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Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities. (English) Zbl 0780.35038
The authors study an elliptic problem of the form $-\Delta u-\lambda u= f(x,u)- h(x) \text{ in } \Omega, \qquad u=0 \text{ on } \partial\Omega. \tag{P}$ The nonlinearity $$f$$ is sufficiently smooth and has sublinear growth at $$+\infty$$, i.e. $$\lim_{s\to+\infty} {{f(x,s)} \over s} =0$$ uniformly for $$x\in\Omega$$. Moreover, $$\lambda\in\mathbb{R}$$ and $$h\in C^{0,\alpha} (\overline {\Omega})$$. The authors prove the following multiplicity result:
Theorem. Let $$\lambda_ 1$$ denote the first eigenvalue of $$-\Delta$$ and let $$\varphi$$ denote a positive eigenfunction of $$-\Delta$$ associated with $$\lambda_ 1$$. Assume there exist $$c,C\in L^ 1(\Omega)$$ such that $$c(x)\leq \liminf_{s\to -\infty} f(x,s)$$, $$C(x)\geq \limsup_{s\to +\infty} f(x,s)$$, uniformly for $$x\in\Omega$$ and such that $\int_ \Omega c(x)\varphi(x)dx> \int_ \Omega h(x)\varphi(x)dx> \int_ \Omega C(x)\varphi(x)dx.$ Then there exists $$\delta>0$$ such that (P) has at least one solution for $$\lambda\leq \lambda_ 1$$ and at least two solutions for $$\lambda_ 1<\lambda<\lambda_ 1+\delta$$.
The paper constitutes a development of J. Mawhin and K. Schmitt [Result. Math. 14, No. 1/2, 138-146 (1988; Zbl 0780.35043)]. It should be noted that no upper bound is imposed on $$f$$, as $$s\to-\infty$$. The technique is based on the use of super- and sub-solutions and the bifurcation theorem from infinity of P. H. Rabinowitz [J. Differ. Equations 14, 462-475 (1973; Zbl 0272.35017)].

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B32 Bifurcations in context of PDEs 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
##### Citations:
Zbl 0780.35043; Zbl 0272.35017
Full Text:
##### References:
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