Related papers: On a superquadratic elliptic system with strongly …
In this article, we will prove the existence of infinitely many positive weak solutions to the following nonlocal elliptic PDE. \begin{align} (-\Delta)^s u&= \frac{\lambda}{u^{\gamma}}+ f(x,u)~\text{in}~\Omega,\nonumber…
We study the Dirichlet problem for systems of the form -\Delta u^k=f^k(x,u)+\mu^k, x\in\Omega, k=1,...,n, where \Omega\subset R^d$ is an open (possibly nonregular) bounded set, \mu^1,...,\mu^n are bounded diffuse measures on \Omega,…
We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(\phi(|\nabla u|)\nabla u\big)+V(x)\phi (|u|)u=f(u)\qquad u\in W^{1,\Phi}(\mathbb{R}^{N}), \] where $\phi$ and $f$ satisfy suitable conditions. The positive…
In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -\Delta_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $\Omega$,} \newline u\geq 0 & \text{in $\Omega$,} \newline u=0 & \text{on $\partial…
We study the fractional Schr\"{o}dinger equations coupled with a neutral scalar field $$ (-\Delta)^s u+V(x)u=K(x)\phi u +g(x)|u|^{q-2}u, \quad x\in \mathbb{R}^3,\qquad (I-\Delta)^t \phi=K(x)u^2, \quad x\in \mathbb{R}^3, $$ where…
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] -…
In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \begin{equation*} \begin{cases} u_t -…
In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: \begin{equation*} \left\{\begin{array}{ll} -\Delta u_j + au_j = \mu_ju_j^3+\beta\sum_{k\ne j}u_k^2u_j, u_j>0\ \ \hbox{in}\ \Omega, u_j=0 \…
Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $\Omega \subset \mathbb{R}^d$ (possibly unbounded), $d\geq 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - \varphi (x,…
In this paper we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \lambda |u|^{q-2}u+f(x,u)+\mu\,\,\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\, \partial\Omega;…
In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-\Delta)^s u = \frac{\alpha}{2_s^*}|u|^{\alpha-2}u|v|^{\beta}+f(x)\;\;\text{in}\;\mathbb{R}^{N},…
We study parameterized elliptic systems on symmetric domains with additional system symmetries. We prove the existence of continua of nontrivial solutions bifurcating from the constant branch determined by a critical point of the potential,…
We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \[ \left\{ \begin{tabular} [c]{l}% $-\Delta u=\mu_{1}|u|^{{2}^{\ast}-2}u+\lambda\alpha|u|^{\alpha-2}|v|^{\beta }u,$\\ $-\Delta…
We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus \Omega)^2} = f(x,u) & \quad \mbox{in } \Omega \\ u = 0 &…
We consider the following problem $ -\Delta_{p}u= h(x,u) \mbox{ in }\Omega$, $u\in W^{1,p}_{0}(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $1<p<N$, with a smooth boundary. In this paper we assume that…
In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$ \left\{ \begin{array}{ll} -\Delta u+u=|u|^{r-2}u &\text{in} \; \Omega,\\ \\ \frac{\partial…
In this paper we study the initial boundary value problem for the system $\Delta v= u_{x_1},\ u_t-\mbox{div}\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\right)=-\nabla…
In this note we construct smooth bounded domains $\Omega \subset \mathbb R^2$, other than disks, for which the overdetermined problem $$ \left\{ \begin{alignedat}{2} \Delta u + \lambda u &= 0 &\qquad& \text{ in } \Omega, \newline u &= b…
We prove that the existence of a solution to a fully nonlinear elliptic equation in a bounded domain $\Omega$ with an overdetermined boundary condition prescribing both Dirichlet and Neumann constant data forces the domain $\Omega$ to be a…
In this paper, we consider the following nonlinear critical Schr\"odinger system: \begin{eqnarray*}\begin{cases} -\Delta u=K_1(y)u^{2^*-1}+\frac{1}{2} u^{\frac{2^*}{2}-1}v^\frac{2^*}{2}, \,\,\,\,\,y\in\Omega,\,\,\,\,\,u>0,\cr -\Delta…