Related papers: A note on the classification of positive solutions…

Under the assumption of finite energy, positive solutions to the critical p-Laplace equation in $\mathbb{R}^n$ for $1< p<n$ have been classified completely by moving plane method. In this paper, the author provide a new approach to obtain…

In this paper we provide the classification of positive solutions to the critical $p-$Laplace equation on $\mathbb{R}^n$, for $1<p<n$, possibly having infinite energy. If $n=2$, or if $n=3$ and $\frac 32<p<2$ we prove rigidity without any…

We provide the classification of the positive solutions to $-\Delta_p u =u^{p^*-1}$ in $\mathcal {D}^{1,p}(\R^N)$ in the case $2<p<N$. Since the case $1<p\leq2$ is already known this provides the complete classification for $1<p<N$.

By providing optimal or nearly optimal integral estimates, we show that every positive, bounded or moderately growing, local weak solution to the critical $p$-Laplace equation in $\mathbb{R}^n$, with $n\geq 3$, and whose infimum over a ball…

For $N\ge2$ and $1<p<N$, we classify all positive $\mathcal{D}^{1,p}(\mathbb{R}^N)$-solutions to $p$-Laplace equations with a critical Hardy-Sobolev exponent and a Hardy potential.

For $1<p<n$, it is well-known that non-negative, energy weak solutions to $\Delta_p u + u^{p^{\ast}-1} =0$ in $\mathbb{R}^n$ are completely classified. Moreover, due to a fundamental result by Struwe and its extensions, this classification…

In this paper, for general $n\geq2$, we classify solutions to $n$-Laplacian Liouville equation with positive nonlinear Neumann boundary condition on the half-space $\mathbb{R}^{n}_{+}$. Under the positive nonlinear Neumann boundary…

In this paper, we study the existence of nonnegative solutions for a class of multivalued $(p,N)$-Laplace problems having discontinuous nonlinearity with critical exponential growth in $\mathbb{R}^N$. To demonstrate the existence results,…

The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation $(-\Delta_p)^s u= |u|^{p^{*}_s -2} u +\lambda f(x,u)$ in a bounded domain with…

In this paper, we are concerned with the critical Hardy-Sobolev equation \begin{equation*} -\Delta_{p}u = \frac{u^{p^{*}_s-1}}{|x|^{s}}, \ \ x\in \mathbb{R}^n \end{equation*} where $p^{*}_s = \frac{(n-s)p}{n-p}$ denotes the critical…

In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation}\label{0.1} -\Delta^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation} without the finite volume constraint for $n \geq…

In this paper, we investigate positive solutions to a class of Laplace equations with a gradient term on a complete, connected, and noncompact Riemannian manifold \((M^n,g)\) with nonnegative Ricci curvature, namely \[-\Delta u =…

We prove the existence of ground state positive solutions for a class of semipositone $p$-Laplacian problems with a critical growth reaction term. The proofs are established by obtaining crucial uniform $C^{1,\alpha}$ a priori estimates and…

We construct clustering positive solutions for a perturbed critical elliptic equation on a closed manifold of dimension $n=4,5$. Such a construction is already available in the literature in dimensions $n\ge 6$ (see for instance…

We study positive supersolutions to an elliptic equation $(*)$: $-\Delta u=c|x|^{-s}u^p$, $p,s\in\bf R$ in cone-like domains in $\bf R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_*<1$ such…

In this paper, we classify the solutions of the critical semilinear problem involving the logarithmic Laplacian $$(E)\qquad \qquad\qquad\qquad\qquad \mathcal{L}_\Delta u= k u\log u,\qquad u\geq0 \quad \ {\rm in}\ \ \mathbb{R}^n,…

In this paper, we investigate the existence of multiple positive solutions to the following multi-critical Schr\"{o}dinger equation \begin{equation} \label{p} \begin{cases} -\Delta u+\lambda V(x)u=\mu…

This article focuses on the study of the existence, multiplicity and concentration behavior of ground states as well as the qualitative aspects of positive solutions for a $(p, N)$-Laplace Schr\"{o}dinger equation with logarithmic…

This work deals with the existence of at least two positive solutions for the class of quasilinear elliptic equations with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*}…

In this paper, positive solutions to the Laplace equation with 1-dimensional circular singularities are investigated. First, we establish $L^p$ integrability estimates for such solutions $u$ near the singularities, in comparison with…