Lyapunov inequality for an elliptic problem with the Robin boundary condition

Abstract We consider a semilinear elliptic problem, driven by the Laplacian with Robin boundary condition. We consider a reaction term which is resonant at {\pm\infty} and at 0. Using variational methods and critical groups, we show that under resonance conditions at {\pm\infty} and at zero the problem has at least two nontrivial smooth solutions.

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Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

Advances in Nonlinear Analysis ◽

10.1515/anona-2017-5001 ◽

2019 ◽

Vol 8 (1) ◽

pp. 1286

Author(s):

Yibin Zhang ◽

Lei Shi

Keyword(s):

Boundary Condition ◽

Elliptic Problem ◽

Robin Boundary Condition ◽

Concentrating Solutions ◽

Robin Boundary

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A Sub-Supersolution Approach for Robin Boundary Value Problems with Full Gradient Dependence

Mathematics ◽

10.3390/math8050658 ◽

2020 ◽

Vol 8 (5) ◽

pp. 658 ◽

Cited By ~ 2

Author(s):

Dumitru Motreanu ◽

Angela Sciammetta ◽

Elisabetta Tornatore

Keyword(s):

Boundary Condition ◽

Boundary Value Problems ◽

Positive Solutions ◽

Elliptic Problem ◽

Robin Boundary Condition ◽

Convection Term ◽

Nonlinear Elliptic Problem ◽

Existence Of A Solution ◽

Nonlinear Elliptic ◽

Robin Boundary

The paper investigates a nonlinear elliptic problem with a Robin boundary condition, which exhibits a convection term with full dependence on the solution and its gradient. A sub- supersolution approach is developed for this type of problems. The main result establishes the existence of a solution enclosed in the ordered interval formed by a sub-supersolution. The result is applied to find positive solutions.

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Solutions for parametric double phase Robin problems

Asymptotic Analysis ◽

10.3233/asy-201598 ◽

2021 ◽

Vol 121 (2) ◽

pp. 159-170 ◽

Cited By ~ 2

Author(s):

Nikolaos S. Papageorgiou ◽

Calogero Vetro ◽

Francesca Vetro

Keyword(s):

Boundary Condition ◽

Existence Theorems ◽

Robin Boundary Condition ◽

Phase Problem ◽

Double Phase ◽

Robin Boundary ◽

Double Phase Problem

We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is ( p − 1 )-superlinear and the solutions produced are asymptotically big as λ → 0 + . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small as λ → 0 + .

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