Volume 6, Issue 3

In this article, we consider a singularly perturbed nonlinear reaction-diffusion equation whose solutions display thin boundary layers near the boundary of the domain. We fully analyse the singular behaviours of the solutions at any given order with respect to the small parameter ε, with suitable asymptotic expansions consisting of the outer solutions and of the boundary layer correctors. The systematic treatment of the nonlinear reaction terms at any given order is novel along the singular perturbation analysis. We believe that the analysis can be suitably extended to other nonlinear problems.

In this article, we take up the existence and the asymptotic behavior of entire bounded positive solutions to the following semilinear elliptic system: -Δ u = a1a_{1}( x )uαu^{\alpha}vrv^{r}, x ∈\inℝn\mathbb{R}^{n} ( n ≥\geq 3), -Δ v = a2a_{2}( x )vβv^{\beta}usu^{s}, x ∈\inℝn\mathbb{R}^{n}, u , v ¿ 0 in ℝn\mathbb{R}^{n}, lim|x|→+∞\lim_{|x|\rightarrow+\infty} u ( x ) = lim|x|→+∞\lim_{|x|\rightarrow+\infty} v ( x )=0, where α,β<1{\alpha,\beta<1}, r,s∈ℝ{r,s\in\mathbb{R}} such that ν:=(1-α)⁢(1-β)-r⁢s>0{\nu:=(1-\alpha)(1-\beta)-rs>0}, and the functions a1a_{1}, a2a_{2} are nonnegative in 𝒞locγ⁢(ℝn){\mathcal{C}^{\gamma}_{\mathrm{loc}}(\mathbb{R}^{n})} (0¡γ¡1) and satisfy some appropriate assumptions related to Karamata regular variation theory.

This paper offers a new perspective to look at the Riesz potential. On the one hand, it is shown that not only 𝔏q,q⁢p-1⁢(n-α⁢p)∩𝔏p,κ-α⁢p\mathfrak{L}^{q,qp^{-1}(n-\alpha p)}\cap\mathfrak{L}^{p,\kappa-\alpha p} contains Iα⁢Lp,κ{I_{\alpha}L^{p,\kappa}} under the conditions 1<p<∞{1<p<\infty}, 1≤q<∞{1\leq q<\infty}, q⁢(κ/p-α)≤κ≤nq(\kappa/p-\alpha)\leq\kappa\leq n, 0<α<min⁡{n,1+κ/p}{0<\alpha<\min\{n,1+\kappa/p\}}, but also 𝔏q,λ{\mathfrak{L}^{q,\lambda}} exists as an associate space under the condition -q<λ<n{-q<\lambda<n}, where Iα⁢Lp,κ{I_{\alpha}L^{p,\kappa}} and 𝔏q,λ{\mathfrak{L}^{q,\lambda}} are the Morrey–Sobolev and Campanato spaces on ℝn{\mathbb{R}^{n}} respectively. On the other hand, a nonnegative Radon measure μ is completely characterized to produce a continuous map Iα:Lp,1→Lμq{I_{\alpha}:L_{p,1}\to L^{q}_{\mu}} under the condition 1<p<min⁡{q,n/α}{1<p<\min\{q,{n}/{\alpha}\}} or 1<q≤p<min⁡{q⁢(n-α⁢p)/(n-α⁢q⁢(q-1)-1),n/α}{1<q\leq p<\min\{{q(n-\alpha p)}/({n-\alpha q(q-1)^{-1}}),{n}/{\alpha}\}}, where Lp,1{L_{p,1}} and Lμq{L^{q}_{\mu}} are the (p,1){(p,1)}-Lorentz and (q,μ){(q,\mu)}-Lebesgue spaces on ℝn{\mathbb{R}^{n}} respectively.

In this article, we study the following fractional elliptic equation with critical growth and singular nonlinearity: (-Δ)s⁢u=u-q+λ⁢u2s*-1,u>0 in ⁢Ω,u=0 in ⁢ℝn∖Ω,(-\Delta)^{s}u=u^{-q}+\lambda u^{{2^{*}_{s}}-1},\qquad u>0\quad\text{in }% \Omega,\qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in ℝn{\mathbb{R}^{n}} with smooth boundary ∂⁡Ω{\partial\Omega}, n>2⁢s{n>2s}, s∈(0,1){s\in(0,1)}, λ>0{\lambda>0}, q>0{q>0} and 2s*=2⁢nn-2⁢s{2^{*}_{s}=\frac{2n}{n-2s}}. We use variational methods to show the existence and multiplicity of positive solutions with respect to the parameter λ.