In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity. We investigate existence or nonexistence properties, power- and exponential-type Sobolev regularity results, and the boundary behaviour of the weak solution, in the light of the interplay between the summability of the datum and the power exponent in singular nonlinearities.

In this paper, we analytically investigate multi-component Cahn–Hilliard and Allen–Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form ∫Ω½Γ∇c : ∇c + ½|∇z|2+Wch(c)+Wel(e,c,z)dx with a polynomial or logarithmic chemical energy density Wch, an inhomogeneous elastic energy density Wel and a quadratic structure of the gradient of damage variable z. For the corresponding elastic Cahn–Hilliard and Allen–Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e.

In this paper, we introduce and consider a new class of complementarity problems, which are called the generalized mixed quasi-complementarity problems in a topological vector space. We show that the generalized mixed quasi-complementarity problems are equivalent to the generalized mixed quasi variational inequalities. Using a new type of KKM mapping theorem, we study the existence of a solution of the generalized mixed quasi-variational inequalities and generalized mixed quasi-complementarity problems. Several special cases are also discussed. The results obtained in this paper can be viewed as extension and generalization of the previously known results.