Multiple solutions for the $p$-Laplacian under global nonresonance

Manuel del Pino, Raúl Manásevich

Type: Article

Publication Date: 1991-01-01

Citations: 46

DOI: https://doi.org/10.1090/s0002-9939-1991-1045589-1

Abstract

Via the study of a simple Dirichlet boundary value problem associated with the one-dimensional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Laplacian, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that in globally nonresonant problems for this differential operator the number of solutions may be arbitrarily large when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p element-of left-parenthesis 1 comma normal infinity right-parenthesis minus StartSet 2 EndSet"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi class="MJX-variant" mathvariant="normal">∖</mml:mi> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>2</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p \in (1,\infty )\backslash \{ 2\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. From this point of view <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> turns out to be a very special case.

Locations

Proceedings of the American Mathematical Society - View - PDF

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