<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper we are concerned with the study of the spectrum for a class of eigenvalue problems driven by two non-homogeneous differential operators with different variable growth and an indefinite potential in the following form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2019-0015_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mo>-</mo><mtext>div</mtext><mrow><mo>[</mo><mrow><mi>𝒣</mi><mo>(</mo><mi>x</mi><mo>,</mo><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo><mo>)</mo><mo>∇</mo><mi>u</mi><mo>+</mo><mi>𝔌</mi><mo>(</mo><mi>x</mi><mo>,</mo><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo><mo>)</mo><mo>∇</mo><mi>u</mi></mrow><mo>]</mo></mrow><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>m</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo></mtd></mtr><mtr><mtd><mo>=</mo><mi>λ</mi><mrow><mo>(</mo><mrow><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><msub><mrow><mi>q</mi></mrow><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><msub><mrow><mi>q</mi></mrow><mn>2</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow><mi>u</mi><mi>   </mi><mtext>in</mtext><mi>   </mi><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></math><tex-math>$$\eqalign{ &amp; - {\rm{div}}\left[ {{\cal H}(x,|\nabla u|)\nabla u + \Im (x,|\nabla u|)\nabla u} \right] + V(x)|u{|^{m(x) - 2}}u = \cr &amp; = \lambda \left( {|u{|^{{q_1}(x) - 2}} + |u{|^{{q_2}(x) - 2}}} \right)u\;{\rm{in}}\;\Omega , \cr}$$</tex-math></alternatives></disp-formula>which is subjected to Dirichlet boundary condition. The proofs rely on variational arguments and they consist in finding two Rayleigh-type quotients, which lead us to an unbounded continuous spectrum on one side, and the nonexistence of the eigenvalues on the other.</abstract>

In this paper we are concerned with the study of the spectrum for a class of eigenvalue problems driven by two non-homogeneous differential operators with different variable growth and an indefinite potential in the following form-div[𝒣(x,|∇u|)∇u+𝔌(x,|∇u|)∇u]+V(x)|u|m(x)-2u==λ(|u|q1(x)-2+|u|q2(x)-2)u   in   Ω,$$\eqalign{ & - {\rm{div}}\left[ {{\cal H}(x,|\nabla u|)\nabla u + \Im (x,|\nabla u|)\nabla u} \right] + V(x)|u{|^{m(x) - 2}}u =  \cr  & = \lambda \left( {|u{|^{{q_1}(x) - 2}} + |u{|^{{q_2}(x) - 2}}} \right)u\;{\rm{in}}\;\Omega , \cr}$$which is subjected to Dirichlet boundary condition. The proofs rely on variational arguments and they consist in finding two Rayleigh-type quotients, which lead us to an unbounded continuous spectrum on one side, and the nonexistence of the eigenvalues on the other.

Multiple solutions for eigenvalue problems involving an indefinite potential and with (<italic>p</italic>1(<italic>x</italic>), <italic>p</italic>2(<italic>x</italic>)) balanced growth

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Multiple solutions for eigenvalue problems involving an indefinite potential and with ((), ()) balanced growth

In this paper we are concerned with the study of the spectrum for a class of eigenvalue problems driven by two non-homogeneous differential operators with...

Multiple solutions for eigenvalue problems involving an indefinite potential and with (p₁(x), p₂(x)) balanced growth

Published Online: Mar 02, 2019

Page range: 289 - 307

Received: Jan 21, 2018

Accepted: Feb 19, 2018

DOI: https://doi.org/10.2478/auom-2019-0015

Keywords
Nonlinear eigenvalue problem, Sign-changing potential, Continuous spectrum, Nonhomogeneous differential operator, Variable exponents

© 2019 Vasile-Florin Uţă, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Multiple solutions for eigenvalue problems involving an indefinite potential and with (p1(x), p2(x)) balanced growth

Published Online: Mar 02, 2019

Page range: 289 - 307

Received: Jan 21, 2018

Accepted: Feb 19, 2018

DOI: https://doi.org/10.2478/auom-2019-0015

KeywordsNonlinear eigenvalue problem, Sign-changing potential, Continuous spectrum, Nonhomogeneous differential operator, Variable exponents

© 2019 Vasile-Florin Uţă, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Multiple solutions for eigenvalue problems involving an indefinite potential and with (p₁(x), p₂(x)) balanced growth

Keywords
Nonlinear eigenvalue problem, Sign-changing potential, Continuous spectrum, Nonhomogeneous differential operator, Variable exponents