Related papers: Multiple solutions for a class of quasilinear prob…
This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem $$ \left\{ \begin{array}{l} {\rm{div}} \left(…
In this note, we deal with a problem of the type $$\cases {-h\left ( \int_{\Omega}|\nabla u(x)|^2dx\right ) \Delta u=f(u) & in $\Omega$\cr & \cr u_{|\partial\Omega}=0\ .\cr}$$ As an application of a new general multiplicity result, we…
In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain…
In the present paper, we establish a multiplicity result for a following class of nonlocal Neumann eigenvalue problems involving the fractional p-Laplacian. \begin{align} \begin{cases} (-\Delta)^{s}_{p}u + a(x) \abs{u}^{p-2}u =\lambda…
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial…
We prove existence and multiplicity results for the nonlinear and nonlocal PDE $$ - \Delta u + (I_\alpha \star |u|^p)\, |u|^{p-2}\, u = f(|x|,u) \quad \textrm{in} \,\,\mathbb {R}^N, $$ where $N \geq 2$, $I_\alpha : \mathbb{R}^N \setminus…
It is established existence and multiplicity of solutions for strongly nonlinear problems driven by the $\Phi$-Laplacian operator on bounded domains. Our main results are stated without the so called $\Delta_{2}$ condition at infinity which…
We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…
For $p \in (1, \infty),$ for an integer $N \geq 2$ and for a bounded Lipschitz domain $\Omega$, we consider the following nonlinear Steklov bifurcation problem \begin{equation*} \begin{aligned} -\Delta_p \phi & = 0 \; \text{in} \ \Omega, \\…
We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-\Delta_p u = \lambda f(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function…
In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is \begin{equation*} \begin{cases} -\Delta_{p} u -\text{div} (c(x)|u|^{p-2}u)) =f & \text{in}\ \Omega, \\ \left( |\nabla…
This article deals with the study of the following nonlinear doubly nonlocal equation: \begin{equation*} (-\Delta)^{s_1}_{p}u+\ba(-\Delta)^{s_2}_{q}u = \la a(x)|u|^{\delta-2}u+ b(x)|u|^{r-2} u,\; \text{ in }\; \Om, \; u=0 \text{ on }…
We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-\Delta)^m u=h(x,u)\quad&\mbox{in }\Omega,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partial\Omega,…
In this paper, we consider the following quasilinear elliptic problem with potential $$(P) \begin{cases} -\mbox{div}(\phi(x,|\nabla u|)\nabla u)+ V(x)|u|^{q(x)-2}u= f(x,u) & \ \ \mbox{ in }\Omega, u=0 & \ \ \mbox{ on } \partial\Omega,…
We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…
We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} \Delta(a(x)\Delta u) = a(x) \left\vert u \right\vert^{p-2-\epsilon} u \ \text{ in } \ \Omega, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial \Omega,…
We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -\Delta u + \mu u=v^p, &\hbox{in } \Omega, \\-\Delta v + \mu v=u^q, &\hbox{in } \Omega, \\\frac{\partial u}{\partial n} = \frac{\partial…
We are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{\Omega\times \Omega} \…
We investigate the problem $$ \left\{ \begin{array}{ll} -\Delta_p u = g(u)|\nabla u|^p + f(x,u) \ & \mbox{in} \ \ \Omega, \ \ \\ u>0 \ &\mbox{in} \ \ \Omega, \ \ u = 0 \ &\mbox{on} \ \ \partial\Omega, \end{array} \right. \leqno{(P)} $$ in a…
In this article, we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (P_\la):\;\quad (-\De)^s u = u^{-q} + \la u^{{2^*_s}-1}, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…