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In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…

In this paper we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schr\"{o}dinger equations with square root of the Laplacian $$ \left\{ \begin{array}{lr}…

We examine the H\'enon equation $ -\Delta u =|x|^\alpha u^p$ in $ \Omega \subset \mathbb{R}^N$ with $u=0$ on $ \partial \Omega$ where $ 0 < \alpha$. We show there exists a sequence $ \{p_k\}_k \subset [ \frac{N+2}{N-2}, p_{\alpha}(N)]$ with…

We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions…

In this paper, we study fully nonlinear second-order elliptic and parabolic equations with Neumann boundary conditions on compact Riemannian manifolds with smooth boundary. We derive oscillation bounds for admissible solutions with Neumann…

We are concerned with the existence of solution of the problem $ -\Delta ^H_pu+|u|^{p-2}u=\lambda|u|^{q-2}u+ |u|^{p^*-2}u\quad \mbox{in}\quad\Omega,$ $u>0\quad \mbox{in}\quad\Omega,$ $a(\nabla u)\cdot \nu =0\quad \mbox{on}\quad\partial…

We study a nonlinear Schr\"{o}dinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- \Delta u+ u + \lambda^2 \left(\frac{1}{\omega|x|^{N-2}}\star \rho u^2\right) \rho(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \]…

We consider the boundary value problem $-\Delta_p u = \lambda c(x) |u|^{p-2}u + \mu(x) |\grad u|^p + h(x)$, $u \in W^{1,p}_0(\Omega) \cap L^{\infty}(\Omega)$, where $\Omega \subset \mathbb R^N$, $N \geq 2$, is a bounded domain with smooth…

In this work, we study the existence and regularity results of anisotropic elliptic equations with a singular lower order term that grows naturally with respect to the gradient and unbounded coefficients. We take up the following model…

We study elliptic and parabolic problems governed by singular elliptic operators \begin{equation*} \mathcal L =\sum_{i,j=1}^{N+1}q_{ij}D_{ij}+\frac c y D_y \end{equation*} in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N,…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $\lambda\to 0$ and…

We prove that all positive solutions of $-\Delta u = u^{\frac{2n}{n-2}}$ on the upper half space $\mathbb{R}^n_{+}$ (for $n \geq 3$) satisfying the boundary condition $D_{x_n}u = -u^{\frac{n}{n-2}}$ are of the form $u(x) = a \left(…

We study local and global properties of solutions of --$\Delta$u = u p + M ||u| q in a domain $\Omega$ of R N , in the range min{p, q} > 1 and M $\in$ R. We prove a priori estimates and existence or non-existence of ground states.

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla…

We study the existence of bound and ground states for a class of nonlinear elliptic systems in $\mathbb{R}^N$. These equations involve critical power nonlinearities and Hardy-type singular potentials, coupled by a term containing up to…

This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type \begin{equation*} \left\{ \begin{aligned} -\partial_{xx}u + (-\Delta)_{y}^{s_{1}} u + u - u^{2_{s_{1}}^{}-1} = \kappa…

In this paper, we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{aligned} &-\Delta u=f(u)+ \lambda u\quad \mbox{in}\ \mathbb{R}^{N},\\ &u\in…

In this paper, we consider the following nonlinear elliptic equation with Dirichlet boundary condition: $-\Delta u=K(x)u^{\frac{n+2}{n-2}},\, u>0$ in $\Omega,\, u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in…

In this paper we prove the existence of a signed ground state solution in the mountain pass level for a class of asymptotically linear elliptic problems, even when the nonlinearity is just continuous in the second variable. The (strongly)…