Related papers: Multiple solutions for elliptic equations driven b…
We study a nonlinear elliptic equation driven by the degenerate fractional p-Laplacian, with Dirichlet type condition and a jumping reaction, i.e., (p-1)-linear both at infinity and at zero but with different slopes crossing the principal…
We study the existence of solutions to the fractional elliptic equation (E1) $(-\Delta)^\alpha u+\epsilon g(|\nabla u|)=\nu $ in a bounded regular domain $\Omega$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $\Omega^c$, where…
We study the following nonlinear and nonlocal elliptic equation in~$\R^n$ $$ (-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {\mbox{ in }}\R^n, $$ where~$s\in(0,1)$, $n>2s$, $\epsilon>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $q\in(0,1)$,…
In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with…
In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special…
This article is concerned with the existence and multiplicity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \|u\|^2\right) (-\Delta)^s u =…
This paper studies the properties of solutions to a class of elliptic and parabolic problems involving the fractional Laplacian. By applying the mountain pass theorem, we prove the existence of bounded classical positive solutions in the…
This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-\Delta)^{s_1}_{p}u+(-\Delta)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) =…
We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves…
It is established the existence and multiplicity of weak solutions for a class of nonlocal equations involving the fractional laplacian, nonlinearities with critical exponential growth and potentials this is which may change sign. The…
In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus \Omega\\…
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 consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…
In this work, we study the existence and multiplicity of solutions for the following problem \begin{equation}\label{probaa1} \left\{ \begin{aligned} -(\Delta)_{p}^{s} u + V(x)|u|^{p-2}u &= \lambda f(u),&x\in\Omega; u&=0,&x\in…
In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of…
In this paper, we consider the elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u=g(x,v)\,\, \textnormal{in}\Omega, & \hbox{} -\Delta v=f(x,u)\,\,\textnormal{in}\Omega, & \hbox{} u=v=0\textnormal{on}\partial\Omega, &…
In this paper, we use Legendre-Fenchel transform and a space decomposition to carry out Fountain theorem and dual Fountain theorem for the following elliptic system of Hamiltonian type: \[ \begin{cases} \begin{aligned} -\Delta u&=H_v(u, v)…
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) -…
It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $\Phi$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-\Delta_\Phi u= g(x,u), &…
We study the quasilinear elliptic system \[ -\textbf{div}(A(x,\boldsymbol u)|D\boldsymbol u|^{p-2}D\boldsymbol u) +\frac{1}{p}\nabla_{\boldsymbol s}A(x,\boldsymbol u)|D\boldsymbol u|^p = \boldsymbol g(x,\boldsymbol u) \quad \text{in }…