Related papers: Normalized solutions for some quasilinear elliptic…

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(x) u=\lambda u+f(x)\left(I_\alpha *\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_\alpha…

We consider the semilinear fractional equation $ (I-\Delta)^s u = a(x) |u|^{p-2}u$ in $\mathbb{R}^N$, where $N \geq 3$, $0<s<1$, $2<p<2N/(N-2s)$ and $a$ is a bounded weight function. Without assuming that $a$ has an asymptotic profile at…

We study the monotonicity and one-dimensional symmetry of positive solutions to the problem $-\Delta_p u = f(u)$ in $\mathbb{R}^N_+$ under zero Dirichlet boundary condition, where $p>1$ and $f:(0,+\infty)\to\mathbb{R}$ is a locally…

In this paper, we consider the existence and multiplicity of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation}\label{Equation1} \left\{\begin{aligned} &-\Delta u-\Delta_q u+\lambda u=g(u),\quad x \in…

We consider the system of coupled elliptic equations \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 \\ -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v \end{cases} \text{in $\mathbb{R}^3$}, \] and study the existence of…

In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities,…

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…

We study the following class of linearly coupled Schr\"{o}dinger elliptic systems $$\left\{ \begin{array}{lr} -\Delta u+V_{1}(x)u=\mu|u|^{p-2}u+\lambda(x)v, & \quad x\in\mathbb{R}^{N}, \\ -\Delta v+V_{2}(x)v=|v|^{q-2}v+\lambda(x)u, &…

We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} \Delta(a(x)\Delta u) = a(x) \left\vert u \right\vert^{p-2-\epsilon} u \ \text{ in } \ \Omega, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial \Omega,…

In this paper, we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \frac{\lambda}{u^{\gamma}}+g(u)+\mu~\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\,…

We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-\Delta)^s u+V(\varepsilon x)u=\lambda…

In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems \begin{align*} \left\{ \begin{aligned} &-\epsilon^2\Delta u+V(x)u=\lambda u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,…

This paper is concerned with the following focusing biharmonic Schr\"{o}dinger equation with mixed dispersion and Sobolev critical growth: $$ \begin{cases} {\Delta}^2u-\Delta u-\lambda u-\mu|u|^{p-2}u-|u|^{4^*-2}u=0\ \ \mbox{in}\…

We are interested in the existence of normalized solutions to the problem \begin{equation*} \begin{cases} (-\Delta)^m u+\frac{\mu}{|y|^{2m}}u + \lambda u = g(u), \quad x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\…

The paper addresses an open problem raised in [Bartsch, Molle, Rizzi, Verzini: Normalized solutions of mass supercritical Schr\"odinger equations with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence of normalized…

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…

The paper concerns the existence of normalized solutions to a large class of quasilinear problems, including the well-known Born-Infeld operator. In the mass subcritical cases, we study a global minimization problem and obtain a ground…

We investigate normalized groundstates for mixed $(p,2)$-Laplacian equations \begin{align*} \begin{cases} -\Delta_p u-\Delta u+\lambda u=f(u) & \text{in } \mathbb{R}^2, \displaystyle \int_{\mathbb{R}^2}|u|^2\,\mathrm{d}x=m, u\in…

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 study positive supersolutions to an elliptic equation $(*)$: $-\Delta u=c|x|^{-s}u^p$, $p,s\in\bf R$ in cone-like domains in $\bf R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_*<1$ such…