Related papers: On Neumann superlinear elliptic problems

We construct multibump nodal solutions of the elliptic equation $$ -\Delta u=a^+[\lambda u+ f(\, \cdot\,, u)]-\mu a^- g(\, \cdot\,, u) $$ in $H^1_0(\Omega)$, when $\mu$ is large, under appropriate assumptions, for $f$ superlinear and…

We study a weakly coupled supercritical elliptic system of the form \begin{equation*} \begin{cases} -\Delta u = |x_2|^\gamma \left(\mu_{1}|u|^{p-2}u+\lambda\alpha |u|^{\alpha-2}|v|^{\beta}u \right) & \text{in }\Omega,\\ -\Delta v =…

We consider the second order semilinear elliptic system $\Delta u= p\left( x\right) v^\alpha,$ $\Delta v= q\left(x\right) u^\beta,$ where $x \in \mathbf{R}^N,$ $N \geq 3,$ $\alpha$ and $\beta$ are positive constants, $p$ and $q$ are…

In this paper we deal with existence and uniqueness of solution to super-linear problems for the Pucci operator: $$ -\M^+(D^2u)+|u|^{s-1}u=f(x) \quad {in} \RR^n, $$ where $s>1$ and $f$ satisfies only local integrability conditions. This…

In this paper we deal with positive solutions for singular quasilinear problems whose model is $$ \begin{cases} -\Delta u + \frac{|\nabla u|^2}{(1-u)^\gamma}=g & \mbox{in $\Omega$,}\newline \hfill u=0 \hfill & \mbox{on $\partial\Omega$,}…

We prove new results concerning the nonlinear scalar field equation \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = g(u)&\quad \hbox{in }\mathbb{R}^N,\; N\geq 3, u\in H^1(\mathbb{R}^N)& \end{array} \right. \end{equation*} with a…

Given $\Omega$ a bounded open subset of $\mathbb{R}^N$, we consider nonnegative solutions to the singular semilinear elliptic equation $-\Delta\,u\,=\,\frac{f}{u^{\beta}}$ in $H^1_{loc}(\Omega)$, under zero Dirichlet boundary conditions.…

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=\Delta u+v^p,\qquad & x\in\Omega,\,\,\,t>0,\\ \partial_t…

The paper concerns with positive solutions of problems of the type $-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $\Omega\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $\Omega$ can be an exterior domain, i.e.…

We introduce a method for solving Calder\'on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain…

In this work we consider the following class of elliptic problems $$- \Delta_A u + u = a(x) |u|^{q-2}u+b(x) |u|^{p-2}u , \mbox{ in } \mathbb{R}^N, $$ $u\in H^1_A (\mathbb{R}^N)$, with $2<q<p<2^*= \frac{2N}{N-2}$, $a(x)$ and $b(x)$ are…

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.

This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem $$ \left\{ \begin{array}{l} {\rm{div}} \left(…

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times…

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nabla v) &{\rm…

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\delta$ be the distance to $\partial \Omega$. We study positive solutions of equation (E) $-L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$ where $L_\mu=\Delta +…

It is known that the elliptic function solutions of the nonlinear Schr\"odinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, $\displaystyle{…

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\…

This paper proves the existence of nontrivial solution for two classes of quasilinear systems of the type \begin{equation*} \left\{\; \begin{aligned} -\Delta_{\Phi_{1}} u&=F_u(x,u,v)+\lambda R_u(x,u,v)\;\text{ in } \Omega& \\…

In this paper, we consider the following parabolic-parabolic-elliptic system } \begin{align*} \left\{\aligned & u_t=\Delta u-\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w)+au-\mu u^{\alpha}, && x\in\Omega, t>0,\\ & v_t=\Delta…