scholarly journals Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient

2019 ◽  
Vol 168 (8) ◽  
pp. 1487-1537 ◽  
Author(s):  
Marie-Françoise Bidaut-Véron ◽  
Marta García-Huidobro ◽  
Laurent Véron
Keyword(s):  
Elliptic Equations ◽  
Reaction Term ◽  
Estimates Of Solutions

scholarly journals Existence and Multiplicity of Solutions for Resonant (p,2)-Equations

2018 ◽  
Vol 18 (1) ◽  
pp. 105-129 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Variational Methods ◽  
Existence Theorem ◽  
Elliptic Equations ◽  
Smooth Solution ◽  
Critical Groups ◽  
Constant Sign ◽  
Smooth Solutions ◽  
Reaction Term ◽  
Existence And Multiplicity ◽  
Multiplicity Theorem

Abstract We consider Dirichlet elliptic equations driven by the sum of a p-Laplacian {(2<p)} and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both {\pm\infty} and at zero. We prove an existence theorem (producing one nontrivial smooth solution) and a multiplicity theorem (producing five nontrivial smooth solutions, four of constant sign and the fifth nodal; the solutions are ordered). Our approach uses variational methods and critical groups.

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