<abstract xmlns="http://www.w3.org/1999/xhtml">

Let Ω ⊂ ℝ<italic>N</italic>, <italic>N</italic> ≥ 2, be a bounded domain with smooth boundary ∂Ω. Consider the following generalized Robin-Steklov eigenvalue problem associated with the operator 𝒜<italic>u</italic> = <italic>−</italic> Δ<italic>pu −</italic> Δ<italic>qu</italic>
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2024-0003_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>{</mo> <mrow><mtable><mtr><mtd><mrow><mi>𝒜</mi><mi>u</mi><mo>+</mo><msub><mrow><mi>ρ</mi></mrow><mn>1</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><msub><mrow><mi>ρ</mi></mrow><mn>2</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>q</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><mi>λ</mi><mi>α</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>∈</mo><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><msub><mrow><mi>v</mi></mrow><mi>A</mi></msub></mrow></mfrac><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mn>1</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mn>2</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>q</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><mi>λ</mi><mi>β</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mi> </mi><mi> </mi><mi>x</mi><mo>∈</mo><mo>∂</mo><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{ {\mathcal{A}u + {\rho _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\rho _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \alpha \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,\,x \in \Omega ,} \cr {{{\partial u} \over {\partial {v_A}}} + {\gamma _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\gamma _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \beta \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,x \in \partial \Omega ,} \cr } } \right.</tex-math>
</alternatives>
</disp-formula>
where <italic>p, q, r</italic> ∈ (1, ∞), <italic>p &lt; q</italic>; α, <italic>ρi </italic>∈ <italic>L</italic>∞(Ω) and β, <italic>γi </italic>∈ <italic>L</italic>∞(∂Ω) are nonnegative functions satisfying ∫Ω <italic>α dx</italic> + ∫∂Ω β <italic>dσ ></italic> 0 and ∫Ω ρ<italic>i dx</italic> + ∫∂Ω γ<italic>i dσ</italic> > 0, <italic>i</italic> = 1, 2.
We show that, if either <italic>r &lt; p</italic> or <italic>r > q</italic> with <italic>r &lt; q</italic>(<italic>N −</italic> 1)<italic>/</italic>(<italic>N − q</italic>) in case <italic>q &lt; N</italic>, then the eigenvalue set (spectrum) of the above problem is precisely (0, ∞). If <italic>r</italic> ∈ {<italic>p, q</italic>} then the corresponding spectrum is a smaller interval (<italic>d,</italic> ∞), <italic>d ></italic> 0. On the other hand, if <italic>r</italic> ∈ (<italic>p, q</italic>) with <italic>r &lt; p</italic>(<italic>N −</italic> 1)<italic>/</italic>(<italic>N − p</italic>) in case <italic>p &lt; N</italic>, then we are able to identify an interval of eigenvalues [λ*, ∞), where λ* is a positive number depending on <italic>r</italic>.
Obviously, the spectrum of the above problem coincides with the spectra of the Neumann-like, Robin-like, and Steklov-like eigenvalue problems corresponding to the cases when some of the functions α, β, γ<italic>i </italic>vanish.
</abstract>

Let Ω ⊂ ℝN, N ≥ 2, be a bounded domain with smooth boundary ∂Ω. Consider the following generalized Robin-Steklov eigenvalue problem associated with the operator 𝒜u = − Δpu − Δqu



{ 𝒜u+ρ1(x)| u |p-2u+ρ2(x)| u |q-2u=λα(x)| u |r-2u,   x∈Ω,∂u∂vA+γ1(x)| u |p-2u+γ2(x)| u |q-2u=λβ(x)| u |r-2u,  x∈∂Ω, 
\left\{ {\matrix{ {\mathcal{A}u + {\rho _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\rho _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \alpha \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,\,x \in \Omega ,} \cr {{{\partial u} \over {\partial {v_A}}} + {\gamma _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\gamma _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \beta \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,x \in \partial \Omega ,} \cr } } \right.


where p, q, r ∈ (1, ∞), p < q; α, ρi ∈ L∞(Ω) and β, γi ∈ L∞(∂Ω) are nonnegative functions satisfying ∫Ω α dx + ∫∂Ω β dσ > 0 and ∫Ω ρi dx + ∫∂Ω γi dσ > 0, i = 1, 2.
We show that, if either r q with r < q(N − 1)/(N − q) in case q < N, then the eigenvalue set (spectrum) of the above problem is precisely (0, ∞). If r ∈ {p, q} then the corresponding spectrum is a smaller interval (d, ∞), d > 0. On the other hand, if r ∈ (p, q) with r < p(N − 1)/(N − p) in case p < N, then we are able to identify an interval of eigenvalues [λ*, ∞), where λ* is a positive number depending on r.
Obviously, the spectrum of the above problem coincides with the spectra of the Neumann-like, Robin-like, and Steklov-like eigenvalue problems corresponding to the cases when some of the functions α, β, γi vanish.

On an eigenvalue problem associated with the (<italic>p, q</italic>) − Laplacian

andreea.laura.burlacu@365.univ-ovidius.ro

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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

On an eigenvalue problem associated with the () − Laplacian

Let Ω ⊂ ℝN, N ≥ 2, be a bounded domain with smooth boundary ∂Ω. Consider the following generalized Robin-Steklov eigenvalue problem associated with the operator 𝒜u = − Δpu − Δqu



{ 𝒜u+ρ1(x)| u |p-2u+ρ2(x)| u |q-2u=λα(x)| u |r-2u,   x∈Ω,∂u∂vA+γ1(x)| u |p-2u+γ2(x)| u |q-2u=λβ(x)| u |r-2u,  x∈∂Ω, 
\left\{ {\matrix{ {\mathcal{A}u + {\rho _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\rho _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \alpha \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,\,x \in \Omega ,} \cr {{{\partial u} \over {\partial {v_A}}} + {\gamma _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\gamma _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \beta \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,x \in \partial \Omega ,} \cr } } \right.


where p, q, r ∈ (1, ∞), p 0 and ∫Ω ρi dx + ∫∂Ω γi dσ > 0, i = 1, 2.
We show that, if either r q with r 0. On the other hand, if r ∈ (p, q) with r < p(N − 1)/(N − p) in case p < N, then we are able to identify an interval of eigenvalues [λ*, ∞), where λ* is a positive number depending on r.
Obviously, the spectrum of the above problem coincides with the spectra of the Neumann-like, Robin-like, and Steklov-like eigenvalue problems corresponding to the cases when some of the functions α, β, γi vanish.

Let Ω ⊂ ℝN, N ≥ 2, be a bounded domain with smooth boundary ∂Ω. Consider the following generalized Robin-Steklov eigenvalue problem associated with the...

On an eigenvalue problem associated with the (p, q) − Laplacian

Published Online: May 27, 2024

Page range: 45 - 64

Received: Feb 14, 2023

Accepted: May 31, 2023

DOI: https://doi.org/10.2478/auom-2024-0003

Keywords
Eigenvalues, ()Laplacian, parametric boundary condition, Nehari manifold, variational methods

© 2024 Luminiţa Barbu, published by Sciendo

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

On an eigenvalue problem associated with the (p, q) − Laplacian

Published Online: May 27, 2024

Page range: 45 - 64

Received: Feb 14, 2023

Accepted: May 31, 2023

DOI: https://doi.org/10.2478/auom-2024-0003

KeywordsEigenvalues, ()Laplacian, parametric boundary condition, Nehari manifold, variational methods

© 2024 Luminiţa Barbu, published by Sciendo

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

Keywords
Eigenvalues, ()Laplacian, parametric boundary condition, Nehari manifold, variational methods