Related papers: Ground state solutions for weighted biharmonic pro…

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 consider the problem \[-\Delta u + W(x)u = ((1/{|x|^{\alpha}} * |u|^{p}) |u|^{p-2}u, u \in H_{0}^{1}(\Omega)\], where $\Omega$ is an exterior domain in $\mathbb{R}^{N}$, $N\geq3,$ $\alpha \in(0,N)$, $p\in[2,(2N-\alpha)/(N-2)$, $W$ is…

We establish the uniqueness of the higher radial bound state solutions of $$ \Delta u +f(u)=0,\quad x\in \RR^n. \leqno(P) $$ We assume that the nonlinearity $f\in C(-\infty,\infty)$ is an odd function satisfying some convexity and growth…

The higher order Kirchhoff type equation $$\int_{\mathbb{R}^{2m}}(|\nabla^m u|^2 +\sum_{\gamma=0}^{m-1}a_{\gamma}(x)|\nabla^{\gamma}u|^2)dx \left((-\Delta)^m u+\sum_{\gamma=0}^{m-1}(-1)^\gamma \nabla^\gamma\cdot(a_\gamma (x)\nabla^\gamma…

In this paper we analyze the semi-linear fractional Laplace equation $$(-\Delta)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\mathbb{R}^N_+=\{x=(x',x_N)\in…

In this paper we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\…

In this paper we consider nodal radial solutions of the problem $$ \begin{cases} -\Delta u=|u|^{2^*-2}u+\lambda u&\text{ in }B,\\ u=0&\text{ on }\partial B \end{cases} $$ where $2^*=\frac{2N}{N-2}$ with $3\le N\le6$ and $B$ is the unit ball…

In this paper, we establish the existence of ground state solutions for Choquard equations \begin{equation}\label{eq 1} - \Delta u + u = q\,(I_\alpha \ast |u|^p) |u|^{q - 2} u+p\,(I_\alpha \ast |u|^q) |u|^{p - 2} u\quad {\rm in }\quad…

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…

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 prove the existence of non-radial solutions to the problem $-\triangle u=f(z,u)$, $u|_{\partial D}=0$ on the unit disc $D:=\{z\in \mathbb C : |z|<1\}$ with $u(z)\in \mathbb R^k$, where $f$ is a sub-linear continuous…

On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $\Delta^2 u -\alpha \Delta u +\beta u = u^q$, where $\alpha$ and $\beta$ are positive constants satisfying that $\alpha^2 \geq 4…

This paper concerns with the existence of nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & = \gamma H_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber u…

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$. In this paper, we consider the following nonlinear elliptic equation of $N$-Laplacian type: $-\Delta_{N}u=f(x,u)$ where $u\in W_{0}^{1,2}\{0}$ when $f$ is of subcritical or critical…

We study the Schr\"odinger equations $-\Delta u + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-\Delta u - \lambda u = f(x,u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and…

In this paper, we study the following coupled nonlinear logarithmic Hartree system \begin{align*} \left\{ \displaystyle \begin{array}{ll} \displaystyle -\Delta u+ \lambda_1 u =\mu_1\left( -\frac{1}{2\pi}\ln(|x|) \ast u^2 \right)u+\beta…

We propose and study a concept of renormalized solution to the problem $\Delta_p u=0$ in $\mathbb{R}^N_+$, $|\nabla u|^{p-2}u_{\nu} + g(u) = \mu$ on $\partial\mathbb{R}^N_+$, where $1<p\leq N$, $N\geq 2$,…

We investigate the problem $$-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \mbox{ in } \Omega, \quad \frac{\partial u}{\partial \mathbf{n}} = 0 \mbox{ on } \partial \Omega, \leqno{(P_\lambda)} $$ where $\Omega$ is a bounded smooth…

We propose a new method for constructing exact solutions to nonlinear delay reaction--diffusion equations of the form $$ u_t=ku_{xx}+F(u,w), $$ where $u=u(x,t)$, $w=u(x,t-\tau)$, and $\tau$ is the delay time. The method is based on…

We show that the elliptic problem $\Delta u+f(u)=0$ in $\mathbb{R}^N$, $N\geq 1$, with $f\in C^1(\mathbb{R})$ and $f(0)=0$ does not have nontrivial stable solutions that decay to zero at infinity, provided that $f$ is nonincreasing near the…