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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
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