This paper is concerned with the existence of normalized solutions of the nonlinear Schr"odinger equation [ -Delta u+V(x)u+lambda u = |u|^{p-2}u qquad ext{in $mathbb{R}^N$} ] in the mass supercritical and Sobolev subcritical case $2+rac{4}{N}<2^*$. We prove the existence of a solution $(u,lambda)in H^1(mathbb{R}^N) imesmathbb{R}^+$ with prescribed $L^2$-norm $|u|_2= ho$ under various conditions on the potential $V:mathbb{R}^N omathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.
Bartsch, T., Molle, R., Rizzi, M., Verzini, G. (2020). Normalized solutions of mass supercritical Schrödinger equations with potential. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS.
Normalized solutions of mass supercritical Schrödinger equations with potential
Molle, R;
2020-08-17
Abstract
This paper is concerned with the existence of normalized solutions of the nonlinear Schr"odinger equation [ -Delta u+V(x)u+lambda u = |u|^{p-2}u qquad ext{in $mathbb{R}^N$} ] in the mass supercritical and Sobolev subcritical case $2+rac{4}{N}<2^*$. We prove the existence of a solution $(u,lambda)in H^1(mathbb{R}^N) imesmathbb{R}^+$ with prescribed $L^2$-norm $|u|_2= ho$ under various conditions on the potential $V:mathbb{R}^N omathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.File | Dimensione | Formato | |
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