Related papers: Singular elliptic problems with unbalanced growth …
In this paper, we consider the regularity of weak solutions (in an appropriate space) to the elliptic partial differential equation \begin{equation*} (-\Delta_{p})^{s} u + (-\Delta_{q})^{s} u = f(x) \quad \text{in} \quad \mathbb{R}^{N},…
We study the non-existence and multiplicity of positive solutions of the nonlinear Choquard type equation $$ -\Delta u+ \varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u, \quad {\rm in} \ \mathbb R^N, \qquad (P_\varepsilon)$$…
We consider the problem $(P)$, $$ -\Delta u =c(x)u+\mu|\nabla u|^2 +f(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega),$$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N \geq 3$, $\mu>0, \, c \in…
We study existence and multiplicity of nontrivial solutions of the following problem $$ \left\{ \begin{array}{rcll} -\Delta_p u+(-\Delta_p)^{s} u & = & \lambda|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }\Omega,\\ u & = & 0 & \mbox{ on }…
We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity \begin{align} \mathfrak{M}\left(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s}…
In this article, we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (P_\la):\;\quad (-\De)^s u = u^{-q} + \la u^{{2^*_s}-1}, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…
We consider the existence of positive solutions to weighted quasilinear elliptic differential equations of the type \[ \begin{cases} - \Delta_{p, w} u = \sigma u^{q} & \text{in $\Omega$}, \\ u = 0 & \text{on $\partial \Omega$} \end{cases}…
In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, we establish existence of at least one weak solution for a parameter dependent singular nonlinearity…
In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R…
The aim of this paper is to establish two results about multiplicity of solutions to problems involving the $1-$Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem $$ \left\{…
We consider a nonlinear parametric Neumann problem driven by the anisotropic $(p,q)$-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive…
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 investigate the existence and multiplicity of abstract weak solutions of the equation $-\Delta_p u -\Delta_q u=\alpha |u|^{p-2}u + \beta |u|^{q-2}u$ in a bounded domain under zero Dirichlet boundary conditions, assuming $1<q<p$ and…
In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P)\left\{\begin{split} -\Delta_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }\Omega,\\ u&>0\text{ in…
In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system {lll} -\Delta_{p(x)}u = a(x)|u|^{p(x)-2}u - b(x)|u|^{\alpha(x)}|v|^{\beta(x)} v + f(x) in \Omega, \Delta_{q(x)}v = c(x) |v|^{q(x)-2}v -…
We study the existence, multiplicity and regularity results of weak solutions for the Dirichlet problem of a semi-linear elliptic equation driven by the mixture of the usual Laplacian and fractional Laplacian \begin{equation*} \left\{%…
In this paper, we deal with the existence and multiplicity of solutions to the nonuniformly elliptic equation of the N-Lapalcian type with a potential and a nonlinear term of critical exponential growth and satisfying the…
In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp…
This paper considers a local and non-local problem characterized by singular nonlinearity and a source term. Specifically, we focus on the following problem: \begin{equation}\label{A}\tag{P} -\Delta_{p} u + (-\Delta)^{s}_{q} u = f(x)…
We are interested in the regularity of weak solutions $u$ to the elliptic equation in divergence form; precisely in their local boundedness and their local Lipschitz continuity under general growth conditions, the so called $p,q-$growth…