This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see (KC) below). We use the variational method in the light of Moser-Trudinger inequality to show the existence of weak solutions to (KC). Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex-concave problem (P λ,M ) below.

In this work, we establish a new Picone identity for anisotropic quasilinear operators, such as the p(x)-Laplacian defined as div(|∇u| p(x)−2 ∇u). Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents. This new Picone identity can be also used to prove some accretivity property to a class of fast diffusion equations involving variable exponents. Using this, we prove for this class of parabolic equations a new weak comparison principle.

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In this article, we study the existence, multiplicity, regularity and asymptotic behavior of the positive solutions to the problem of half-Laplacian with singular and exponential growth nonlinearity in one dimension (see below [Formula: see text]). We prove two results regarding the existence and multiplicity of solutions to the problem [Formula: see text]. In the first result, existence and multiplicity have been proved for classical solutions via bifurcation theory while in the latter result multiplicity has been proved for critical exponential nonlinearity by variational methods. An independent question of symmetry and monotonicity properties of classical solution has been answered in the paper. To characterize the behavior of large solutions, we further study isolated singularities for the singular semi linear elliptic equation in [Formula: see text], [Formula: see text] involving exponential growth nonlinearities in the more general framework of [Formula: see text] operator and for all [Formula: see text] (see below [Formula: see text]).

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