Related papers: Existence and Nonexistence of Positive Solutions o…

Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u = h(x) u^{p-1}$ for $2<p$. The function $h$ may have an indefinite sign. Key ingredients are a $h$-dependent…

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…

We consider the existence and nonexistence of positive solution for the following Br\'ezis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -\Delta u={\left|u\right|}^{{2}^{\ast }-2}u+\lambda u+\mu u\log…

We consider the existence and multiplicity of positive solutions for the following critical problem with logarithmic term: \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -\Delta u={\mu\left|u\right|}^{{2}^{\ast }-2}u+\nu…

We prove new results on the existence of positive radial solutions of the elliptic equation $-\Delta u= \lambda h(|x|,u)$ in an annular domain in $\mathbb{R}^{N}, N\geq 2$. Existence of positive radial solutions are determined under the…

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,…

In this paper we prove the existence of a positive solution of the nonlinear and nonlocal elliptic equation in $\mathbb{R}^n$ \[ (-\Delta)^s u =\varepsilon h u^q+u^{2_s^*-1} \] in the convex case $1\leq q<2_s^*-1$, where $…

Let $G=(V,E)$ be a locally finite graph, whose measure $\mu(x)$ have positive lower bound, and $\Delta$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some…

We consider positive solution to the weighted elliptic problem \begin{equation*} \left \{ \begin{array}{ll} -{\rm div} (|x|^\theta \nabla u)=|x|^\ell u^p \;\;\; \mbox{in $\mathbb{R}^N \backslash {\overline B}$},\\ u=0 \;\;\; \mbox{on…

In this paper, we consider the existence (and nonexistence) of solutions to \[ -\mathcal{M}_{\lambda,\Lambda}^\pm (u'') + V(x) u = f(u) \quad {\rm in} \ \mathbf{R} \] where $\mathcal{M}_{\lambda,\Lambda}^+$ and…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous…

In the present work, we establish the existence of two positive solutions for singular nonlocal elliptic systems. More precisely, we consider the following nonlocal elliptic problem: $$\left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u =…

For $N\geq 3$, by the seminal paper of Brezis and V\'eron (Arch. Rational Mech. Anal. 75(1):1--6, 1980/81), no positive solutions of $-\Delta u+u^q=0$ in $\mathbb R^N\setminus \{0\}$ exist if $q\geq N/(N-2)$; for $1<q<N/(N-2)$ the existence…

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp…

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{ in $\Omega$}, \\ \quad u>0 &\text{ in $\Omega$}, \\ \quad u=0 & \text{ on $\partial \Omega$}, \end{cases} \] where $L(v)=-{\rm…

We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-\Delta_p u = \lambda f(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function…

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}…

This paper studies the properties of solutions to a class of elliptic and parabolic problems involving the fractional Laplacian. By applying the mountain pass theorem, we prove the existence of bounded classical positive solutions in the…

We study the existence of positive solutions on $\R^{N+1}$ to semilinear elliptic equation $-\Delta u+u=f(u)$ where $N\geq 1$ and $f$ is modeled on the power case $f(u)=|u|^{p-1}u$. Denoting with $c$ the mountain pass level of $\f(u)=\tfrac…

In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \[ \displaystyle \left\{\begin{array}{rcll} (-\Delta)_p^s(u) &=& \lambda f(u) \qquad & \text{in} \ \ \Omega \\u &=& 0 & \text{in} \…