Related papers: Nodal Sets for Broken Quasilinear Partial Differen…

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, \begin{equation*} \operatorname{div}(A_s(x,u)\nabla u)=\operatorname{div}{\vec f}(x), \end{equation*} where…

We are concerned with the nodal set of solutions to equations of the form \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- >…

This paper deals with solutions to the equation \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in (0,1)$,…

We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem \[ -\Delta u = \lambda_+(u^+)^{p-1}-\lambda_-(u^-)^{q-1} \] where $1 \le p<q<2$, $\lambda_+ >0$, $\lambda_- \ge 0$. The equation is…

We study the nodal set of solutions to equations of the form $$ (-\Delta)^s u = \lambda_+ (u_+)^{q-1} - \lambda_- (u_-)^{q-1}\quad\text{in $B_1$}, $$ where $\lambda_+,\lambda_->0, q \in [1,2)$, and $u_+$ and $u_-$ are respectively the…

In this paper, we study the parabolic equations $\partial_t u=\partial_j\left(a^{ij}(x,t)\partial_iu\right)+b^j(x,t)\partial_ju+c(x,t)u$ in a domain of $\mathbb{R}^n$ under the condition that $a^{ij}$ are Lipschitz continuous. Consider the…

We establish the existence of weak solutions of coupled systems of elliptic partial differential equations with quasimonotone nonlinearities in the domain interior and on the boundary. When the nonlinearities satisfy some monotonicity…

We consider singular quasilinear elliptic systems with homogeneous Dirichlet boundary condition. Using Leray-Schauder topological degree, combined with the sub-supersolutions method and suitable truncation arguments, we establish the…

In this paper we study quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary. Under very general assumptions our main result says that each weak solution of such systems belongs…

The aim of this paper is investigating the existence of weak bounded solutions of the gradient-type quasilinear elliptic system $$(P)\qquad \left\{ \begin{array}{ll} - {\rm div} ( a_i(x, u_i, \nabla u_i) ) + A_{i, t} (x, u_i, \nabla u_i) =…

The purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation \begin{equation}\label{00} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^{\alpha} u+u^p=0\ \ \ \ &\ {\rm in}\ \…

We study the existence and regularity of weak solutions to the following quasilinear elliptic system: \[ -\mathrm{div}(A_k(x, u_k) |\nabla u_k|^{p_k - 2} \nabla u_k) + \dfrac{1}{p_k} D_s A_k(x, u_k) |\nabla u_k|^{p_k} = g_k(x, u) \quad…

In this work, we study the removability of boundary singular sets for certain classes of quasilinear elliptic equations in domains $\Omega$ of an $n$-dimensional Finsler manifold ( $\mathcal{M}, F, \vartheta$ ). We work with Lipschitz…

We establish partial regularity result for vector-valued solutions to second order elliptic system in divergence form. The coefficients safisfies Dini condition respect to $(x,u)$ with growth order lager than 2. We prove $C^1$-regularity of…

We study the Dirichlet problem for the semi--linear partial differential equations ${\rm div}\,(A\nabla u)=f(u)$ in simply connected domains $D$ of the complex plane $\mathbb C$ with continuous boundary data. We prove the existence of the…

We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to…

We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) -…

n this paper, we prove existence of nodal solutions for singular semilinear elliptic systems without variational structure where its both components are of sign changing. Our approach is based on sub-supersolutions method combined with…

In this paper we establish a new existence result for the quasilinear elliptic problem \[ -{\rm div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x,u)|\nabla u|^p + V(x)|u|^{p-2} u = g(x,u)\quad\mbox{ in } \mathbb{R}^N, \] with $N\ge 2$,…

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: \begin{equation*} \begin{cases} \partial_t \overline{u} - y^{-a} \nabla…