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A nonsymmetric asymptotically linear elliptic problem. (English) Zbl 0844.35035
Let $\Omega$ be a bounded domain in $\bbfR^N$. 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 $\alpha, \beta, t\in \bbfR$. 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 $\bbfR^2$, 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.

35J65Nonlinear boundary value problems for linear elliptic equations
58E05Abstract critical point theory
35J20Second order elliptic equations, variational methods