Related papers: Multiple positive solutions to elliptic boundary b…

In this article, we investigate the existence and multiplicity of solutions to the Robin problem \begin{equation*} \begin{cases} -\Delta u = \lambda f(u) & \text{in } \Omega, \frac{\partial u}{\partial \nu} + \gamma u=0 & \text{on }…

In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain $\Omega\subset \R^N$, $N> 2s$: % \begin{eqnarray*} (P_\lambda)\left\{\begin{array}{lll}…

The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017] classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy--Sobolev growth $$-\Delta u=|x|^{-s}…

We prove that a class of superlinear indefinite problems with homogeneous Neumann boundary conditions admits an arbitrarily high number of positive solutions, provided that the parameters of the problem are adequately chosen. The…

We build infinitely many geometrically distinct non-radial sign-changing solutions for the Hamiltonian-type elliptic systems $$ -\Delta u =|v|^{p-1}v\ \hbox{in}\ \mathbb{R}^N,\ -\Delta v =|u|^{q-1}u\ \hbox{in}\ \mathbb{R}^N,$$ where the…

We prove the existence of multiple positive BV-solutions of the Neumann problem $$ \begin{cases} \displaystyle -\left(\frac{u'}{\sqrt{1+u'^2}}\right)'=a(x)f(u)\quad&\mbox{in }(0,1), u'(0)=u'(1)=0,& {cases} $$ where $a(x) > 0$ and $f$…

We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^{q}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes…

We establish the existence and multiplicity of positive solutions to the problems involving the fractional Laplacian: \begin{equation*} \left\{\begin{array}{lll} &(-\Delta)^{s}u=\lambda u^{p}+f(u),\,\,u>0 \quad &\mbox{in}\,\,\Omega,\\…

We prove the monotonicity of positive solutions to the problem $-\Delta u = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some…

In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…

In this paper we shall classify all positive solutions of $ \Delta u =a u^p$ on the upper half space $ H =\Bbb{R}_+^n$ with nonlinear boundary condition $ {\partial u}/{\partial t}= - b u^q $ on $\partial H$ for both positive parameters $a,…

In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-\Delta_p)^s u&=& \frac{\lambda}{u^\gamma}+u^q~\text{in}~\Omega,\nonumber…

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 =…

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -\Delta u+ \lambda V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in…

In this work we study the existence of nodal solutions for the problem $$ -\Delta u = \lambda u e^{u^2+|u|^p} \text{ in }\Omega, \; u = 0 \text{ on }\partial \Omega, $$ where $\Omega\subseteq \mathbb R^2$ is a bounded smooth domain and…

We consider the elliptic equation with boundary singularities \begin{equation} \begin{cases} -\Delta u=-\lambda |x|^{-s_{1}}|u|^{p-2}u+|x|^{-s_{2}}|u|^{q-2}u &\text { in } \varOmega , u(x)=0 &\text { on } \partial \varOmega , \end{cases}…

The paper deals with the equation $-\Delta u+a(x) u +b(x)u^q -u^p = 0$, $u \in H^1(\R^N)$, whith $N\ge 2$, $1<q<p,\ p<{N+2\over N-2}$ if $N\ge 3$, $\inf a>0$, $a(x)\to a_\infty$ and $b(x)\to 0$ as $|x|\to\infty$. When $a(x)\le a_\infty$ and…

We verify the existence of radial positive solutions for the semi-linear equation $$ -\,\Delta u=u^{p}\,-\,V(y)\,u^{q},\,\quad\quad u>0,\quad\quad\mbox{ in }\mathbb{R}^N$$ where $N\geq 3$, $p$ is close to $p^*:=(N+2)/(N-2)$, and $V$ is a…

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm…

We consider the problem -{\epsilon}^2\Delta_gu+u = |u|^{p-2}u in M, where (M,g) is a symmetric Riemannian manifold. We give a multiplicity result for antisymmetric changing sign solutions.