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Multiple solutions for a class of quasilinear problems

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  • In this paper we establish the existence of positive and multiple solutions for the quasilinear elliptic problem

    $-\Delta_p u = g(x,u)$  in  $\Omega$
    $u = 0 $  on  $\partial \Omega$,

    where $\Omega \subset \mathbb{R}^N$ is an open bounded domain with smooth boundary $\partial \Omega$, $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carathéodory function such that $g(x,0)=0$ and which is asymptotically linear. We suppose that $g(x,t)/t$ tends to an $L^r$-function, $r>N/p$ if 1 < p ≤ N and $r=1$ if $p>N$, which can change sign. We consider both the resonant and the nonresonant cases.

    Mathematics Subject Classification: 35J25 (58E05).


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