Related papers: The Dirichlet Problem For the Logarithmic p-Laplac…

Let $\Omega$ be a bounded, smooth domain. Supposing that $\alpha(p) + \beta(p) = p$, $\forall\, p \in \left(\frac{N}{s},\infty\right)$ and $\displaystyle\lim_{p \to \infty} \alpha(p)/{p} = \theta \in (0,1)$, we consider two systems for the…

In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the…

We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we…

We study the first nonzero eigenvalues for the $p$-Laplacian on quaternionic K\"ahler manifolds. Our first result is a lower bound for the first nonzero closed (Neumann) eigenvalue of the $p$-Laplacian on compact quaternionic K\"ahler…

In this work we introduce volume constraint problems involving the nonlocal operator $(-\Delta)_{\delta}^{s}$, closely related to the fractional Laplacian $(-\Delta)^{s}$, and depending upon a parameter $\delta>0$ called horizon. We study…

We establish the solvability of the $L^p$-Dirichlet and $L^{p^\prime}$-Neumann problems for the Laplacian for $p\in (\frac{n}{n-1}-\varepsilon,\frac{2n}{n-1}]$ for some $\varepsilon>0$ in $2$-sided chord-arc domains with unbounded boundary…

We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that…

In the present paper we study perturbation theory for the $L^p$ Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic…

We consider the first Robin eigenvalue $\l_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of…

In this paper, we study the first eigenvalue of a nonlinear elliptic system involving $p$-Laplacian as the differential operator. The principal eigenvalue of the system and the corresponding eigenfunction are investigated both analytically…

We introduce the discrete poly-Laplace operator on a subgraph with Dirichlet boundary condition. We obtain upper and lower bounds for the sum of the first $k$ Dirichlet eigenvalues of the poly-Laplace operators on a finite subgraph of…

We obtain a criterion for the existence of solutions of the problem $$ \Delta_p u = 0 \quad \mbox{in } M \setminus \partial M, \quad \left. u \right|_{ \partial M } = h, $$ with the bounded Dirichlet integral, where $M$ is an oriented…

We prove the existence of an unbounded branch of solutions to the non-linear non-local equation $$ (-\Delta)^s_p u=\lambda |u|^{p-2}u + f(x,u,\lambda) \quad\text{in}\quad \Omega,\quad u=0 \quad\text{in}\quad \mathbb{R}^n\setminus\Omega, $$…

A new idea to approximate the second eigenfunction and the second eigenvalue of $p$-Laplace operator is given. In the case of the Dirichlet boundary condition, the scheme has the restriction that the positive and the negative part of the…

We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian and the corresponding eigenfunctions upon perturbation of the domain and we generalize a few results known for the standard $p$-Laplacian. In particular, we prove…

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-\Delta_p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$,…

We deal with the first eigenvalue for a system of two $p-$Laplacians with Dirichlet and Neumann boundary conditions. If $\Delta_{p}w=\mbox{div}(|\nabla w|^{p-2}w)$ stands for the $p-$Laplacian and $\frac{\alpha}{p}+\frac{\beta}{q}=1,$ we…

In this paper we study the nonlinear Dirichlet problem involving p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using nonsmooth critical point theory for locally Lipschitz functionals due to Chang and the…

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb R^n \setminus \mathbb R^d$…

We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the…