Related papers: On the critical $p$-Kirchhoff equation

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the…

We study a class of Schr\"{o}dinger-Kirchhoff system involving critical exponent. We aim to find suitable conditions to assure the existence of a positive ground state solution of Nehari-Poho\u{z}aev type $u_{\varepsilon}$ with exponential…

This paper deals with a fourth order elliptic equation on compact Riemannian manifolds.We establish the existence of solutions to the equation with critical Sobolev growth which is the subject of the first theorem. In the second one, we…

This paper has two main purposes. In the first part, combining the nondegeneracy of the ground state with the Lyapunov--Schmidt reduction method, we prove the existence of multi-peak positive solutions to the singularly perturbed problem…

In this paper, we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator. With the help of the subcritical approach from variational method, we obtain the non-existence,…

In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff problem, which involves the $p(x)$-fractional Laplacian equations of variable order. The problem is stated as follows: \begin{eqnarray*} \left\{ \begin{array}{ll}…

We consider a nonlocal differential equation of Kirchhoff type with a convolution coefficient involving variable growth. The novelty of our work lies in allowing a variable exponent in the nonlocal term. By relating the variable growth…

By using the fibering method jointly with Nehari manifold techniques, we obtain the existence of multiple solutions to a fractional $p$-Laplacian system involving critical concave-convex nonlinearities provided that a suitable smallness…

We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*}…

In this paper we will prove the existence of three nontrivial weak solutions of the following problem involving a nonlinear integro-differential operator and a term with critical exponent. \begin{align*} \begin{split} -\mathscr{L}_\Phi u &…

We investigate a class of variable growth nonlocal differential equations of Kirchhoff-type having the general form \(-A\!\left(\int_0^1 b(1-s)\,\big(u(s)\big)^{p(s)}\,ds\right)\,u''(t) = \lambda\,f(t,u(t))\) for \(t\in(0,1)\), where \(A\)…

This article presents a comprehensive study of \textit{Kirchhoff-type Critical Elliptic Equations} involving $p$-sub-Laplacian Operators on the \textit{Heisenberg Group} $\mathcal{H}_{n}$. It delves into the mathematical framework of…

We prove that a Hamiltonian $p\in C^\infty(T^*{\bf R}^n)$ is locally integrable near a non-degenerate critical point $\rho_0$ of the energy, provided that the fundamental matrix at $\rho_0$ has no purely imaginary eigenvalues. This is done…

In this paper we prove new multiplicity results for a critical growth anisotropic quasilinear elliptic system that is coupled through a subcritical perturbation term. We identify a certain scaling for the system and a parameter {\gamma}…

On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the…

In this article, we study an elliptic problem of mixed order with both local and nonlocal aspects involving singular nonlinearity in combination with critical Hartree-type nonlinearity. Using variational methods together with the critical…

In this work, our interest lies in proving the existence of critical values of the following Rayleigh-type quotients $$Q_{\mathbf p}(u) = \frac{\|\nabla u\|_{\mathbf p}}{\|u\|_{\mathbf p}},\quad\text{and}\quad Q_{\mathbf s,\mathbf p}(u) =…

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob}…

In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u =…

In this paper we study some nonlinear elliptic equations in $\R^n$ obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$ (-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {{in}}\R^n,$$ where $s\in(0,1)$,…