Related papers: Normalized Solutions to Kirchhoff Equation with No…

In this paper, we study normalized solutions for the following critical Schr\"odinger-Bopp-Podolsky system: $$-\Delta u + q(x)\phi u = \lambda u + |u|^{p-2}u + \bigl(I_\alpha * |u|^{3+\alpha}\bigr)|u|^{1+\alpha}u,\quad \text{in }…

We consider the singular elliptic problem of the form \[ -\Delta u + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] where the coefficients are allowed to have low regularity. Under natural…

We consider the following singularly perturbed Kirchhoff type equations $$-\varepsilon^2 M\left(\varepsilon^{2-N}\int_{\R^N}|\nabla u|^2 dx\right)\Delta u +V(x)u=|u|^{p-2}u~\hbox{in}~\R^N, u\in H^1(\R^N),N\geq 1,$$ where $M\in…

In this paper, we study the following biharmonic Schr\"odinger equation with potential and mixed nonlinearities \begin{equation*} \left\{\begin{array}{ll}\Delta^2 u +V(x,y)u+\lambda u =\mu|u|^{p-2}u+|u|^{q-2}u,\ (x, y) \in \Omega_r \times…

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},…

We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^p\Delta_p u+V(x)|u|^{p-2}u=\lambda…

In this paper, we study the existence of normalized solutions for the following quasilinear Schr\"odinger equation with Sobolev critical exponent: \begin{eqnarray*} -\Delta u-u\Delta (u^2)+\lambda…

We study the existence of normalized solutions to the following logarithmic Schr\"{o}dinger equation \begin{equation*}\label{eqs01} -\Delta u+\lambda u=\alpha u\log u^2+\mu|u|^{p-2}u, \ \ x\in\R^N, \end{equation*} under the mass constraint…

In this paper the existence of solutions, $(\lambda,u)$, of the problem $$-\Delta u=\lambda u -a(x)|u|^{p-1}u \quad \hbox{in }\Omega, \qquad u=0 \quad \hbox{on}\;\;\partial\Omega,$$ is explored for $0 < p < 1$. When $p>1$, it is known that…

In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schr\"{o}dinger equation given by \begin{align*} \left\{ \begin{aligned} &-\epsilon^2 \Delta u+V( x)u=\lambda u+u \log u^2,…

By introducing some new tricks, we prove that the nonlinear problem of Kirchhoff-type \begin{equation*} \left\{ \begin{array}{ll} -\left(a+b\int_{\R^3}|\nabla u|^2\mathrm{d}x\right)\triangle u+V(x)u=f(u), & x\in \R^3; u\in H^1(\R^3),…

In this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)\Delta…

We consider the following autonomous Kirchhoff-type equation \begin{equation*} -\left(a+b\int_{\mathbb{R}^N}|\nabla{u}|^2\right)\Delta u= f(u),~~~~u\in H^1(\mathbb{R}^N), \end{equation*} where $a\geq0,b>0$ are constants and $N\geq1$. Under…

In this paper, we consider the existence and multiplicity of normalized solutions for the following $p$-Laplacian critical equation \begin{align*} \left\{\begin{array}{ll} -\Delta_{p}u=\lambda\lvert u\rvert^{p-2}u+\mu\lvert…

In this paper, we study the existence and non-existence of normalized solutions to the lower critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -\Delta u+\lambda u=\gamma…

In this paper, we consider solutions to the following nonlinear Schr\"odinger equation with competing Hartree-type nonlinearities, $$ -\Delta u + \lambda u=\left(|x|^{-\gamma_1} \ast |u|^2\right) u - \left(|x|^{-\gamma_2} \ast |u|^2\right)…

Two main results are presented: 1) a new class of applied problems that lead to equations with $(p,q)$-Laplace is presented; 2) a method for solving nonlinear boundary value problems involving $(p,q)$-Laplace with measurable unbounded…

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert…

We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schr\"odinger equation $$ \begin{array}{c} -\Delta u +V(x)u=\lambda a(x)u^{-\gamma}+b(x)u^{p}~\mbox{in}~ \mathbb{R}^{N}, \end{array}…

We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth \[\left\{\begin{gathered} - \Bigl({\varepsilon ^2}a + \varepsilon b\int_{{\R^3}} {{{\left| {\nabla u} \right|}^2}}…