Quasilinear problems under local Landesman–Lazer condition
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This paper presents results on the existence and multiplicity of solutions for quasilinear problems in bounded domains involving the p-Laplacian operator under local versions of the Landesman–Lazer condition. The main results do not require any growth restriction at infinity on the nonlinear term which may change sign. The existence of solutions is established by combining variational methods, truncation arguments and approximation techniques based on a compactness result for the inverse of the p-Laplacian operator. These results also establish the intervals of the projection of the solution on the direction of the first eigenfunction of the p-Laplacian operator. This fact is used to provide the existence of multiple solutions when the local Landesman–Lazer condition is satisfied on disjoint intervals.
Mathematics Subject Classification35J20 35J92 47J30
This work was done while the second and third authors were visiting the Departamento de Análisis Matemático, Universidad de Granada. They would like to present their gratitude for the warm hospitaltiy of the whole members of that department.
First author is supported by FEDER-MEC (Spain) PGC2018-096422-B-I00 and Junta de Andalucía FQM-116. Third author is supported by CNPq (Brazil) 311808/2014-0 and 312060/2018-1.
- 14.Castro, A.: Reduction Methods via Minimax. First Latin American School of Differential Equations (São Paulo, Brazil, 1981), Lecture Notes in Mathematics, Vol. 957, pp. 1–20. Springer, Berlin (1982)Google Scholar
- 22.Ladyzenskaya, O., Uralt’seva, N.: Linear and Quasilinear Elliptic Equations. Translated by Scripta Technica. Academic Press, New York (1968)Google Scholar
- 23.Landesman, E.M., Lazer, A.C.: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19, 609–623 (1969/1970)Google Scholar
- 27.Peral, I.: Multiplicity of Solutions for the p-Laplacian. Second School on Nonlinear Functional Analysis and Applications to Differential Equations (1997)Google Scholar
- 28.Rabinowitz, P.H.: Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations. Nonlinear Analysis (a Collection of Papers in Honor of Erich Röthe), pp. 161–177. Academic Press, New York (1978)Google Scholar
- 29.Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence (1986)Google Scholar