相关论文: Global bifurcation techniques for Yamabe type equa…

In this paper, we consider the existence and multiplicity of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation}\label{Equation1} \left\{\begin{aligned} &-\Delta u-\Delta_q u+\lambda u=g(u),\quad x \in…

In this paper we revisit the $\sigma_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $\sigma_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe…

Let $\Omega=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(\Omega)$, $\lambda,\mu>0$ be real parameters, and $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive…

In this paper, we employ the Nash-Moser iteration technique to study local and global properties of positive solutions to the equation $$\Delta_pv+bv^q+cv^r =0$$ on complete Riemannian manifolds with Ricci curvature bounded from below,…

In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type $$ (P_\lambda^\mu)\left\{ \begin{array}{lll} (-\Delta)^s u&=& \lambda(u^{q}-1)+\mu u^r \mbox{ in } \Omega\\ u&>&0 \mbox{ in…

For the reduction ordinary differential equation due to Baird and Kamissoko \cite{BK} for biharmonic maps from a Riemannian manifold $(M^m,g)$ into another one $(N^n,h)$, we show that this ODE has no global positive solution for every…

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet boundary…

We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2} u + \beta |u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in $\Omega$.…

We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations \begin{align*}\tag{$P_\lambda$} -\mathrm{div}(A(x)Du)=c_\lambda(x)u+( M(x)Du,Du)+h(x),\qquad u\in…

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on }…

For a Riemannian manifold with dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator.

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta…

We establish the multiplicity of positive solutions to a quasilinear Neumann problem in expanding balls and hemispheres with critical exponent in the boundary condition.

Given a conformally variational scalar Riemannian invariant $I$, we identify a sufficient condition for a compact Riemannian manifold to admit finite regular coverings with many nonhomothetic conformal rescalings with $I$ constant. We also…

In this paper we study the existence of solution for the following class of system of elliptic equations $$ \left\{ \begin{array}{lcl} -\Delta u=\left(a-\int_{\Omega}K(x,y)f(u,v)dy\right)u+bv,\quad \mbox{in} \quad \Omega -\Delta…

On a complete Riemannian manifold $(M,g)$, we consider $L^{p}_{loc}$ distributional solutions of the the differential inequality $-\Delta u + \lambda u \geq 0$ with $\lambda >0$ a locally bounded function that may decay to $0$ at infinity.…

We consider a Neumann boundary value problem driven by the anisotropic $(p,q)$-Laplacian plus a parametric potential term. The reaction is ``superlinear". We prove a global (with respect to the parameter) multiplicity result for positive…

The main goal of this paper is to address an important conjecture in the field of differential equations in the presence of a harmonic potential. While in the subcritical case, the uniqueness of positive solution has been addressed by…

Let $\Delta_k$ be the Dunkl Laplacian on $\mathbb R^d$ associated with a reflection group $W$ and a multiplicity function $k$. The purpose of this paper is to establish necessary and sufficient condition under which there exists a positive…

We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -\Delta u &= f_\lambda(|x|,u,|\nabla u|) &&\text{in }\Omega, u &= 0 &&\text{on }\partial\Omega, \end{aligned}…