Related papers: Infinitely many sign-changing solutions for Kirchh…
This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities \begin{equation*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u(x)|^{2}\right) \Delta u =\lambda…
In this paper we study the Kirchhoff problem \begin{equation*} \left \{ \begin{array}{ll} -m(\| u \|^{2})\Delta u = f(u) & \mbox{in $\Omega$,} u=0 & \mbox{on $\partial\Omega$,} \end{array}\right. \end{equation*} in a bounded domain,…
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 paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_{\Omega}\vert\nabla u\vert^2dx)\Delta u+\lambda u=\vert u\vert^{p-2}u & \text{…
We consider the problem -{\epsilon}^2\Delta_gu+u = |u|^{p-2}u in M, where (M,g) is a symmetric Riemannian manifold. We give a multiplicity result for antisymmetric changing sign solutions.
We show the existence of infinitely many positive solutions, defined on the real line, for the nonlinear scalar ODE \[ \ddot u + (a^+(t) - \mu a^-(t)) u^3 = 0, \] where $a$ is a periodic, sign-changing function, and the parameter $\mu>0$ is…
We show the linking-type result which allows us to study strongly indefinite problems with sign-changing nonlinearities. We apply the abstract theory to the singular Schr\"{o}dinger equation $$ -\Delta u + V(x)u + \frac{a}{r^2} u = f(u) -…
In this paper we study the following class of nonlocal {problems} involving Caffarelli-Kohn-Nirenberg type critical growth \begin{align*} L(u)&-\lambda h(x)|x|^{-2(1+a)}u=\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\;\; \text{in } \mathbb R^N,…
We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-\Delta u=|v|^{q-2}v,\quad -\Delta v=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical…
In this paper we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $L_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary…
In this paper, we consider the following Kirchhoff type problem $$\left\{\aligned&-\biggl(a + b\int_{\mathbb{R}^N} |\nabla u|^2 dx \biggr) \Delta u + V(x) u = |u|^{p-2}u &\text{ in } \mathbb{R}^N,\cr &u\in H^1(\mathbb{R}^N),…
This paper is dedicated to studying the existence of nontrivial positive solutions for a Kirchhoff-type problem with sign change nonlinearities and a singular term, Using the Nehari manifold and EkelandS variational principle we prove that…
For the $p$-Laplace Dirichlet problem (where $\varphi (t)=t|t|^{p-2}$, $p>1$) \[ \varphi(u'(x))'+ f(u(x))=0 \;\;\;\; \mbox{for $-1<x<1$}, \;\; u(-1)=u(1)=0 \] assume that $f'(u)>(p-1)\frac{f(u)}{u}>0$ for $u>\gamma>0$, while $\int_u^\gamma…
In this paper we are concerned with some $p$-Kirchhoff type problems involving sign-changing weight functions. We prove the existence of multiple positive solutions of the problem via the Nehari manifold approach.
We consider the following nonlinear Schrodinger equation [{l} \Delta u-(1+\delta V)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $ f(u)$ is a superlinear nonlinearity satisfying…
In this paper, we first establish the uniqueness and non-degeneracy of positive solutions to the fractional Kirchhoff problem \begin{equation*}…
In this paper, we consider the following Kirchhoff problem $$ \left\{\aligned -\bigg(a+b\int_{\Omega}|\nabla u|^2dx\bigg)\Delta u&= \lambda u^{q-1} + \mu u^{2^*-1}, &\quad \text{in }\Omega, \\ u&>0,&\quad\text{in }\Omega,\\…
In this paper, we prove the existence of infinitely many solutions of a doubly critical Choquard-Kirchhoff type equation \begin{equation*} \begin{split}…
In this paper, we consider the existence of multiple nodal solutions of the nonlinear Choquard equation \begin{equation*} \ \ \ \ (P)\ \ \ \ \begin{cases} -\Delta u+u=(|x|^{-1}\ast|u|^p)|u|^{p-2}u \ \ \ \text{in}\ \mathbb{R}^3, \ \ \ \ \\…
We prove that the superlinear indefinite equation \begin{equation*} u" + a(t)u^{p} = 0, \end{equation*} where $p > 1$ and $a(t)$ is a $T$-periodic sign-changing function satisfying the (sharp) mean value condition $\int_{0}^{T} a(t)~\!dt <…