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We consider the inhomogeneous Allen-Cahn equation $$ \epsilon^2\Delta u\,+\,V(y)(1-u^2)\,u\,=\,0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu}\,=\,0\quad \mbox{on}\ \partial \Omega, $$ where $\Omega$ is a bounded domain in…

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following elliptic Dirichlet problem $$ \begin{cases} -\Delta\upsilon= e^{\upsilon}-s\phi_1-4\pi\alpha\delta_p-h(x)\,\,\,\,…

Let $\Omega$ be a bounded open set and $p,q,r>1$. The main observation of the present work is the following: $W_0^{1,p}(\Omega)$-solutions of the equation $-\Delta_p u = \mu |u|^{q-2}u + |u|^{r-2}u$ parameterized by $\mu$ are in bijection…

In this paper we find a positive weak solution for a semipositone $p(\cdot )$- Laplacian problem. More precisely, we find a solution for the problem \[ \left\{ \begin{array}{cc} -\Delta _{p(\cdot )}u=f(u)-\lambda & \text{in }\Omega \\ u>0 &…

Let $G=(V,E)$ be a locally finite graph, $\Omega\subset V$ be a bounded domain, $\Delta$ be the usual graph Laplacian, and $\lambda_1(\Omega)$ be the first eigenvalue of $-\Delta$ with respect to Dirichlet boundary condition. Using the…

We study similarity solutions to the multidimensional aggregation equation $u_t+\Div(uv)=0$, $v=-\nabla K*u$ with general power-law kernels $K(x)=|x|^\alpha,\alpha\in (2-d,2)$. We analyze the equation in different regimes of the parameter…

We consider the equation $-\epsilon^{2}\Delta u + u = u^ {p}$ in a bounded domain $\Omega\subset\R^{3}$ with edges. We impose Neumann boundary conditions, assuming $1<p<5$, and prove concentration of solutions at suitable points of…

We study the self-similar solutions of the equation \[ u_{t}-div(| \nabla u| ^{p-2}\nabla u)=0, \] in $\mathbb{R}^{N},$ when $p>2.$ We make a complete study of the existence and possible uniqueness of solutions of the form \[ u(x,t)=(\pm…

We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: \begin{equation*} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) \Delta u+\mu V(x)u=\lambda…

In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~\Omega, u>0& \mbox{in}~\Omega, u= 0& \mbox{on}~\partial\Omega, \end{array}…

In this paper we consider the following coupled gradient-type quasilinear elliptic system \begin{equation*} \left\{ \begin{array}{ll} - {\rm div} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &\hbox{ in $\Omega$,}\\[10pt] -…

We study the following Brezis-Nirenberg problem of Kirchhoff type $$ \left\{\aligned &-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u = \lambda|u|^{q-2}u + \delta |u|^{2}u, &\quad \text{in}\ \Omega, \\ &u=0,& \text{on}\ \partial\Omega,…

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 are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{\Omega\times \Omega} \…

We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation $$P_\alpha u_\alpha=\Delta_g^k u_\alpha+\hbox{lot}=|u_\alpha|^{2^\star-2-\epsilon_\alpha} u_\alpha\hbox{ in }M$$ that behave like…

In this paper we extend the classical sub-supersolution Sattinger iteration method to $1$-Laplace type boundary value problems of the form \begin{equation*} \begin{cases} \displaystyle -\Delta_1 u = F(x,u) & \text{in}\;\Omega,\\ \newline…

We study the problem% \[ -\Delta v+\lambda v=| v| ^{p-2}v\text{ in }\Omega ,\text{\qquad}v=0\text{ on $\partial\Omega$},\text{ }% \] for $\lambda\in\mathbb{R}$ and supercritical exponents $p,$ in domains of the form% \[…

In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{…

The aim of this paper is to establish two results about multiplicity of solutions to problems involving the $1-$Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem $$ \left\{…

We study the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on…