Search results for " Weak solution"
showing 2 items of 12 documents
First and second critical exponents for an inhomogeneous Schrödinger equation with combined nonlinearities
2022
AbstractWe study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation $$\begin{aligned} iu_t+\Delta u=\lambda |u|^p+\mu |\nabla u|^q+w(x),\quad t>0,\, x\in {\mathbb {R}}^N, \end{aligned}$$ i u t + Δ u = λ | u | p + μ | ∇ u | q + w ( x ) , t > 0 , x ∈ R N , where $$N\ge 1$$ N ≥ 1 , $$p,q>1$$ p , q > 1 , $$\lambda ,\mu \in {\mathbb {C}}$$ λ , μ ∈ C , $$\lambda \ne 0$$ λ ≠ 0 , and $$u(0,\cdot ), w\in L^1_{\mathrm{loc}}({\mathbb {R}}^N,{\mathbb {C}})$$ u ( 0 , · ) , w ∈ L loc 1 ( R N , C ) . We consider both the cases where $$\mu =0$$ μ = 0 and $$\mu \ne 0$$ μ ≠ 0 , respectively. We establish existence/nonexistence of global weak solutions. In ea…
On the regularity of very weak solutions for linear elliptic equations in divergence form
2020
AbstractIn this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .