Related papers: A note on radial solutions to the critical Lane-Em…

In this paper, we will study the following parabolic problem $u_t - div(\omega(x) \nabla u)= h(t) f(u) + l(t) g(u)$ with non-negative initial conditions pertaining to $C_b(\mathbb{R}^N)$, where the weight $\omega$ is an appropriate function…

In this paper, we prove that there exists at most one positive radial weak solution to the following quasilinear elliptic equation with singular critical growth \[ \begin{cases} -\Delta_{p}u-{\displaystyle…

We consider the critical focusing wave equation $(-\partial_t^2+\Delta)u+u^5=0$ in $\R^{1+3}$ and prove the existence of energy class solutions which are of the form [u(t,x)=t^\frac{\mu}{2}W(t^\mu x)+\eta(t,x)] in the forward lightcone…

We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 & \textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega \subset…

We find radial and nonradial solutions to the following nonlocal problem $$-\Delta u +\omega u= \big(I_\alpha\ast F(u)\big)f(u)-\big(I_\beta\ast G(u)\big)g(u) \text{ in } \mathbb{R}^N$$ under general assumptions, in the spirit of Berestycki…

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…

We are concerned with the existence and boundary behaviour of positive radial solutions for the system \begin{equation*} \left\{ \begin{aligned} \Delta u&=|x|^{a}v^{p} &&\quad\mbox{ in } \Omega, \\ \Delta v&=|x|^{b}v^{q}f(|\nabla u|)…

We study a nonlinear Schr\"{o}dinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- \Delta u+ u + \lambda^2 \left(\frac{1}{\omega|x|^{N-2}}\star \rho u^2\right) \rho(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \]…

We study a semilinear elliptic problem with a singular nonlinear term of the type $g(u)=-u^{-1}$, using a variational approach. Note that the minus sign is important since the corresponding term in the Euler-Lagrange functional is concave.…

We consider the following fractional Schr\"{o}dinger equation involving critical exponent: \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u+V(|y'|,y'')u=u^{2^*_s-1} \ \hbox{ in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N,…

We prove existence of solutions to a nonlinear degenerate elliptic equation of the form \[ \begin{cases} -\Delta_{1} u+ \frac{|D u|}{(1-u)^{\gamma}}=g & \mbox{in $\Omega$,}\\ u=0 \hfill & \mbox{on $\partial\Omega$,} \end{cases} \] in a…

In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \Delta u - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2…

Given (M, g0) we consider the problem -{\epsilon}^2Delta_{g0+h}u + u = (u+)^{p-1} with ({\epsilon}, h) \in (0, {\epsilon}0) \times B{\rho}. Here B{\rho} is a ball centered at 0 with radius {\rho} in the Banach space of all Ck symmetric…

In this paper, we consider radial distributional solutions of the quasilinear equation $-\Delta_N u=f(u)$ in the punctured open ball $ B_R\backslash\{0\}\subset \RR^N$, $N \geq 2$. We obtain sharp conditions on the nonlinearity $f$ for…

We prove scattering below the mass-energy threshold for the focusing inhomogeneous nonlinear Schr\"odinger equation \begin{equation} iu_t + \Delta u + |x|^{-b}|u|^{p-1}u=0, \end{equation} when $b \geq 0$ and $N > 2$ in the intercritical…

The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane-Emden's equation $-\Delta u=|u|^{p-1} u$ with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution $u$…

We investigate the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+|u|^{p-1}u=0$ with $1+\frac{4}{N}<p<1+\frac{4}{N-2}$ (when $N=1, 2$, $1+\frac{4}{N}<p<\infty$) in energy space $H^1$ and study the divergent property of…

We prove existence of radially symmetric solutions and validity of Euler-Lagrange necessary conditions for a class of variational problems such that neither direct methods nor indirect methods of Calculus of Variations apply. We obtain…

In this paper, we study a class of generalized extensible beam equations with a superlinear nonlinearity \begin{equation*} \left\{ \begin{array}{ll} \Delta ^{2}u-M\left( \Vert \nabla u\Vert _{L^{2}}^{2}\right) \Delta u+\lambda V(x) u=f(…

In this paper, we consider the following Schr\"{o}dinger equation: \begin{equation*} \begin{cases} -\Delta u=\lambda u+\mu|u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u(x)|^2dx=a,\quad u\in H^1(\mathbb{R}^N),\\…