Marius Ghergu

We are concerned with the study of $$C^{2, \gamma }_{loc}$$ C loc 2 , γ -positive solutions to the Choquard equation $$ \displaystyle -\Delta u=\lambda (|x|^{-\alpha }*u^p)u^q \quad \text{ in } \mathbb {R}^N_+=\mathbb {R}^{N-1}\times (0, \infty )\, , N\ge 2, $$ - Δ u = λ ( | x | - α ∗ u p ) u q in R + N = R N - 1 × ( 0 , ∞ ) , N ≥ 2 , subject to the Robin boundary condition $$\beta u+\frac{\partial u}{\partial \nu }=g$$ β u + ∂ u ∂ ν = g on $$\partial \mathbb {R}^N_+$$ ∂ R + N . Here $$p,q, \lambda >0$$ p , q , λ > 0 , $$\alpha \in (0,N)$$ α ∈ ( 0 , N ) , $$\beta \in \mathbb {R}$$ β ∈ R and $$0\le g\in C^{1, \gamma }_{loc}(\mathbb {R}^{N-1})$$ 0 ≤ g ∈ C loc 1 , γ ( R N - 1 ) , $$\gamma \in (0,1)$$ γ ∈ ( 0 , 1 ) . First we show that if $$\beta <0$$ β < 0 then no solutions exist. Further, in the case $$\beta =0$$ β = 0 we prove that the existence of a $$C^{2, \gamma }_{loc}$$ C loc 2 , γ solution is closely related to the rate at which g decays at infinity. To this aim, we assume $$g(x')\simeq (1+|x'|)^{-m}$$ g ( x ′ ) ≃ ( 1 + | x ′ | ) - m , $$m>0$$ m > 0 , and determine the exact range of $$p, q, \alpha , m$$ p , q , α , m for which a $$C^{2, \gamma }_{loc}$$ C loc 2 , γ solution exists. A similar discussion arises in the case $$\beta >0$$ β > 0 . Our approach combines integral estimates and integral representation of superharmonic functions together with a new sub and supersolution method devised in a nonlocal setting.

The purpose of this article is two-fold. First, we investigate the inequality $$ -Δu+V(x) u\geq f\quad\mbox{ in } B_1\setminus\{0\}\subset \mathbb{R}^N , N \geq 2, $$ where $f\in L^1_{loc}(B_1)$. If $V\geq 0$ is radially symmetric, we provide optimal conditions for which any solution $0\leq u\in \mathcal{C}^2(B_1\setminus\{0\})$ of the above inequality satisfies $u, Δu, V(x)u\in L^1_{loc}(B_1)$. This extends a result of H. Brezis and P.-L. Lions (1982), originally established for constant potentials $V$. Second, we investigate the equation $$\displaystyle -Δu + λV(x) u = (K_{α, β} * u^p) u^q \quad\text{in } B_1 \setminus \{0\},$$ where $0\leq V\in \mathcal{C}^{0, ν}( \overline B_1\setminus\{0\})$, $0<ν0$ and $$K_{α, β}(x) = |x|^{-α}\log^β\frac{2e}{|x|}, \quad\text{where } 0 \leq α< N, β\in \mathbb{R}.$$ For $N \geq 3$, we establish sharp conditions on the exponents $α, β, p, q$ under which singular solutions exist and exhibit the asymptotic behavior $u(x) \simeq |x|^{2-N}$ near the origin. For $N = 2$, we provide a classification of the existence and boundedness of solutions based on the local behavior of the potential $V(x)$ near the origin.

In this work, we present a novel stationary Gierer-Meinhardt system incorporating non-local proliferation rates, defined as follows: $$ \begin{cases} \displaystyle -Δu+λu=\frac{J*u^p}{v^q}+ρ(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 1,\\[0.1in] \displaystyle -Δv+μv=\frac{J*u^m}{v^s} &\quad\mbox{ in }\mathbb{R}^N.\\[0.1in] \end{cases} $$ This system emerges in various contexts, such as biological morphogenesis, where two interacting chemicals, identified as an activator and an inhibitor, are described, and in ecological systems modelling the interaction between two species, classified as specialists and generalists. The non-local interspecies interactions are represented by the terms $J*u^p, J*u^m$ where the $*$-symbol denotes the convolution operation in $\mathbb{R}^N$ with a kernel $J\in C^1(\mathbb{R}^N\setminus\{0\})$. In the system, we assume that $0<ρ\in C^{0, γ}(\mathbb{R}^N)$ with $γ\in (0,1)$, while the parameters satisfy $λ, μ, q,m,s>0$ and $p>1$. Under various integrability conditions on the kernel $J$, we establish the existence and non-existence of classical positive solutions in the function space $C^{2, δ}_{loc}(\mathbb{R}^N).$ These results further highlight the influence of the non-local terms, particularly the proliferation rates, in the proposed model.

We discuss the existence of positive superharmonic functions $u$ in $\mathbb{R}^N_+=\mathbb{R}^{N-1}\times (0, \infty)$, $N\geq 3$, in the sense $-Δu=μ$ for some Radon measure $μ$, so that $u$ satisfies the nonlocal boundary condition $$ \frac{\partial u}{\partial n}(x',0)=λ\int\limits_{\mathbb{R}^{N-1}}\frac{u(y',0)^p}{|x'-y'|^k}dy' \quad\mbox{ on }\partial \mathbb{R}^N_+, $$ where $p,λ>0$ and $k\in (0, N-1)$. First, we show that no solutions exist if $0<k\leq 1$. Next, if $1<kp^*$ and discuss the existence of regular solutions, case in which we identify a second critical exponent given by $p^{**}=2\cdot \frac{N-1}{k-1}-1$. Our approach combines various integral estimates with the properties of the newly introduced $α$-lifting operator and fixed point theorems.

We discuss the existence of positive superharmonic functions $ u $ in $ \mathbb R^N_+ = \mathbb R^{N-1}\times (0, \infty) $, $ N\geq 3 $, in the sense $ -\Delta u = \mu $ for some Radon measure $ \mu $, so that $ u $ satisfies the nonlocal boundary condition $ \begin{align} \frac{\partial u}{\partial n}(x',0) = \lambda \int\limits_{ \mathbb R^{N-1}}\frac{u(y',0)^p}{|x'-y'|^k}dy' \quad\text{ on }\partial \mathbb R^N_+, \end{align} $ where $ p,\lambda>0 $ and $ k\in (0, N-1) $. First, we show that no solutions exist if $ 0<k\leq 1 $. Next, if $ 1<kp^* $ and discuss the existence of regular solutions, case in which we identify a second critical exponent given by $ p^{**} = 2\cdot \frac{N-1}{k-1}-1 $. Our approach combines various integral estimates with the properties of the newly introduced $ \alpha $-lifting operator and fixed point theorems.

We are concerned with the study of positive solutions to the integral equation $ u(x) = \int\limits_{ \mathbb R^N}\frac{K(y) u(y)^{-p}}{|x-y|^{N-\alpha}} dy\quad\mbox{ in } \mathbb R^N (N\geq 1), $ where $ p>0 $, $ \alpha\in (0, N) $ and $ K\in C(0, \infty) $ is a positive and continuous function which satisfies $ C_1(1+|x|)^{-\beta}\leq K(x)\leq C_2(1+|x|)^{-\beta} $ in $ \mathbb R^N $, for some $ \beta>0 $ and $ C_2>C_1>0 $. We discuss the existence, uniqueness, and asymptotic behaviour of solutions to the above equation in terms of $ \alpha, \beta $ and $ p $. Solutions with finite and infinite total mass are also considered.