Related papers: Normalized solution for Kirchhoff equation with up…

We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda…

We study normalized solutions for the nonlinear Schrodinger (NLS) equation with potential and Sobolev critical nonlinearity. By establishing suitable assumptions on the potential, together with new techniques, we find a mountain-pass type…

This paper deals with the existence of ground states for degenerative ($a=0$) and non-degenerative ($a>0$) double weighted critical Kirchhoff equation \begin{eqnarray*} \left\{ \begin{array}{ll} \displaystyle-\left(a+b\int_B |\nabla…

We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*}…

In this paper, we are concerned with solutions to the following nonlinear Schr\"odinger equation with combined inhomogeneous nonlinearities, $$ -\Delta u + \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\, \R^N,…

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrodinger equations with mixed nonlinearities. The standard approach based on the Pohozaev identity to obtain normalized…

We consider the Choquard system with both linear and nonlinear couplings $-\Delta u + \mu_1 u =\lambda_1 ( I_\alpha * |u|^{r_1} ) |u|^{r_1-2} u + \beta p( I_\alpha * |v|^q)|u|^{p-2} u + \kappa v,$ $-\Delta v + \mu_2 v =\lambda_2 ( I_\alpha…

In the present paper, we establish the uniqueness and nondegeneracy of positive energy solutions to the Kirchhoff equation \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+u=|u|^{p-1}u & & \text{in…

In this paper, we consider the existence of solutions for the linearly coupled Choquard system with potentials \begin{align*} \left\{\begin{aligned} &-\Delta u+\lambda_1 u+V_1(x)u=\mu_1(I_{\alpha}\star|u|^p)|u|^{p-2}u+\beta(x) v,\\ &-\Delta…

In this paper, we study the fractional Kirchhoff equation with critical nonlinearity \begin{align*} \left(a+b\int_{\mathbb R^N}|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^su+u=f(u)\ \ \mbox{in}\ \ \mathbb R^N, \end{align*} where $N>2s$…

We study the Sobolev critical Schr\"odinger-Bopp-Podolsky system \begin{gather*} -\Delta u+\phi u=\lambda u+\mu|u|^{p-2}u+|u|^4u\quad \text{in }\mathbb{R}^3, -\Delta\phi+\Delta^2\phi=4\pi u^2\quad \text{in } \mathbb{R}^3, \end{gather*}…

We look for normalized solutions to the nonlinear Schr\"{o}dinger equation with mixed fractional Laplacians and combined nonlinearities $$ \left\{\begin{array}{ll} (-\Delta)^{s_{1}} u+(-\Delta)^{s_{2}} u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u…

We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho,…

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schr\"{o}dinger equation with general nonlinearity: Mass super-critical case: \[\begin{cases} -\Delta u+V(x)u+\lambda…

We study the Choquard equation with a local perturbation \begin{equation*} -\Delta u=\lambda u+(I_\alpha\ast|u|^p)|u|^{p-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*}…

This paper is concerned with the following fractional Schr\"{o}dinger equations involving critical exponents: \begin{eqnarray*} (-\Delta)^{\alpha}u+V(x)u=k(x)f(u)+\lambda|u|^{2_{\alpha}^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N},…

This paper aims to establish the existence of a weak solution for the non-local problem: \begin{equation*} \left\{\begin{array}{ll} -a\left(\int_{\Omega}\mathcal{H}(x,|\nabla u|)dx \right) \Delta_{\mathcal{H}}u &=f(x,u) \ \ \hbox{in} \ \…

We study the Schr\"{o}dinger-Poisson-Slater equation \begin{equation*}\left\{\begin{array}{lll} -\Delta u + \lambda u + \big(|x|^{-1} \ast |u|^{2}\big)u = V(x) u^{ p_{\varepsilon}-1 }, \, \text{ in } \mathbb{R}^{3},\\[2mm]…

In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb…

For $N\ge 3$ and $2<p<N$, we find normalised solutions to the equation \begin{align*} -\Delta_p u+(1+V(x))|u|^{p-2}u+\lambda u&=|u|^{q-2}u\qquad\text{in $\mathbb{R}^N$}\\ \|u\|_2&=\rho \end{align*} in the mass supercritical and Sobolev…