Related papers: Fundamental solutions for a class of three-dimensi…

In this paper, we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \frac{\lambda}{u^{\gamma}}+g(u)+\mu~\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\,…

We study the regularity of the extremal solution of the semilinear biharmonic equation $\beta \Delta^2 u-\tau \Delta u=\frac{\lambda}{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=\Delta u=0$ on $\partial B$,…

In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems: \begin{equation}…

Let $\mathcal{U(\alpha, \lambda)}$, $0<\alpha <1$, $0 < \lambda <1$ be the class of functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda$$ in the unit disc ${\mathbb…

The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-\Delta u =\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{\alpha,…

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by $$ -\Delta u= h(u){f} \ \ \text{in}\,\ \Omega, $$ where $f$ is an irregular datum,…

Let $n\geq2$ and $ \Omega\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$\Delta u + u^p = 0 \quad\hbox{in}\, \Omega,$$ which vanish in a suitable trace sense on…

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in…

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative…

We study a weakly coupled supercritical elliptic system of the form \begin{equation*} \begin{cases} -\Delta u = |x_2|^\gamma \left(\mu_{1}|u|^{p-2}u+\lambda\alpha |u|^{\alpha-2}|v|^{\beta}u \right) & \text{in }\Omega,\\ -\Delta v =…

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob}…

We study regularity properties for solutions to elliptic equations that are degenerate or singular along orthogonal hyperplanes. The degenerate ellipticity is carried out by a weight term which is the monomial product of different powers of…

We consider the q-hypergeometric equation with q^{N}=1 and $\alpha, \beta, \gamma \in {\Bbb Z}$. We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the…

We examine the equation \[\Delta^2 u = \lambda f(u) \qquad \Omega, \] with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $ \lambda$. We obtain similar results for…

In this paper, first we study carefully the positive solutions to $\Delta u+\lambda_{1}u\ln u +\lambda_{2}u^{b+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as…

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…

This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation*} %\tag{$\mathcal P$}\label{MAT1} \begin{cases} (-\Delta_p)^s u =|u|^{p^*_s-2}u+…

We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+\lambda u=f+\mathrm{div } \mathbf{F}\quad \text{in }…

In this paper we consider the phase retrieval problem for Herglotz functions, that is, solutions of the Helmholtz equation $\Delta u+\lambda^2u=0$ on domains $\Omega\subset\mathbb{R}^d$, $d\geq2$. In dimension $d=2$, if $u,v$ are two such…