Perturbed asymptotically linear problems |
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Authors: | Rossella Bartolo Anna Maria Candela Addolorata Salvatore |
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Institution: | 1. Dipartimento di Meccanica Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125, Bari, Italy 2. Dipartimento di Matematica, Università degli Studi di Bari “Aldo Moro”, Via E. Orabona 4, 70125, Bari, Italy
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Abstract: | The aim of this paper is to investigate the existence of solutions of the semilinear elliptic problem $$\left\{\begin{array}{ll} -\Delta u\ =\ p(x, u) + \varepsilon g(x, u)\quad {\rm in}\,\, \Omega, \\ u=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\,\,{\rm on}\,\, \partial\Omega, \end{array} \right. \quad\quad\quad(0.1) $$ where Ω is an open bounded domain of ${\mathbb{R}^N}$ , ${\varepsilon\in\mathbb{R}, p}$ is subcritical and asymptotically linear at infinity, and g is just a continuous function. Even when this problem has not a variational structure on ${H^1_0(\Omega)}$ , suitable procedures and estimates allow us to prove that the number of distinct critical levels of the functional associated to the unperturbed problem is “stable” under small perturbations, in particular obtaining multiplicity results if p is odd, both in the non-resonant and in the resonant case. |
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