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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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