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A nonsymmetric asymptotically linear elliptic problem. (English) Zbl 0844.35035
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\). The paper is concerned with the semilinear elliptic problem \[ \Delta u+ g(x, u)= te_1 \quad \text{in }\Omega, \qquad u= 0 \quad \text{on }\partial\Omega,\tag \(*\) \] where \(g(x, u)= \alpha u^++ \beta u^-+ g_0(x, u)\), \(g_0(x, u)/u\to 0\) as \(|u|\to \infty\), \(e_1\) is the positive eigenvalue of the Laplacian and \(\alpha, \beta, t\in \mathbb{R}\). To \((*)\) there corresponds a functional \[ f_t(u)= \int_\Omega (\textstyle{{1\over 2}} |\nabla u|^2- G(x, u)+ te_1 u)dx \] in \(H^1_0(\Omega)\) and critical points of \(f_t\) are solutions of \((*)\). It is shown that for \((\alpha, \beta)\) in certain regions of \(\mathbb{R}^2\), if \(t\) is large enough, then \((*)\) has at least one, two, three, respectively four solutions. Existence of one solution is shown by using a variant of the saddle point theorem of Rabinowitz. Two and three solutions are obtained by linking-type arguments where careful estimates are needed in order to show that certain linking levels are different. An additional argument gives a fourth critical point. It should also be noted that a rather general sufficient condition for \(f_t\) to satisfy the Palais-Smale condition is given in this paper.

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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