Multiple solutions to logarithmic Schrodinger equations with periodic potential

We study a class of logarithmic Schrodinger equations with periodic potential which come from physically relevant situations and obtain the existence of infinitely many geometrically distinct solutions.Comment: 3 pages, corrigendum to version

Ground states of critical and supercritical problems of Brezis-Nirenberg type

We study the existence of symmetric ground states to the supercritical problem $$-\Delta v=\lambda v+\left\vert v\right\vert ^{p-2}v\text{ \ in }\Omega,\qquad v=0\text{ on }\partial\Omega,$$ in a domain of the form $$\Omega=\{(y,z)\in\mathbb{R}^{k+1}\times\mathbb{R}^{N-k-1}:\left( \left\vert y\right\vert ,z\right) \in\Theta\},$$ where $\Theta$ is a bounded smooth domain such that $\overline{\Theta} \subset\left( 0,\infty\right) \times\mathbb{R}^{N-k-1},$ $1\leq k\leq N-3,$ $\lambda\in\mathbb{R},$ and $p=\frac{2(N-k)}{N-k-2}$ is the $(k+1)$-st critical exponent. We show that symmetric ground states exist for $\lambda$ in some interval to the left of each symmetric eigenvalue, and that no symmetric ground states exist in some interval $(-\infty,\lambda_{\ast})$ with $\lambda_{\ast}>0$ if $k\geq2.$ Related to this question is the existence of ground states to the anisotropic critical problem $$-\text{div}(a(x)\nabla u)=\lambda b(x)u+c(x)\left\vert u\right\vert ^{2^{\ast }-2}u\quad\text{in}\ \Theta,\qquad u=0\quad\text{on}\ \partial\Theta,$$ where $a,b,c$ are positive continuous functions on $\overline{\Theta}.$ We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of $\lambda,$ and obtain a bifurcation result for ground states

Sharp constant in the curl inequality and ground states for curl-curl problem with critical exponent

Let $\Omega\subset\mathbb{R}^3$ be a Lipschitz domain and let $S_{\text{curl}}(\Omega)$ be the largest constant such that $$\int_{\mathbb{R}^3}|\nabla\times u|^2dx\geq S_{\text{curl}}(\Omega)\inf_{w\in W_0^\sigma(\text{curl};\mathbb{R}^3)\\\nabla\times w=0}\left(\int_{\mathbb{R}^3}|u+w|^6dx\right)^{\frac{1}{3}}$$ for any $u$ in $W_0^6(\text{curl};\Omega)\subset W_0^6(\text{curl};\mathbb{R}^3)$ where $W_0^6(\text{curl};\Omega)$ is the closure of $C_0^\infty(\Omega, \mathbb{R}^3)$ in ${u \in L^6 (\Omega, \mathbb{R}^3):\nabla\times u \in L^2(\Omega, \mathbb{R}^3)}$ with respect to the norm $(|u|^2_6+|\nabla\times u|^2_2)^{1/2}$. We show that $S_{\text{curl}}(\Omega)$ is strictly larger than the classical Sobolev constant $S$ in $\mathbb{R}^3$ . Moreover, $S_{\text{curl}}(\Omega)$ is independent of $\Omega$ and is attained by a ground state solution to the curl-curl problem $$\nabla\times(\nabla\times u)=|u|^4u$$ if $\Omega = \mathbb{R}^3$. With the aid of those results, we also investigate ground states of the Brezis-Nirenberg-type problem for the curl-curl operator in a bounded domain $\Omega$ $$\nabla\times(\nabla\times u)+\lambda u=|u|^4u \text{ }\text{ }\text{ in }\Omega$$ with the so-called metallic boundary condition $\nu\times u=0$ on $\partial\Omega$ where $\nu$ is the exterior normal to $\partial\Omega$

A Sobolev-Type Inequality for the Curl Operator and Ground States for the Curl–Curl Equation with Critical Sobolev Exponent

Let Ω⊂R$^{3}$ be a Lipschitz domain and let S$_{curl}$(Ω) be the largest constant such that ∫$_{R^{3}}$|∇×u|$^{2}$dx≥S$_{curl}$(Ω) infw ∈W$^{6}$$_{0}$ (curl;R$^{3}$)∇×w=0(∫$_{R^{3}}$|u+w|$^{6}$dx)$^{1/3}$ for any u in W$^{6}$$_{0}$(curl;Ω)⊂W$^{6}$$_{0}$(curl;R$^{3}$), where W$^{6}$$_{0}$(curl;Ω) is the closure of C$^{∞}_{0}$(Ω,R$^{3}$) in {u∈L$^{6}$(Ω,R$^{3}$):∇×u∈L$^{2}$(Ω,R$^{3}$)} with respect to the norm (|u|$^{2}_{6}$+|∇×u|$^{2}_{2}$)$^{1/2}$. We show that S$_{curl}$(Ω) is strictly larger than the classical Sobolev constant S in R$^{3}$. Moreover, S$_{curl}$(Ω) is independent of Ω and is attained by a ground state solution to the curl–curl problem ∇×(∇×u)=|u|$^{4}$u if Ω=R$^{3}$. With the aid of these results we also investigate ground states of the Brezis–Nirenberg-type problem for the curl–curl operator in a bounded domain Ω ∇×(∇×u)+λu=|u|$^{4}$u in Ω, with the so-called metallic boundary condition ν×u=0 on ∂Ω, where ν is the exterior normal to ∂Ω

A Sobolev-type inequality for the curl operator and ground states for the curl-curl equation with critical Sobolev exponent

Let $\Omega\subset \mathbb{R}^3$ be a Lipschitz domain and let $S_\mathrm{curl}(\Omega)$ be the largest constant such that $$\int_{\mathbb{R}^3}|\nabla\times u|^2\, dx\geq S_{\mathrm{curl}}(\Omega) \inf_{\substack{w\in W_0^6(\mathrm{curl};\mathbb{R}^3)\\ \nabla\times w=0}}\Big(\int_{\mathbb{R}^3}|u+w|^6\,dx\Big)^{\frac13}$$ for any $u$ in $W_0^6(\mathrm{curl};\Omega)\subset W_0^6(\mathrm{curl};\mathbb{R}^3)$ where $W_0^6(\mathrm{curl};\Omega)$ is the closure of $\mathcal{C}_0^{\infty}(\Omega,\mathbb{R}^3)$ in $\{u\in L^6(\Omega,\mathbb{R}^3): \nabla\times u\in L^2(\Omega,\mathbb{R}^3)\}$ with respect to the norm $(|u|_6^2+|\nabla\times u|_2^2)^{1/2}$. We show that $S_{\mathrm{curl}}(\Omega)$ is strictly larger than the classical Sobolev constant $S$ in $\mathbb{R}^3$. Moreover, $S_{\mathrm{curl}}(\Omega)$ is independent of $\Omega$ and is attained by a ground state solution to the curl-curl problem $$\nabla\times (\nabla\times u) = |u|^4u$$ if $\Omega=\mathbb{R}^3$. With the aid of those results, we also investigate ground states of the Brezis-Nirenberg-type problem for the curl-curl operator in a bounded domain $\Omega$ $$\nabla\times (\nabla\times u) +\lambda u = |u|^4u\quad\hbox{in }\Omega$$ with the so-called metallic boundary condition $\nu\times u=0$ on $\partial\Omega$, where $\nu$ is the exterior normal to $\partial\Omega$