Nonresonance for problems involving $(p,q)$-Laplacian equations with nonlinear perturbations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908445864624128 |
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| author | Lopera, Emer Mavinga, Nsoki Sanchez, Diana |
| author_facet | Lopera, Emer Mavinga, Nsoki Sanchez, Diana |
| contents | We study the solvability of $(p,q)$-Laplacian problems with nonlinear reaction terms and non-homogeneous Neumann boundary conditions. First, we provide a complete description of the spectrum of the eigenvalue problem involving the $(p,q)$-Laplacian with weights and a spectral parameter present in both the differential equation and on the boundary. Then, using variational methods and critical point theory, we prove the existence of weak solutions for the nonlinear problem when the nonlinearities involved remain asymptotically, in some sense, below the first eigenvalue of the $q$-Laplacian problem with weights and a spectral parameter present in both the differential equation and on the boundary. We also establish an existence result for the nonlinear problem when the nonlinearities involved remain asymptotically below the first Steklov-Neumann eigenvalue-line, which is a line connecting the first Steklov and first Neumann eigenvalues for $q$-Laplacian problems with weights and a spectral parameter present either in the differential equation or on the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonresonance for problems involving $(p,q)$-Laplacian equations with nonlinear perturbations Lopera, Emer Mavinga, Nsoki Sanchez, Diana Analysis of PDEs Functional Analysis 35A15, 35B38, 35J66, 35J92, 35P30 We study the solvability of $(p,q)$-Laplacian problems with nonlinear reaction terms and non-homogeneous Neumann boundary conditions. First, we provide a complete description of the spectrum of the eigenvalue problem involving the $(p,q)$-Laplacian with weights and a spectral parameter present in both the differential equation and on the boundary. Then, using variational methods and critical point theory, we prove the existence of weak solutions for the nonlinear problem when the nonlinearities involved remain asymptotically, in some sense, below the first eigenvalue of the $q$-Laplacian problem with weights and a spectral parameter present in both the differential equation and on the boundary. We also establish an existence result for the nonlinear problem when the nonlinearities involved remain asymptotically below the first Steklov-Neumann eigenvalue-line, which is a line connecting the first Steklov and first Neumann eigenvalues for $q$-Laplacian problems with weights and a spectral parameter present either in the differential equation or on the boundary. |
| title | Nonresonance for problems involving $(p,q)$-Laplacian equations with nonlinear perturbations |
| topic | Analysis of PDEs Functional Analysis 35A15, 35B38, 35J66, 35J92, 35P30 |
| url | https://arxiv.org/abs/2507.08229 |