Related papers: Eigenvalue problem for a p-Laplacian equation with…

This article considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian \begin{align*} \begin{cases} \left(\vert u'\vert^{p-2}u'\right)'+\left(\lambda+r(x)\right)\vert u\vert ^{p-2}u=0,\,\, x\in (0,\pi_{p}),\\…

We consider a number of boundary value problems involving the $p$-Laplacian. The model case is $-\Delta_p u=V|u|^{p-2}u$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R}^n$. We derive necessary conditions for the existence of…

In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(.)$-Laplacian type equation with logarithmic nonlinearity: $$ u_t-\Delta u_t-\mbox{div}\left(\left\vert \nabla…

In this paper, we consider the existence of normalized solutions for the following $p$-Laplacian equation \begin{equation*} \left\{\begin{array}{ll} -\Delta_{p}u-V(x)\lvert u\rvert^{p-2}u+\lambda\lvert u\rvert^{p-2}u=\lvert…

We are interested in the existence and asymptotical behavior for the least energy solutions of the following fractional eigenvalue problem \begin{equation*} (P)\quad (-\Delta)^{s}u+V(x)u=\mu u+am(x)|u|^{\frac{4s}{N}}u,\quad…

In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equations \begin{equation} \begin{cases} u_{t}\left(x,t\right)=\mbox{div}(|\nabla u\left(x,t\right)|^{p-2}\nabla u(x,t))+f(u(x,t)), &…

In this paper, we study the problem of minimizing the first eigenvalue of the $p-$Laplacian plus a potential with weights, when the potential and the weight are allowed to vary in the class of rearrangements of a given fixed potential $V_0$…

We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\Delta u= \lambda V(x) e^u-4\pi N \delta_0\;\mbox{ in } B_1,\quad u=0 \;\mbox{ on }\partial B_1,$$ where $B_1$ is the unit ball in…

This paper concerns the initial-boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation. By constructing a family of potential wells, we first present the explicit expression for the depth of potential well, and then…

In this article we prove that the first eigenvalue of the $\infty-$Laplacian $$ \left\{ \begin{array}{rclcl} \min\{ -\Delta_\infty v,\, |\nabla v|-\lambda_{1, \infty}(\Omega) v \} & = & 0 & \text{in} & \Omega v & = & 0 & \text{on} &…

Let $\Omega$ be a bounded smooth domain in $\RR^N$. We consider the problem $u_t= \Delta u + V(x) u^p$ in $\Omega \times [0,T)$, with Dirichlet boundary conditions $u=0$ on $\partial \Omega \times [0,T)$ and initial datum $u(x,0)= M \phi…

In this paper, we are concerned with the existence of least energy solutions for the following biharmonic equations: $$\Delta^2 u+(\lambda V(x)-\delta)u=|u|^{p-2}u \quad in\quad \mathbb{R}^N$$ where $N\geq 5, 2<p\leq\frac{2N}{N-4},…

We consider the initial boundary value problem of a pseudo-parabolic equation with singular potential and the exponent $p(x,t)$ depending on both spatial and temporal variables. We prove the finite time blow up and estimate the upper and…

We consider the eigenvalue problem for the fractional $p \& q-$Laplacian \begin{equation} \left\{\begin{aligned} (- \Delta)_p^{s}\, u + \mu(- \Delta)_q^{s}\, u+ |u|^{p-2}u+\mu|u|^{q-2}u=\lambda\ V(x)|u|^{p-2}u\quad & \text{in } \Omega\\…

This paper deals with the following Petrovsky equation with damping and nonlinear source \[u_{tt}+\Delta^2 u-M(\|\nabla u\|_2^2)\Delta u-\Delta u_t+|u_t|^{m(x)-2}u_t=|u|^{p(x)-2}u\] under initial-boundary value conditions, where $M(s)=a+…

This paper investigates the initial boundary value problem for a fractional pseudo-parabolic equation with singular potential. The global existence and blow-up of solutions to the initial boundary value problem are obtained at low initial…

We consider two eigenvalue problems for Laplacian on some specific doubly connected domain. In particular, we study the following two eigenvalue problems. Let $B_1$ be an open ball in $\mathbb{R}^n$ and $B_0$ be a ball contained in $B_1$.…

Let $M$ be a closed hypersurface in $\mathbb{R}^{n}$ and $\Omega$ be a bounded domain such that $M= \partial\Omega$. In this article, we obtain an upper bound for the first non-zero eigenvalue of the following problems. \begin{itemize}…

Let $G=(V,E)$ be a locally finite connected weighted graph, $\Delta$ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation $u_t=\Delta u + f(u)$ on $G$. The blow-up phenomenons of…

We study the existence and multiplicity of positive normalized solutions with prescribed $L^{2}$-norm for the Sobolev critical Schr\"odinger equation $-\Delta U + V(x) U = \lambda U + |U|^{2^*-2} U$ in $\mathbb{R}^N$, $\int_{\mathbb{R}^N}…