Related papers: Poincar\'{e} Sobolev equations in the Hyperbolic s…

In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -\Delta u - \mu \frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n), \end{equation} where $x = (y,z) \in \mathbb{R}^h…

In this article we will study the existence and nonexistence of sign changing solutions for the Brezis-Nirenberg type problem in the Hyperbolic space. We will also establish sharp asymptotic estimates for the solutions and the compactness…

We study the existence and the profile of sign-changing solutions to the slightly subcritical problem $$ -\De u=|u|^{2^*-2-\eps}u \hbox{in} \cB, \quad u=0 \hbox{on}\partial \cB, $$ where $\cB$ is the unit ball in $\rr^N$, $N\geq 3$,…

On the hyperbolic space, we study a semilinear equation with non-autonomous nonlinearity having a critical Sobolev exponent. The Poincar\'e-Sobolev equation on the hyperbolic space explored by Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa…

We study the existence and multiplicity of sign changing solutions of the following equation $ \begin{cases} -\Delta u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^t}+a(x)u \quad\text{in}\quad \Omega, u=0…

We consider semilinear elliptic problems on two-dimensional hyperbolic space involving critical growth. We first establish the Palais-Smale(P-S) condition and using (P-S) condition we obtain existence of solutions. In addition, we also…

We consider the log-perturbed Br\'ezis-Nirenberg problem on the hyperbolic space \begin{align*} \Delta_{\mathbb{B}^N}u+\lambda u +|u|^{p-1}u+\theta u \ln u^2 =0, \ \ \ \ u \in H^1(\mathbb{B}^N), \ u > 0 \ \mbox{in} \ \mathbb{B}^N,…

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$ -\Delta_{\mathbb{B}^N} u-\lambda u=a(x)u^{p-1} \, + \, \varepsilon u^{2^*-1}…

In this article we establish the radial symmetry of positive solutions of a p- Laplace equation in the Hyperbolic space, which is the Euler Lagrange equation of the p- Poincare Sobolev inequality in the Hyperbolic space. We will also…

We build infinitely many geometrically distinct non-radial sign-changing solutions for the Hamiltonian-type elliptic systems $$ -\Delta u =|v|^{p-1}v\ \hbox{in}\ \mathbb{R}^N,\ -\Delta v =|u|^{q-1}u\ \hbox{in}\ \mathbb{R}^N,$$ where the…

We study the nonlinear eigenvalue problem for the p-Laplacian, and more general problem constituting the Fucik spectrum. We are interested in some vanishing properties of sign changing solutions to these problems. Our method is applicable…

We study the higher-order Schr\"odinger equation with critical Sobolev exponent on the hyperbolic space $\mathbb{H}^n$: $$P_m u + a(x)\,u = |u|^{q-2}u, \quad u \in D^{m,2}(\mathbb{H}^n),$$ where $P_m$ is the GJMS operator of order $2m$, $q…

We study the existence of sign changing solutions to the following problem $$ (P) \quad \quad \quad \left\{ \begin{array}{ll} \Delta u+|u|^{p-1}u=0 \quad & {\rm in} \quad \Omega_\epsilon; u=0 \quad & {\rm on} \quad\partial \Omega_\epsilon,…

We consider a priori estimates of possibly sign-changing solutions to superlinear parabolic problems and their applications (blow-up rates, energy blow-up, continuity of blow-up time, existence of nontrivial steady states etc). Our…

This article investigates the multiplicity of solutions to the Brezis-Nirenberg problem on smooth bounded domains in the hyperbolic space $\mathbb{B}^N$ for $N \ge 4$. Specifically, we study the critical semilinear equation…

We study the existence of positive solutions for the following class of scalar field problem on the hyperbolic space $$ -\Delta_{\mathbb{H}^N} u - \lambda u = a(x) |u|^{p-1} \, u\;\;\text{in}\;\mathbb{B}^{N}, \quad u \in…

We establish the interior and boundary H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{p-2}u\big)-\Delta_p u=0,\quad p>1. \] The proof…

In this note we prove that: \begin{theorem} for $2\leq s<\frac{n}{2}$ or $1\leq s<\frac{2n}{n+1}$ or $1\leq s<\frac{n}{2}$ but n is even, $(-\Delta)^{s}(u)=|u|^{q-2}u,q=\frac{2n}{n-2s}$ has infinitely many sign changing solutions or…

We consider the Schr\"odinger-Poisson-Slater (SPS) system in $\R^3$ and a nonlocal SPS type equation in balls of $\mathbb R^3$ with Dirichlet boundary conditions. We show that for every $k\in\mathbb N$ each problem considered admits a nodal…

A classical result owing to Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] asserts that all positive solutions of the Poincar\'e-Sobolev equation on the hyperbolic space $$ -\Delta_{\mathbb{B}^n} u-\lambda u =…