Related papers: Three solutions to a Steklov problem involving the…
We prove the existence of at least three solutions for a weighted $p$-Laplacian operator involving Dirichlet boundary condition in a weighted Sobolev space. The main tool we use here is a three solution theorem in reflexive Banach spaces…
The purpose of this paper is to investigate the existence of three different weak solutions to a nonlinear elliptic problem that is governed by the weighted {\varphi}-Laplacian operator and subjected to Dirichlet boundary conditions. We…
In this paper we prove the~existence of two non-trivial weak solutions of Dirichlet boundary value problem for p-Laplacian problem with a~singular part and two disturbances satisfying the~proper assumptions. The~abstract existence result we…
In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(.),p(.)}\left( \Omega ,\vartheta _{0},\vartheta \right) $ with respect to two different weight functions. Also, we investigate the basic properties of this…
We consider a class of nonlinear Dirichlet problems involving the $p(x)$--Laplace operator. Our framework is based on the theory of Sobolev spaces with variable exponent and we establish the existence of a weak solution in such a space. The…
We consider the Steklov problem associated with the weighted p-Laplace operator and $(p,q)$-Laplacian on submanifolds with the boundary of Euclidean spaces and prove Reilly-type upper bounds for their first eigenvalues.
In this paper we study the following boundary value problem involving the weak p-Laplacian. \begin{equation*} \quad -M(\|u\|_{\mathcal{E}_p}^p)\Delta_p u = h(x,u) \; \text{in}\; \mathcal{S}\setminus\mathcal{S}_0; \quad u = 0 \; \mbox{on}\;…
We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some…
We study a non-homogeneous boundary value problem in a smooth bounded domain in $\mathbb{R}^N$. We prove the existence of at least two nonnegative and non-trivial weak solutions. Our approach relies on Orlicz-Sobolev spaces theory combined…
We study the solvability of $(p,q)$-Laplacian problems with nonlinear reaction terms and non-homogeneous Neumann boundary conditions. First, we provide a complete description of the spectrum of the eigenvalue problem involving the…
We consider a number of boundary value problems involving the $p$-Laplacian. The model case is $-\Delta_p u=V|u|^{p-2}u$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R}^n$. We derive necessary conditions for the existence of…
In this paper, we consider Dirichlet boundary value problem involving the anisotropic $p(x)$-Laplacian, where $p(x)= (p_1(x), ..., p_n(x))$, with $p_i(x)> 1$ in $\overline{\Omega}$. Using the topological degree constructed by Berkovits, we…
We consider a non-local boundary value problem for the Laplace equation in unbounded studding the weak and strong solvability of that problem in the framework of the weighted Sobolev space $W^{1,p}_\nu$, with a Muckenhoupt weight. We proved…
This paper is concerned with the study of a nonlinear problems involving the fractional p(x)-Laplacian operator. By means of the Berkovits degree theory, we prove the existence of nontrivial weak solutions for this problem. The appropriate…
We study a superlinear elliptic boundary value problem involving the $p$-laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the…
In this article, we consider the nonlinear Steklov eigenvalue problem in outward cuspidal domains. Using the compactness of the weighted trace embedding we obtain the variational characterization of the first non-trivial eigenvalue and…
We study the boundary value problem $-{\rm div}((|\nabla u|^{p\_1(x) -2}+|\nabla u|^{p\_2(x)-2})\nabla u)=f(x,u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\RR^N$. We focus on the cases when…
The present paper studies the existence of weak solutions for \begin{equation*} (\mathcal{P}) \left\{\begin{aligned} (-\Delta)^{s_1}_{p_1} u &=\la f_1\,(x,u,v) +g_1(x,u) \,\mbox{ in }\, \Om, \\ (-\Delta)^{s_2}_{p_2} v &=\la f_2\,(x,u,v)…
The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous $ p ( x ) -$laplacian equations, managed by non polynomial growth operator in the framework of variable exponent Sobolev spaces on Riemannian…
We investigate existence and uniqueness of solutions to second-order elliptic boundary value problems containing a power nonlinearity applied to a fractional Laplacian. We detect the critical power separating the existence from the…