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Unilateral Global Bifurcation, Half-Linear Eigenvalues and Constant Sign Solutions for a Fractional Laplace Problem

Bian-Xia Yang

College of Science, Northwest A&F University, Yangling, Shaanxi 712100, P. R. China

Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

E-mail Address: g-yangb@utrgv.edu

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Hong-Rui Sun

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China

E-mail Address: hrsun@lzu.edu.cn

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, and

Zhaosheng Feng

School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

E-mail Address: zhaosheng.feng@utrgv.edu

Corresponding author.

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https://doi.org/10.1142/S0218127417500158Cited by:3 (Source: Crossref)

Abstract

In this paper, we are concerned with the unilateral global bifurcation structure of fractional differential equation

{(−Δ)αu(x)=λa(x)u(x)+F(x,u,λ),x∈Ω,u=0,inℝN\Ω<math display="block" altimg="eq-00001.gif"><mrow><mfenced separators="" open="{" close=""><mrow><mtable style="text-align:axis" equalrows="false" equalcolumns="false" class="array"><mtr><mtd class="array" columnalign="left"></mtd></mtr><mtr><mtd class="array" columnalign="left"><msup><mrow><mo class="MathClass-open" stretchy="false">(</mo><mo stretchy="false" class="MathClass-bin">−</mo><mi mathvariant="normal">Δ</mi><mo class="MathClass-close" stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup><mi>u</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">=</mo><mi>λ</mi><mi>a</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mi>u</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">+</mo><mi>F</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-punc">,</mo><mi>u</mi><mo class="MathClass-punc">,</mo><mi>λ</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace><mi>x</mi><mo class="MathClass-rel">∈</mo><mi mathvariant="normal">Ω</mi><mo class="MathClass-punc">,</mo></mtd></mtr><mtr><mtd class="array" columnalign="left"><mi>u</mi><mo class="MathClass-rel">=</mo><mn>0</mn><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace><mstyle><mtext class="textrm" mathvariant="normal">in</mtext></mstyle><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>N</mi></mrow></msup><mo stretchy="false" class="MathClass-bin">\</mo><mi mathvariant="normal">Ω</mi></mtd></mtr></mtable></mrow></mfenced></mrow></math>

with nondifferentiable nonlinearity

$F$ . It shows that there are two distinct unbounded subcontinua

$𝒞_{+}$ and

$𝒞_{-}$ consisting of the continuum

$𝒞$ emanating from

$[λ_{1} - d, λ_{1} + d] \times {0}$ , and two unbounded subcontinua

$𝒟_{+}$ and

$𝒟_{-}$ consisting of the continuum

$𝒟$ emanating from

$[λ_{1} - \bar{d}, λ_{1} + \bar{d}] \times {\infty}$ . As an application of this unilateral global bifurcation results, we present the existence of the principal half-eigenvalues of the half-linear fractional eigenvalue problem. Finally, we deal with the existence of constant sign solutions for a class of fractional nonlinear problems. Main results of this paper generalize the known results on classical Laplace operators to fractional Laplace operators.

Keywords:

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