The paper deals with a new class of nonlinear fourth-order problems driven by nonhomogeneous operators with variable exponent. The main feature is the presence of a no-flux boundary condition, which is motivated by models describing surfaces which are impermeable to some contaminants. The presence of a variable exponent corresponds to anisotropic media. Another feature of this paper is that the variable exponent is assumed to satisfy weak regularity properties. The paper discusses in a rigorous manner the weak solvability of the problem and the authors develop a refined variational analysis, which combines variational and topological tools. Under natural hypotheses, the authors establish the existence of weak solutions in the case of high perturbations (with respect to a suitable real parameter). Next, it is proved a multiplicity property under weak assumptions on the variable exponent. The mountain pass theorem of Ambrosetti and Rabinowitz plays an important role in the arguments developed in this paper. This is the first paper dealing with no-flux boundary conditions for fourth-order problems with variable exponent and the methods developed in this work can be extended to other classes of nonlinear anisotropic problems. The study developed in this paper complements the results included in the 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]
This paper is concerned with a nonlinear elliptic multivalued problem driven by a nonhomogeneous operator and whose term is the sum of a convex subdifferential and a convection (gradient) term. The content of this paper is at the interplay between convex and nonsmooth analysis. The convex subdifferential term incorporates problems with unilateral constraints. The study developed in this paper is motivated by models arising in nonsmooth mechanics. The main purpose of the present paper is to establish (under weak hypotheses) sufficient conditions for the existence of smooth solutions. The proofs combine several tools, including topological methods and the Moreau-Yosida approximation of the subdifferential term. Another basic ingredient in this paper is the theory of pseudo-monotone operators introduced by Brezis and Browder. The content of this paper is in strong relationship with variational inequalities (in the sense of Hartmann and Stampacchia) and hemivariational inequalities (in the sense of Panagiotopoulos). The topological methods introduced in this paper can be applied in the study of several classes of unilateral stationary problems.
This paper is concerned with the mathematical analysis of a new class of problems, namely nonlinear elliptic equations on Carnot groups depending of a positive parameter and involving a critical nonlinearity. As a special case of the results developed in this paper we establish the existence of a nontrivial solution for a subelliptic critical equation defined on a smooth bounded domain of the Heisenberg group. The proofs combine several refined tools such as Cayley maps on Heisenberg groups, properties of Folland-Stein spaces or concentration-compactness principles. Since the techniques used in this paper do not require any Lie group structure, the results established in the present paper are valid for more general operators than the sub-Laplacians on Carnot groups. This paper is at the interplay between several dimensional quantum mechanical systems (Heisenberg groups) and the existence of Riemannian metrics with constant scalar curvature (Yamabe problem).