Related papers: Ground state solution for a Kirchhoff problem with…

We obtain the existence of ground state solution for the nonlocal problem $$ m\left(\int_{\mathbb{R}^2}(|\nabla u|^2 + b(x)u^2) \textrm{d}x\right)(-\Delta u + b(x)u) = A(x)f(u) \ \ \ \textrm{in} \ \ \ \mathbb{R}^2, $$ where $m$ is a…

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 the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho,…

In this paper we prove the existence of positive ground state solution for a class of linearly coupled systems involving Kirchhoff-Schr\"odinger equations. We study the subcritical and critical case. Our approach is variational and based on…

The higher order Kirchhoff type equation $$\int_{\mathbb{R}^{2m}}(|\nabla^m u|^2 +\sum_{\gamma=0}^{m-1}a_{\gamma}(x)|\nabla^{\gamma}u|^2)dx \left((-\Delta)^m u+\sum_{\gamma=0}^{m-1}(-1)^\gamma \nabla^\gamma\cdot(a_\gamma (x)\nabla^\gamma…

In this paper, we study the following class of nonlinear equations: $$ -\Delta u+V(x) u = \left[|x|^{-\mu}*(Q(x)F(u))\right]Q(x)f(u),\quad x\in\mathbb{R}^2, $$ where $V$ and $Q$ are continuous potentials, which can be unbounded or vanishing…

This paper deals with the Klein-Gordon-Maxwell system when the nonlinearity exhibits critical growth. We prove the existence of positive ground state solutions for this system when a periodic potential V is introduced. The method combines…

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…

We consider the existence of ground state solutions for a class of zero-mass Chern-Simons-Schr\"{o}dinger systems \[ \left\{ \begin{array}{ll} \displaystyle -\Delta u +A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u)-a(x)|u|^{p-2}u, \newline…

We are concerned with the following Kirchhoff type equation $$-\varepsilon^2 M \left(\varepsilon^{2-N} \int_{\mathbb{R}^N} | \nabla u|^2\, \mathrm{d} x\right) \Delta u+V(x)u = f(u),\ x \in \mathbb{R}^N,\ \ N\ge2, $$ where $M \in…

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $\lambda\to 0$ and…

In this paper, we are concerned with the ground state solutions of nonlinear fractional Schr\"odinger equation involving critical growth. Without Ambrosetti-Rabinowitz condition and monotonicity condition on the nonlinearity, we get the…

We study the existence of ground states to a nonlinear fractional Kirchhoff equation with an external potential $V$. Under suitable assumptions on $V$, using the monotonicity trick and the profile decomposition, we prove the existence of…

In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness for a class of nonlinear functionals in $H^{2}(\mathbb{R}^4)$. Using this result and the principle of symmetric criticality, we can…

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: (a+b\ds\int_{\R^3}|D u|^2\right)\Delta u+V(x)u=|u|^{p-1}u, u\in H^1(\R^3), u>0, $x\in \R^3, where $a,$ $b>0$ are constants, $2<p<5$…

In this paper, we consider Kirchhoff-Schrodinger equations with singular exponential nonlinearities in R^4,using singular Adams inequality and variational techniques, we get the existence of ground state solutions. Moreover, we also get the…

In this paper, we apply the method of the Nehari manifold to study the Kirchhoff type equation \begin{equation*} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \end{equation*} submitted to Dirichlet boundary conditions. Under a…

In this work, we establish the existence of solutions that change sign at low energy for a non-local weighted Kirchhoff problem in the set $\mathbb{R}^{N}, N>2$. The non-linearity of the equation is assumed to have exponential growth in…

This paper deals with the existence of ground states for degenerative ($a=0$) and non-degenerative ($a>0$) double weighted critical Kirchhoff equation \begin{eqnarray*} \left\{ \begin{array}{ll} \displaystyle-\left(a+b\int_B |\nabla…

In this article, we study the following non local problem $$g\big(\int_{B}w(x) |\Delta u|^{2}\big)\Delta(w(x)\Delta u) =|u|^{q-2}u +\ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial u}{\partial n}=0 \quad\mbox{ on } \quad\partial…