Related papers: Existence and multiplicity results for a new $p(x)…
This paper is concerned with the multiplicity results to a class of $p$-Kirchhoff type elliptic equation with the homogeneous Neumann boundary conditions by an abstract linking lemma due to Br\'{e}zis and Nirenberg. We obtain the twofold…
In this paper, we study the following nonlinear Kirchhoff problem involving critical growth: $$ \left\{% \begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=|u|^4u+\lambda|u|^{q-2}u, u=0\ \ \text{on}\ \ \partial\Omega, \end{array}%…
We are concerned with a class of Kirchhoff type equations in $\mathbb{R}^{N}$ as follows: \begin{equation*} \left\{ \begin{array}{ll} -M\left( \int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right) \Delta u+\lambda V\left( x\right) u=f(x,u) &…
This paper is devoted to the study of the following nonlocal equation: \begin{equation*} -\left(a+b\|\nabla u\|_{2}^{2(\theta-1)}\right) \Delta u =\lambda u+\alpha (I_{\mu}\ast|u|^{q})|u|^{q-2}u+(I_{\mu}\ast|u|^{p})|u|^{p-2}u \ \hbox{in} \…
In this paper we investigate a class of elliptic problems involving a nonlocal Kirchhoff type operator with variable coefficients and data changing its sign. Under appropriated conditions on the coefficients, we have shown existence and…
We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} \displaystyle -\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{\theta(p-1)}}\right) = h(u)f…
In the present paper, we apply a global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions $(\lambda_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the following Kirchhoff problem $$…
Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of…
This paper deals with a class of nonlocal variable $s(.)$-order fractional $p(.)$-Kirchhoff type equations: \begin{eqnarray*} \left\{ \begin{array}{ll}…
Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in…
This article deals with the study of the following nonlinear doubly nonlocal equation: \begin{equation*} (-\Delta)^{s_1}_{p}u+\ba(-\Delta)^{s_2}_{q}u = \la a(x)|u|^{\delta-2}u+ b(x)|u|^{r-2} u,\; \text{ in }\; \Om, \; u=0 \text{ on }…
In this paper, we consider the existence, multiplicity and nonexistence of solutions for the following equation \begin{equation*} \begin{cases} \begin{aligned} &-\Delta u+\omega u=\mu u^{p-1}+u^{q-1},~ u>0 \quad &&\text { in } \Omega, \\…
In this paper, we are going to study the existence of solution for the following Kirchhoff problem $$ \left\{ \begin{array}{l} M\biggl(\displaystyle\int_{\mathbb{R}^{3}}|\nabla u|^{2} dx +\displaystyle\int_{\mathbb{R}^{3}} \lambda…
In this paper, we show the existence and multiplicity of positive solutions of the following fractional Kirchhoff system\\ \begin{equation} \left\{ \begin{array}{rllll} \mc L_M(u)&=\lambda f(x)|u|^{q-2}u+…
This paper aims to establish the existence of a weak solution for the non-local problem: \begin{equation*} \left\{\begin{array}{ll} -a\left(\int_{\Omega}\mathcal{H}(x,|\nabla u|)dx \right) \Delta_{\mathcal{H}}u &=f(x,u) \ \ \hbox{in} \ \…
We study the existence and multiplicity of solutions for a class of fractional Schr\"{o}dinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \begin{gather*} \begin{cases}…
In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{…
We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ A special feature of this paper is that the nonlinearity $f$ and the potential $V$ are…
Even without a variational background, a multiplicity result of positive solutions with ordered $L^{p}(\Omega)$-norms is provided to the following boundary value problem \begin{equation*} \left \{ \begin{array}{ll}…
In this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)\Delta…