## A note on the existence of two nontrivial solutions of a resonance problem.(English)Zbl 0824.35038

Existence of two nontrivial solutions of a semilinear problem at resonance is proved in this paper. The problem $$- \Delta u = \lambda_ 1u + g(x,u)$$ in $$G$$, $$u = 0$$ on $$\partial G$$ is studied, where $$G$$ is a smooth bounded domain in $$\mathbb{R}^ N$$, $$\Delta$$ is the usual Laplacian, $$\lambda_ 1$$ is the first eigenvalue of $$- \Delta$$ and $$g(x,u)$$ is a Carathéodory function such that $$g(x,0) = 0$$, $$| g(x,s) | \leq a | s |^ p + b$$, with $$a,b > 0$$ and $$0 < p < (N + 2)/(N - 2)$$ if $$N \geq 3$$, and $$| G(x,s) | \leq k(x)$$ (with $$G(x,s) = \int^ s_ 0 g(x,t)dt)$$ for some $$k \in L^ 1(G)$$. If, moreover, $$\lim_{s \to 0} G(x,s)/s^ 2 = m(x)$$ in the $$L^ 1 (G)$$ sense with $$m \geq 0$$, $$\int_ G \limsup_{| s | \to \infty} G(x,s)dx \leq 0$$ and $$G(x,s) \leq (\lambda_ 2 - \lambda_ 1)s^ 2/2$$ for all $$s$$ (here $$\lambda_ 2$$ is the second eigenvalue) then the problem has (at least) two nontrivial solutions. The method of proof is variational: the associated functional is $$C^ 1$$ on $$H^ 1_ 0 (G)$$ and bounded from below, but it is not coercive. However, it is possible to show that the Palais-Smale condition holds on some interval and this together with a deformation lemma allows to conclude by using minimax arguments.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 35J20 Variational methods for second-order elliptic equations
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