Related papers: Existence, uniqueness and optimal regularity resul…

This is a simplification of our prior work on the existence theory for the Rosseland-type equations. Inspired by the Rosseland equation in the conduction-radiation coupled heat transfer, we use the locally arbitrary growth conditions…

In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of…

We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum…

We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space $\mathbb{R}^N$. We assume that the nonlinear term satisfies the locally super-$(m_1,m_2)$…

We prove an existence and uniqueness result for solutions to linear $X$-elliptic equations with $L^1$ data and zero Dirichlet boundary conditions. Such solutions depend continuously on the datum. Moreover, we show that an improvement in the…

This paper studies the Sobolev regularity estimates for weak solutions of a class of degenerate, and singular quasi-linear elliptic problems of the form $\text{div}[\mathbf{A}(x,u, \nabla u)]= \text{div}[\mathbf{F}]$ with non-homogeneous…

We consider elliptic problems with nonclassical boundary conditions that contain additional unknown functions on the border of the domain of the elliptic equation and also contain boundary operators of higher orders with respect to the…

We study a general class of quasilinear elliptic equations with nonstandard growth to prove the existence of a very weak solution to such a problem. A key ingredient in the proof is a priori global weighted gradient estimate of a very weak…

This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $\Phi$-Laplacian operator. The proof of existence is based on a…

It has been well known that if $\Omega$ is a bounded $C^1$-domain in $\R^n,\ n \ge 2$, then for every Radon measure $f$ on $\Omega$ with finite total variation, there exists a unique weak solution $u\in W_0^{1,1}(\Omega )$ of the Poisson…

We establish several results related to existence, nonexistence or bifurcation of positive solutions for a Dirichlet boundary value problem with in a smooth bounded domain. The main feature of this paper consists in the presence of a…

This work resolves the open problem of strong singularity ($\alpha(z)> 1$) in nonlocal Kirchhoff-type equations with variable exponents through five original theorems that collectively establish a comprehensive theory. Beginning with…

We present recent advances in the regularity theory for weak solutions to some classes of elliptic and parabolic equations with strongly singular or degenerate structure. The equations under consideration satisfy standard $p$-growth and…

Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition…

In a cylindrical space-time domain with a convex, spatial base, we establish a local Lipschitz estimate for weak solutions to parabolic systems with Uhlenbeck structure up to the lateral boundary, provided homogeneous Dirichlet data are…

To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for…

We study the continuity of weak solutions for quasilinear elliptic systems with source terms of critical growth arising from a transport-energy structure. The latter occurs frequently in connection with the first balance principles of…

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption…

In this paper we consider a quasilinear singular anisotropic elliptic problem with a p-sublinear perturbation. We prove uniqueness of weak solutions and we provide necessary and sufficient conditions for the existence of weak solutions in…

In this paper we give a comprehensive treatment of a two-penalty boundary obstacle problem for a divergence form elliptic operator, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and…