This talk was concerned with some recent contributions to the study of nonlinear problems with variable exponents and the content is strictly related with the objectives and goals of this research project. Under general hypotheses, we considered several classes of nonlinear Dirichlet problems driven by non-homogeneous operators. The corresponding abstract setting corresponds to Lebesgue and Sobolev spaces with variable exponent. By using variational (mountain pass, linking, Ekeland’s principle) and topological (deformation, critical groups, Morse theory) tools, we established sufficient conditions for the existence of nontrivial weak solutions. A key role in the arguments was also played by energy estimates. The abstract methods can be extended to further classes of nonlinear non-homogeneous problems, for instance to biharmonic equations with one or more variable exponents. The content of this talk complements the results included in our monograph V. Radulescu and D. Repovš, Partial differential equations with variable exponents: Variational methods and qualitative analysis, (Monographs and research notes in mathematics). Boca Raton; London; New York: CRC Press, 2015. [COBISS.SI-ID 17325401]
B.04 Guest lectureCOBISS.SI-ID: 17772377
This talk was concerned with a unitary view of several classes of nonlinear problems described by nonlocal or fractional Laplace operators. Some of the results extend theorems established in the framework of the standard Laplace operator. Other results are new and are strictly related with nonlocal problems. One of the results of this talk reported on the papers [COBISS.SI-ID 17460313] and [COBISS.SI-ID 17334617]. One of these papers gives a solution to an open problem of B. Ricceri concerning a characterization property in a sublinear elliptic framework and corresponding to the Sierpinski gasket and other self-similar fractal domains (a related result is in the press in the Israel Journal of Mathematics). During the talk, the author raised some open problems. The audience of the Politecnico di Milano raised several questions connected with the problems discussed in this Analysis Seminar. The content of this talk complements the results included in the monograph G. Molica Bisci, V. Radulescu and R. Servadei, Variational methods for nonlocal fractional problems, (Encyclopedia of mathematics and its applications, 162). Cambridge: Cambridge University Press, 2016. [COBISS.SI-ID 17642841].
B.04 Guest lectureCOBISS.SI-ID: 17591641
This conference was dedicated to the memory and mathematical achievements of James Serrin, one of the greatest researchers in nonlinear analysis of the 20th century, with pioneering and deep contributions to this field. The talk included two original and powerful results concerning the generalized Pucci-Serrin maximum principle and the Keller-Osserman condition corresponding to the existence of blow-boundary solutions. In both cases it was pointed out that the original monotonicity assumptions are not necessary and the results are described only by the growth behavior of the nonlinear term, near the origin, respectively at infinity. These results opened numerous perspectives for the study of several classical results in nonlinear analysis, due to the fact that it is expected (in many cases) that the monotonicity hypotheses are no longer necessary. The talk included several open problems connected with pioneering contributions involving monotonicity assumptions like the Ambrosetti-Brezis-Cerami problem or the Brezis-Kamin sublinear equation. The Honorary Chairman of this conference has been Prof. Louis Nirenberg (2015 Abel Prize).
B.04 Guest lectureCOBISS.SI-ID: 17896025