Search results for "mountain pass"
showing 2 items of 12 documents
Multiple nodal solutions for semilinear robin problems with indefinite linear part and concave terms
2017
We consider a semilinear Robin problem driven by Laplacian plus an indefinite and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal solutions which converge to zero in $C^1(\overline{\Omega})$. If the coefficient of the concave term is sign changing, then again we produce a sequence of smooth solutions converging to zero in $C^1(\overline{\Omega})$, but we cannot claim that they are nodal.
Multiple positive normalized solutions for nonlinear Schrödinger systems
2018
We consider the existence of multiple positive solutions to the nonlinear Schr\"odinger systems sets on $H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N)$, \[ \left\{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta r_1 |u_1|^{r_1-2} u_1|u_2|^{r_2}, -\Delta u_2 &= \lambda_2 u_2 + \mu_2 |u_2|^{p_2 -2}u_2 + \beta r_2 |u_1|^{r_1} |u_2|^{r_2 -2} u_2, \end{aligned} \right. \] under the constraint \[ \int_{\mathbb{R}^N}|u_1|^2 \, dx = a_1,\quad \int_{\mathbb{R}^N}|u_2|^2 \, dx = a_2. \] Here $a_1, a_2 >0$ are prescribed, $\mu_1, \mu_2, \beta>0$, and the frequencies $\lambda_1, \lambda_2$ are unknown and will appear as Lagrange multipliers. Two cases are studied, the first …