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Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

Shaya Shakerian

Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

https://doi.org/10.1142/S021919972050008XCited by:2 (Source: Crossref)

Abstract

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities:

{(−Δ)α2u−γu|x|α=λf(x)|u|q−2u+g(x)|u|p−2u|x|sinΩ,u=0inℝn\Ω,<math display="block" altimg="eq-00001.gif"><mrow><mfenced separators="" open="{" close=""><mrow><mtable equalrows="false" columnlines="" equalcolumns="false" class="array"><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><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mi>u</mi><mo stretchy="false" class="MathClass-bin">−</mo><mi>γ</mi><mfrac><mrow><mi>u</mi></mrow><mrow><mo class="MathClass-rel">|</mo><mi>x</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac><mo class="MathClass-rel">=</mo><mi>λ</mi><mi>f</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mrow><mo class="MathClass-rel">|</mo><mi>u</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>q</mi><mo stretchy="false" class="MathClass-bin">−</mo><mn>2</mn></mrow></msup></mrow><mi>u</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>g</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mfrac><mrow><mo class="MathClass-rel">|</mo><mi>u</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>p</mi><mo stretchy="false" class="MathClass-bin">−</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow><mrow><mo class="MathClass-rel">|</mo><mi>x</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>s</mi></mrow></msup></mrow></mfrac><mspace width="1em" class="quad"></mspace><mstyle><mtext class="textrm" mathvariant="normal">in</mtext></mstyle><mspace width=".275em"></mspace><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><mspace width="1em" class="quad"></mspace><mstyle><mtext class="textrm" mathvariant="normal">in</mtext></mstyle><mspace width=".275em"></mspace><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup><mo stretchy="false" class="MathClass-bin">\</mo><mi mathvariant="normal">Ω</mi><mo class="MathClass-punc">,</mo></mtd></mtr></mtable></mrow></mfenced></mrow></math>

where $Ω \subset ℝ^{n}$ is a smooth bounded domain in $ℝ^{n}$ containing $0$ in its interior, and $f, g \in C (\bar{Ω})$ with $f^{+}, g^{+} ≢ 0$ which may change sign in $\bar{Ω}$ . We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for $λ$ sufficiently small. The variational approach requires that $0 < α < 2,$ $0 < s < α < n,$ $1 < q < 2 < p \leq 2_{α}^{*} (s) : = \frac{2 (n - s)}{n - α}$ , and $γ < γ_{H} (α)$ , the latter being the best fractional Hardy constant on $ℝ^{n}$ .

Keywords:

AMSC: 35S15, 35J20, 49J35

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Vol. 23, No. 02

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History

Received 6 April 2018

Revised 13 September 2019

Accepted 19 December 2019

Published: 3 March 2020

Keywords

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Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

Abstract

References

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System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

Abstract

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