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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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