<abstract xmlns="http://www.w3.org/1999/xhtml">We consider the nonlocal eigenvalue problem of the following form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0006_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mi>𝒫</mi><mi>k</mi><mo>)</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><msubsup><mrow><mi>𝒧</mi></mrow><mi>K</mi><mrow><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></msubsup><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><msup><mrow><mrow><mrow><mo>|</mo><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></mrow><mrow><mover><mi>p</mi><mo>¯</mo></mover><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow><mi>λ</mi><msup><mrow><mrow><mrow><mo>|</mo><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></mrow><mrow><mi>r</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd><mrow><mi>i</mi><mi>n</mi></mrow></mtd><mtd><mrow><mi>Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mi>u</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mi>i</mi><mi>n</mi></mrow></mtd><mtd><mrow><msup><mrow><mtext>𝕉</mtext></mrow><mi>N</mi></msup><mo>\</mo><mi>Ω</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>$$(\mathcal{P}k)\left\{ {\matrix{ {\mathcal{L}_K^{p(x)}u(x) + {{\left| {u(x)} \right|}^{\bar p(x) - 2}}u(x)} \hfill &amp; = \hfill &amp; {\lambda {{\left| {u(x)} \right|}^{r(x) - 2}}u(x)} \hfill &amp; {in} \hfill &amp; {\Omega ,} \hfill \cr u \hfill &amp; = \hfill &amp; 0 \hfill &amp; {in} \hfill &amp; {{{\rm\mathbb{R}}^N}\backslash \Omega ,} \hfill \cr } } \right.$$</tex-math></alternatives></disp-formula>where Ω is a smooth open and bounded set in 𝕉<italic>N</italic> (<italic>N</italic> ⩾ 3), λ <italic>></italic> 0 is a real number, <italic>K</italic> is a suitable kernel and <italic>p</italic>, <italic>r</italic> are two bounded continuous functions on ̄Ω. The main result of this paper establishes that any λ <italic>></italic> 0 sufficiently small is an eigenvalue of the above nonhomogeneous nonlocal problem. The proof relies on some variational arguments based on Ekeland's variational principle.</abstract>

We consider the nonlocal eigenvalue problem of the following form(𝒫k){𝒧Kp(x)u(x)+|u(x)|p¯(x)-2u(x)=λ|u(x)|r(x)-2u(x)inΩ,u=0in𝕉N\Ω,$$(\mathcal{P}k)\left\{ {\matrix{  {\mathcal{L}_K^{p(x)}u(x) + {{\left| {u(x)} \right|}^{\bar p(x) - 2}}u(x)} \hfill & = \hfill & {\lambda {{\left| {u(x)} \right|}^{r(x) - 2}}u(x)} \hfill & {in} \hfill & {\Omega ,} \hfill \cr  u \hfill & = \hfill & 0 \hfill & {in} \hfill & {{{\rm\mathbb{R}}^N}\backslash \Omega ,} \hfill \cr } } \right.$$where Ω is a smooth open and bounded set in 𝕉N (N ⩾ 3), λ > 0 is a real number, K is a suitable kernel and p, r are two bounded continuous functions on ̄Ω. The main result of this paper establishes that any λ > 0 sufficiently small is an eigenvalue of the above nonhomogeneous nonlocal problem. The proof relies on some variational arguments based on Ekeland's variational principle.

Nonlocal eigenvalue problems with variable exponent

Laboratory of Mathematical Analysis and Applications

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Nonlocal eigenvalue problems with variable exponent"}

We consider the nonlocal eigenvalue problem of the following form(𝒫k){𝒧Kp(x)u(x)+|u(x)|p¯(x)-2u(x)=λ|u(x)|r(x)-2u(x)inΩ,u=0in𝕉N\Ω,$$(\mathcal{P}k)\left\{...

Nonlocal eigenvalue problems with variable exponent

Published Online: Apr 02, 2019

Page range: 46 - 61

Received: Feb 06, 2019

Accepted: Mar 17, 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0006

Keywords
2010 Mathematics Subject Classiffication. Primary: 35R11, 35P30, 35J20, 35S05

© 2018 Elhoussine Azroul et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Nonlocal eigenvalue problems with variable exponent

Published Online: Apr 02, 2019

Page range: 46 - 61

Received: Feb 06, 2019

Accepted: Mar 17, 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0006

Keywords2010 Mathematics Subject Classiffication. Primary: 35R11, 35P30, 35J20, 35S05

© 2018 Elhoussine Azroul et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
2010 Mathematics Subject Classiffication. Primary: 35R11, 35P30, 35J20, 35S05