue and \(h(x)\) is orthogonal to the first eigenfunction. We give some conditions of existence of positive solutions and of multiple solutions in terms of the primitive of \(f\) with respect to \(u\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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