Existence of solutions for 4p-order PDES

Abstract

<jats:title>Abstract</jats:title>
<jats:p>In this paper, we study the following nonlinear eigenvalue problem:
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<jats:tex-math>\left\{ {\matrix{ {{\Delta ^{2p}}u = \lambda m\left( x \right)u\,\,\,in\,\,\Omega ,} \cr {u = \Delta u = \ldots {\Delta ^{2p - 1}}u = 0\,\,\,\,on\,\,\partial \Omega .} \cr } } \right.</jats:tex-math>
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Where Ω is a bounded domain in ℝ<jats:italic>
<jats:sup>N</jats:sup>
</jats:italic> with smooth boundary ∂ Ω, <jats:italic>N ≥</jats:italic> 1, <jats:italic>p</jats:italic> ∈ ℕ<jats:sup>*</jats:sup>, <jats:italic>m</jats:italic> ∈ <jats:italic>L<jats:sup>∞</jats:sup>
</jats:italic> (Ω), <jats:italic>µ</jats:italic>{<jats:italic>x</jats:italic> ∈ Ω: <jats:italic>m</jats:italic>(<jats:italic>x</jats:italic>) &gt; 0} ≠ 0, and Δ<jats:sup>2</jats:sup>
<jats:italic>
<jats:sup>p</jats:sup>u</jats:italic> := Δ (Δ...(Δ<jats:italic>u</jats:italic>)), 2<jats:italic>p</jats:italic> times the operator Δ.</jats:p>
<jats:p>Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.</jats:p>