Related papers: Nonlinear elliptic equations with measures revisit…

We shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form, \begin{equation}\label{con-c} \left \{ \begin{array}{ll} -\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, & x \in \Omega\\ u=0, & x…

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-\Delta)_{p(\cdot)}^{s}…

In this paper we consider positive supersolutions of the nonlinear elliptic equation \[- \Delta u = \rho(x) f(u)|\nabla u|^p, \qquad \hfill \mbox{ in } \Omega,\] where $0\le p<1$, $ \Omega$ is an arbitrary domain (bounded or unbounded) in $…

Let $\Omega\subset \mathbb{R}^N$ be a bounded regular domain, $0<s<1$ and $N>2s$. We consider $$ (P)\left\{ \begin{array}{rcll} (-\Delta)^s u &= & \frac{u^{q}}{d^{2s}} & \text{ in }\Omega , \\ u &> & 0 & \text{in }\Omega , \\ u & = & 0 &…

We analyze the semilinear elliptic equation $\Delta u=\rho(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that…

We study existence and nonexistence of strictly positive solutions for the elliptic problems of the form $Lu=m\left( x\right) u^{p}$ in a bounded open interval, with zero boundary conditions, where $L$ is a strongly uniformly elliptic…

In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^{2,2}_{\textrm{loc}}(\Omega)$ of the nonlinear elliptic problem \begin{equation} \tag{P} \label{abstr} \left\{ \begin{array}{ll}…

This paper addresses the following problem. \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u=\lambda I_\alpha*_\Omega u+|u|^{2^*-2}u\mbox{ in }\Omega ,\nonumber u\in H_0^1(\Omega).\nonumber \end{array} \right. \end{equation} Here,…

We consider least energy solutions to the nonlinear equation $-\Delta_g u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space $\mathbb H^n$ as well as manifolds with…

We consider equations of the form $-L_\mu u +f(u)=0$ in a smooth domain $\Omega$, where $L_\mu=\Delta + \mu\delta^{-2}$ and $\delta(x)$ denotes the distance of the point $x$ to the boundary of the domain. The nonlinear term $f$ is positive,…

We consider the problem -\Delta u - g(u) = \lambda u, u \in H^1(\R^N), \int_{\R^N} u^2 = 1, \lambda\in\R, in dimension $N\ge2$. Here $g$ is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where…

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…

We discuss the existence and regularity of solutions to the following Dirichlet problem: $$\begin{equation} \begin{cases} -\textrm{div}\left(\frac{Du}{(1+|u|)^{\theta}}\right)= -\textrm{div}\left(u^{\gamma}E(x)\right)+f(x) \qquad & \mbox{in…

In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -\Delta u= h(x,u)\quad \ &{\rm in}\ \Omega,\\[1.5mm] \phantom{ -\Delta }…

We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-\Delta u = (1-u) u^m - \lambda u^n$ in a bounded domain $\Omega \subset…

In this paper, we focus our attention on the positive solutions to second-order nonlinear ordinary differential equations of the form $u''+q(t)g(u)=0$, where $q$ is a sign-changing weight and $g$ is a superlinear function. We exploit the…

In this paper, we consider existence of positive solutions for the Schr\"odinger quasilinear elliptic problem $$ \left\{ \begin{array}{l} \Delta_pu+\Delta_p(|u|^{2\gamma})|u|^{2\gamma-2}u = a(x)g(u)~ \mbox{on}~ \mathbb{R}^N,\\ u>0\…

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_\alpha\ast u^p)u^q$ in $\Omega$, where $\Omega\subset \mathbb{R}^N,…

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $\Omega$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f…

In this paper, first we study carefully the positive solutions to $\Delta u+\lambda_{1}u\ln u +\lambda_{2}u^{b+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as…