Related papers: $n$Kirchhoff type equations with exponential nonli…
This paper focuses on the critical Kirchhoff equation with concave perturbation \begin{align*} \begin{cases} \displaystyle -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=|u|^4u+\lambda|u|^{q-2}u\ \ &\mbox{in}\ \Omega, \displaystyle u=0\ \…
Let $\Omega:=\left( a,b\right) \subset\mathbb{R}$, $m\in L^{1}\left( \Omega\right) $ and $\lambda>0$ be a real parameter. Let $\mathcal{L}$ be the differential operator given by $\mathcal{L}u:=-\phi\left( u^{\prime}\right) ^{\prime}+r\left(…
In this paper, we consider the existence, multiplicity and the asymptotic behavior of prescribed mass solutions to the following nonlinear Kirchhoff equation with mixed nonlinearities: -(a+b\int|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u+\lambda…
We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega…
For $1<p<\infty$, we consider the following problem $$ -\Delta_p u=f(u),\quad u>0\text{ in }\Omega,\quad\partial_\nu u=0\text{ on }\partial\Omega, $$ where $\Omega\subset\mathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is…
We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…
In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)\Delta u+V(x)u+\lambda u=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying…
In this paper, we study the existence of normalized solutions to the following Kirchhoff equation with a perturbation: $$ \left\{ \begin{aligned} &-\left(a+b\int _{\mathbb{R}^{N}}\left | \nabla u \right|^{2} dx\right)\Delta u+\lambda…
In this paper, we investigate the existence of infinitely many solutions for the following elliptic boundary value problem with $(p,q)$-Kirchhoff type \begin{eqnarray*} \begin{cases} -\Big[M_1\left(\int_\Omega|\nabla u_1|^p…
We show the existence of a nodal solution with two nodal domains for a generalized Kirchhoff equation of the type $$ -M\left(\displaystyle\int_\Omega \Phi(|\nabla u|)dx\right)\Delta_\Phi u = f(u) \ \ \mbox{in} \ \ \Omega, \ \ u=0 \ \…
We are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{\Omega\times \Omega} \…
We consider the following $(p, q)$-Laplacian Kirchhoff type problem \begin{align*} \begin{split} &-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{p}\, dx \right)\Delta_{p}u - \left(c+d\int_{\mathbb{R}^{3}}|\nabla u|^{q}\, dx \right ) \Delta_{q}u…
On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $\Delta^2 u -\alpha \Delta u +\beta u = u^q$, where $\alpha$ and $\beta$ are positive constants satisfying that $\alpha^2 \geq 4…
In this paper, we study a solvability result for the nonlinear problem $$ \mbox {div } \left ( \vert \nabla_\omega u\vert^{p-2}\nabla_\omega u \right )+v(x) u^{q-1}+\mu u^{\gamma-1}=0, \quad z\in \Omega, \quad u \Big \vert_{\partial…
Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:\Omega\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem $-\Delta…
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 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}…
The paper concerns with positive solutions of problems of the type $-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $\Omega\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $\Omega$ can be an exterior domain, i.e.…
We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where…
In this work we analyze the existence of solutions to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-\Delta )^s u &=&|\nabla u|^{\alpha}+ f &\inn \Omega_T\equiv\Omega\times (0,T),\\ u(x,t)&=&0 &…