Related papers: Classification of the stable solution to biharmoni…

We consider the equation $-\Delta_p u=f(u)$ in a smooth bounded domain of $\mathbb{R}^n $, where $\Delta_p$ is the $p$-Laplace operator. Explicit examples of unbounded stable energy solutions are known if $n\geq p+4p/(p-1)$. Instead, when…

In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven for $N\geq 2$ and…

We study the regularity of the extremal solution of the semilinear biharmonic equation $\beta \Delta^2 u-\tau \Delta u=\frac{\lambda}{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=\Delta u=0$ on $\partial B$,…

We classify the stable solutions (positive or sign-changing, radial or not) to the following nonlocal Lane-Emden equation: $(-\Delta)^s u=|u|^{p-1}u$ in $\mathbb{R}^n$ for $2<s<3$.

We consider the biharmonic equation $\Delta^2 u = u^\alpha$ in $\mathbf R^n$ with $n \geqslant 1$. It was proved that this equation has a positive classical solution if, and only if, either $\alpha \leqslant 1$ with $n \geqslant 1$ or…

We are concerned with the following semi-linear polyharmonic equation with integral constraint \begin{align} \left\{\begin{array}{rl} &(-\Delta)^pu=u^\gamma_+ ~~ \mbox{ in }{\mathbb{R}^n},\\ \nonumber…

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is the polyharmonic…

In this paper we prove the Liouville type theorem for stable at infinity solutions of the following equation $$\Delta_{m}^{3}u =|u|^{\theta-1}u\;\;\; \mbox{in}\,\, \mathbb{R}^N,$$ for $1<m-1<\theta<\theta_{s, m}:=\frac{N(m-1)+3m }{N-3m}.$…

\noindent Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form \begin{equation*} \Delta \left(r_{n}\left(\Delta \left(x_{n}+p_{n}x_{n-k}\right) \right)…

In this paper we study the semilinear elliptic problem $$ -\Delta u -k^2u=Q|u|^{p-2}u\quad\text{ in }\mathbb{R}^2, $$ where $k>0$, $p\geq 6$ and $Q$ is a bounded function. We prove the existence of real-valued $W^{2,p}$-solutions, both for…

Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$ \left\{ \begin{array}{l} \Delta^2 u +(\lambda V(x)+1)u = f(u), \quad \mbox{in} \quad \mathbb{R}^{4}, u \in…

We examine the equation given by \begin{equation} \label{eq_abstract} -\Delta u + a(x) \cdot \nabla u = u^p \qquad \mbox{in $ \IR^N$,} \end{equation} where $p>1$ and $ a(x)$ is a smooth vector field satisfying some decay conditions. We show…

This paper is devoted to the study of stable radial solutions of $-\Delta u=f(u) \mbox{ in } \mathbb{R}^N\setminus B_1=\{ x\in \mathbb{R}^N : \vert x\vert\geq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We prove that such solutions…

In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation $$ \left(a+b\left(\int_{\mathbb R^N}|\Delta u|^p dx\right)^{\theta-1}\right) \Delta_{p}^{2}u = \alpha…

For the following Neumann problem in a ball $$\begin{cases} -\Delta_p u+u^{p-1}=u^{q-1}\quad&\text{in }B,\\ u>0,\,u\text{ radial}\quad&\text{in }B,\\ \frac{\partial u}{\partial \nu}=0\quad&\text{on }\partial B, \end{cases}$$ with…

In this paper, we are concerned with stable solutions to the fractional elliptic equation $$ (-\Delta)^s u=e^u\mbox{ in }\mathbb R^{N}, $$ where $(-\Delta)^s$ is the fractional Laplacian with $0<s<1$. We establish the nonexistence of stable…

We consider the problem -{\epsilon}^2\Delta_gu+u = |u|^{p-2}u in M, where (M,g) is a symmetric Riemannian manifold. We give a multiplicity result for antisymmetric changing sign solutions.

We discuss the existence of positive superharmonic functions $u$ in $\mathbb{R}^N_+=\mathbb{R}^{N-1}\times (0, \infty)$, $N\geq 3$, in the sense $-\Delta u=\mu$ for some Radon measure $\mu$, so that $u$ satisfies the nonlocal boundary…

We consider the class of stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain of $\mathbb{R}^n$. Since 2010 an interior a priori $L^\infty$ bound for stable solutions is known to hold in dimensions $n \leq 4$…

In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \Delta u - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2…