Related papers: Explicit solutions and multiplicity results for so…

In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution,…

We consider the critical $p$-Laplacian system \begin{equation}\label{92} \begin{cases}-\Delta_p u-\frac{\lambda a}{p}|u|^{a-2}u|v|^b =\mu_1|u|^{p^\ast-2}u+\frac{\alpha\gamma}{p^\ast}|u|^{\alpha-2}u|v|^{\beta}, &x\in\Omega,\\ -\Delta_p…

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}\big(|\nabla u|^{p-2} \nabla u+ \mu(x) |\nabla u|^{q-2} \nabla u\big)=\lambda|u|^{\vartheta-2}u+|u|^{p^*-2}u$ in $\Omega$ and $u= 0$ on…

For $1<p<2$ and $q$ large, we prove the existence of two positive, nonconstant, radial and radially nondreacreasing solutions of the supercritical equation \[-\Delta_p u+u^{p-1}=u^{q-1}\] under Neumann boundary conditions, in the unit ball…

In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by \begin{equation*} \label{1} \left\{\begin{split} \mathcal{L}u\: &= \lambda u^{q} + u^{p}, \quad u>0 ~~ \text{in} ~\Omega,…

We study analytical and computational aspects for Dirichlet problem on the unit ball $B$: $|x|<1$ in $R^n$, modeled on the equation \[ \Delta u +\lambda \left(u^p+u^q \right)=0, \;\; \mbox{in $B$}, \;\; u=0 \s \mbox{on $\partial B$}, \]…

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{on} \;\; \mathbb{R}^n \] in the…

We prove the existence of $N - 1$ distinct pairs of nontrivial solutions of the scalar field equation in ${\mathbb R}^N$ under a slow decay condition on the potential near infinity, without any symmetry assumptions. Our result gives more…

In this paper, we first study a class of discrete $p$-Laplacian systems with logarithmic coupling on locally finite graphs. The system is specifically designed to capture the variational interplay between nonlinear diffusion and logarithmic…

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 paper, we study {existence and multiplicity} of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation*}\label{Eq-Equation1} \left\{\begin{array}{l} -\Delta u-\Delta_q u+\lambda u=f(u) \quad x \in…

We consider the problem of uniqueness of positive solutions to boundary value problems containing the equation: -\Delta_p u =K(|x|)f(u), p>1. f is positive, is locally Lipschitz and satisfies some superlinear growth condition after u_0, a…

Let $\Omega \subset \mathbb{R}^d$ be a bounded open set containing zero, $s \in (0,1)$ and $p \in (1, \infty)$. In this paper, we first deal with the existence, non-existence and some properties of ground-state solutions for the following…

We consider the boundary value problem \begin{equation} - \Delta u = \lambda c(x)u+ \mu(x) |\nabla u|^2 + h(x), \qquad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega), \leqno{(P_{\lambda})} \end{equation} where $\Omega \subset \R^N, N \geq 3$…

In this paper we study existence of ground state solution to the following problem $$ (- \Delta)^{\alpha}u = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^{\alpha}(\mathbb R^N) $$ where $(-\Delta)^{\alpha}$ is the fractional Laplacian,…

We prove a result of existence of positive solutions of the Dirichlet problem for $-\Delta_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $\Delta_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight…

We prove an existence result for a $p$-Laplacian problem set in the whole Euclidean space and exhibiting a critical term perturbed by a singular, convective reaction. The approach used combines variational methods, truncation techniques,…

We study the existence of solutions with multiple concentration to the following boundary value problem $$-\Delta u=\e^2 e^u-4\pi \sum_{p\in Z}\alpha_p \delta_{p}\;\hbox{in} \Omega,\quad u=0 \;\hbox{on}\partial \Omega,$$ where $\Omega$ is a…

In this article we prove the global existence of weak solutions to an initial boundary value problem with an exponential and p-Laplacian nonlinearity. The equation is a continuum limit of a family of kinetic Monte Carlo models of crystal…

Given $k\in\mathbb N$, we define a class of continuous piecewise functions $f$ having abrupt but controlled magnitude changes so that the problem $$\Delta u +f(u)=0,\quad x\in \mathbb R^N, N> 2, $$ has at least $k$ radially symmetric ground…