Abstract
Abstract We consider Dirichlet elliptic equations driven by the sum of a p-Laplacian ( 2 < p ) {(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.
Highlights
Let Ω ⊆ RN be a bounded domain with a C 2 -boundary ∂Ω
Using tools from Morse theory and variational methods based on the critical point theory, we prove existence and multiplicity theorems for resonant (p, 2)equations
The excision property of singular homology implies that the above definition of critical groups is independent of the particular choice of the neighborhood U
Summary
Let Ω ⊆ RN be a bounded domain with a C 2 -boundary ∂Ω. In this paper we study the following nonlinear, nonhomogeneous Dirichlet problem:. The aforementioned works, either do not consider resonant at ±∞ equations (see Aizicovici, Papageorgiou and Staicu [3], Cingolani and Degiovanni [11], Sun [34], Sun, Zhang and Su [35]) or the resonance is with respect to the principal eigenvalue (see Papageorgiou and Rădulescu [26, 28], Papageorgiou, Rădulescu and Repovš [30], Papageorgiou and Winkert [31]). For p 6= 2, we not have a complete knowledge of the spectrum of (−∆p , W01,p (Ω)), the eigenspaces are not linear subspaces of W01,p (Ω) and the Sobolev space W01,p (Ω) cannot be expressed as a direct sum of the eigenspaces All these negative facts make difficult the study of problems with resonance at higher Key words and phrases. Compared with the work of Papageorgiou and Rădulescu [26], the resonance is with respect to any variational eigenvalue of (−∆p , W01,p (Ω)), the principal one. We briefly recall the main mathematical tools which will be used in the sequel
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
