Related papers: Note on the multiplicity of solutions for nonlinea…

We study the existence of solutions for the nonlinear scalar field equation $$-\Delta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\},$$ where the potential $-\frac{(N-2)^2}{4|x|^2}$ is the critical Hardy…

We prove the existence of an infinite sequence of distinct non-radial nodal $G-$invariant solutions for the following critical nonlinear elliptic problem: $({\mathrm{P}})\quad {*{20}c} {-\Delta u = |u|^{4/(n-2)}u},\quad u\in…

In this paper the question of finding infinitely many solutions to the problem $-\Delta u+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and the potential $a(x)$ is a…

We study the following nonlinear scalar field equation $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ Here $f\in C(\mathbb{R},\mathbb{R})$, $m>0$ is a given constant and…

We study the existence of solutions of the following nonlinear Schr\"odinger equation $$ -\Delta u+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic…

We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-\Delta_p)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in W^{s,p}(\mathbb{R}^N),$$ where $s\in (0,1)$, $2 \leq p < \infty$, $sp…

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(\mu x))^{2}u= \mu |u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ \Omega, \ \ \ \ u \in…

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem: \begin{equation*} \left\{\begin{array}{rlll} -\Delta u&=|u|^{\frac{4}{N-2}}u, &\hbox{ in }\Omega,\\ u&=0, &\hbox{ 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…

We consider the nonlinear elliptic equation \begin{equation*} -\Delta u + V(x)u = f(u), \qquad u\in D^{1,2}_0(\Omega), \end{equation*} in an exterior domain $\Omega$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at…

We consider the following nonlinear fractional Schr\"{o}dinger equation $$ (-\Delta)^su+u=K(|x|)u^p,\ \ u>0 \ \ \hbox{in}\ \ R^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0<s<1$, $1<p<\frac{N+2s}{N-2s}$. Under some…

In this paper we study a class of critical Choquard equations with a symmetric potential, i.e. we consider the equation $$-\Delta u +V(|x|) u =\left(|x|^{-\mu}* |u|^{2^\star_\mu}\right)|u|^{2^\star_\mu-2}u,\quad\mbox{in}\quad\mathbb R^N$$…

We consider the following prescribed scalar curvature problem on $ S^N$ (*)$$\left\{\begin{array}{l} - \Delta_{S^N} u + \frac{N(N-2)}{2} u = \tilde{K} u^{\frac{N+2}{N-2}} {on} S^N, u >0 \end{array}\right. $$ where $ \tilde{K}$ is positive…

In this work, the following fractional Laplacian problem with pure critical nonlinearity is considered \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u=|u|^{\frac{4s}{N-2s}}u, &\mbox{in}\ \mathbb{R}^N, \\ u\in…

Many existence and nonexistence results are known for nonnegative radial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left|x\right| ^{-\alpha }dx)$ to the equation \[ -\triangle u+\dfrac{A}{\left| x\right| ^{\alpha…

We consider positive solutions $u$ of the semilinear biharmonic equation $\Delta^2 u = |x|^{-\frac{n+4}{2}} g(|x|^\frac{n-4}{2} u)$ in $\mathbb R^n \setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions…

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.

In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under $N \geq 2$. We also show the…

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation $$\begin{cases} -\Delta u-\displaystyle\frac \gamma{|x|^2}u=\displaystyle\frac{1}{|x|^s}|u|^{p_s-2}u & \text{ in }…

We consider the following critical semilinear nonlocal equation involving the fractional Laplacian $$ (-\Delta)^su=K(|x|)|u|^{2^*_s-2}u,\ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N>2+2s$, $0<s<1$, and…