Related papers: Nonlinear Schr\"odinger equations with a critical,…

We consider the problem -\Delta u+V(x)u = f'(u)+g(x) in RN, under the assumption limx \rightarrow \infty V (x) = 0, and with the non linear term f with a double power behavior. We prove the existence two solutions when g is sufficiently…

In the present paper, we prove the existence of solutions $(\lambda, u)\in \R\times H^1(\R^N)$ to the following elliptic equations with potential $\displaystyle -\Delta u+(V(x)+\lambda)u=g(u)\;\hbox{in}\;\R^N, $ satisfying the normalization…

We study a nonlinear Schr\"{o}dinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- \Delta u+ u + \lambda^2 \left(\frac{1}{\omega|x|^{N-2}}\star \rho u^2\right) \rho(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \]…

We show the strong instability of radial ground state standing waves for the focusing $L^2$-supercritical nonlinear Schr\"odinger equation with inverse-square potential \[ i\partial_t u + \Delta u + c|x|^{-2} u = - |u|^{\alpha} u, \quad…

In the present paper, we study the normalized solutions for the following quasilinear Schr\"odinger equations: $$-\Delta u-u\Delta u^2+\lambda u=|u|^{p-2}u \quad \text{in}~\mathbb R^N,$$ with prescribed mass $$\int_{\mathbb R^N} u^2=a^2.$$…

We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \ast |u|^p)|u|^{p - 2} u= |u|^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_\alpha$ is the Riesz potential of…

In this paper we consider the following quasilinear Schr\"odinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - \Delta u +\phi u = f(u) &\ \mbox{in } \Omega, -\Delta \phi - \varepsilon^{4}\Delta_4…

We study the existence of solutions $(\underline u,\lambda_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -\Delta u + \lambda u = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and…

We consider the following singularly perturbed nonlinear elliptic problem: $$-\e^2\Delta u+V(x)u=f(u),\ u\in H^1(\mathbb{R^N}),$$ where $N\ge 3$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a solution $u_\e$…

In this paper, we are concerned with the ground state solutions of nonlinear fractional Schr\"odinger equation involving critical growth. Without Ambrosetti-Rabinowitz condition and monotonicity condition on the nonlinearity, we get the…

We consider the defocusing energy-critical nonlinear Schr\"odinger equation with inverse-square potential $iu_t = -\Delta u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a>-\frac14…

We show that ground state solutions to the nonlinear, fractional problem \begin{align*} \left\{ \begin{array}{ll} (-\Delta)^{s} u + V(x) u = f(x,u) &\quad \mathrm{in} \ \Omega, \newline u = 0 &\quad \mathrm{in} \ \mathbb{R}^N \setminus…

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schr\"odinger equation \begin{equation*} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=|u|^{p-2}u \quad\mathrm{in}\ \mathbb{R}^{N},…

We consider the focusing nonlinear Schr\"odinger equation with inverse square potential \[ i\partial_t u + \Delta u + c|x|^{-2} u = - |u|^\alpha u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d \geq…

This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N},…

A class of nonlinear Schroedinger equations with critical power-nonlinearities and potentials exhibiting multiple anisotropic inverse square singularities is investigated. Conditions on strength, location, and orientation of singularities…

We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…

We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 1$,} \] having…

We consider nonlinear Choquard equation $$ - \Delta u + V u = \bigl(I_\alpha \ast |u|^{\frac{\alpha}{N}+1}\bigr) |u|^{\frac{\alpha}{N}-1} u\quad\text{in (\mathbb{R}^N)},$$ where $N \ge 3$, $V \in L^\infty (\mathbb{R}^N)$ is an external…

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schr\"odinger equation $$\left( \sqrt{-c^2\Delta + m^2 c^4}-mc^2\right) u + \mu u = |u|^{p-1}u\quad…