Related papers: Superlinear elliptic inequalities on manifolds

Given $\Omega(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $\Omega$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear…

In this paper we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \lambda |u|^{q-2}u+f(x,u)+\mu\,\,\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\, \partial\Omega;…

We prove the existence of a positive {\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \begin{equation} \begin{split} (-\Delta)^su&= \frac{1}{u^\gamma}+\lambda…

In this work, we study the of positive ground state solution for the semilinear elliptic problem $$ \left\{ \begin{array} [c]{ll}% -\Delta u=u^{p(x)-1},\quad u>0 & \mathrm{in}\,G\subseteq\mathbb{R}^{N}% ,\,N\geq3\\ u\in D_{0}^{1,2}(G), &…

In this paper, we establish conditions on the weights that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of a semilinear elliptic system.

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,…

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\…

Two sharp comparison results are derived for three-dimensional complete noncompact manifolds with scalar curvature bounded from below. The first one concerns the Green's function. When the scalar curvature is nonnegative, it states that the…

In this paper we prove some symmetry results for entire solutions to the semilinear equation $-\Delta u=f(u)$, with $f$ nonincreasing in a right neighbourhood of the origin. We consider solutions decaying only in some directions and we give…

The aim of this paper is to prove the existence of solution for a partial differential equation involving a singularity with a general nonnegative, Radon measure $\mu$ as its nonhomogenous term which is given as \begin{eqnarray} -\Delta…

In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by \begin{equation*} \label{1} \left\{\begin{split} \mathcal{L}u\: &= \lambda u^{q} + u^{p}, \quad u>0 ~~ \text{in} ~\Omega,…

Let $(M^n,g)$ be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\Delta u+au\log u=0,$$ where $a$ is a nonzero…

In this paper, firstly, we study gradient estimates for positive solution of the following equation \begin{equation*} \Delta_\xi(u)-\partial_t u- q u =A(u),t\in (-\infty,\infty) \end{equation*} on metric measure space $…

We study the semilinear indefinite elliptic problem \[ -\Delta u = Q_\Omega |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $Q_\Omega = \chi_\Omega - \chi_{\mathbb{R}^N \setminus \Omega}$, $\Omega \subset \mathbb{R}^N$ is a bounded…

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem \[ \begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases} \] that branch…

We study degenerate quasilinear elliptic equations on Riemannian manifolds and obtain several Liouville theorems. Notably, we provide rigorous proof asserting the nonexistence of positive solutions to the subcritical Lane-Emden-Fowler…

Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation Pu = 0 in M admits a positive supersolution which is not a…

In this paper, we give a clear cut relation between the the volume growth $V(r)$ and the existence of nonnegative solutions to parabolic semilinear problem \begin{align}\tag{*}\label{*} \left\{ \begin{array}{ll} \Delta u - \partial_t u +…

We are concerned with global existence for semilinear parabolic equations on Riemannian manifolds with negative sectional curvatures. A particular attention is paid to the class of initial conditions which ensure existence of global…

Let (M^n,g) be a n-dimensional complete, non-compact and connected Riemannian manifold, with Ricci tensor Ricc_g and sectional curvature Sec_g. Assume Ricc_g\geq (1-n)B^2, and either p>2 and Sec_g(x)=o(dist^2(x,a)) when dist^2(x,a)\to\infty…