Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

Cave, Linda Maria De; Durastanti, Riccardo; Oliva, Francescantonio

doi:10.1007/s00030-018-0509-7

Cited by 32 publications

(26 citation statements)

References 40 publications

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“…These kinds of problems were investigated in the linear case -i.e. when the operator A is linear -in [3] and in the case where µ is a measure in [1], [4], [5], [7], [8], [9], [13], [15], [16], [17]. Our note borrows some ideas contained in [3] however the nonlinearity of the operators induces unexpected difficulties.…”

Section: Introduction and Notationmentioning

confidence: 99%

On some singular nonlinear problems for monotone elliptic operators

Chipot

2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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Section: Introduction and Notationmentioning

confidence: 99%

On some singular nonlinear problems for monotone elliptic operators

Chipot

2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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“…On the other hand, if γ > 1, solutions u p have in general only locally finite energy, that is u p ∈ W 1,p loc (Ω), so that the boundary datum needs to be assigned through a suitable weaker condition than the usual trace sense (see [11,18,19,25,37,38] for further remarks on this fact) eventually producing (1.2) in the limit as p → 1 + . Recently, existence of solutions to (1.4) has been shown in case of a general function h and a measure datum f ( [20]). Note that this approximation approach with respect to the parameter p requires a priori estimates on the solutions u p of (1.4) that are independent of p, and that will be proven (see Section 4.4) provided (1.3) holds.…”

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confidence: 99%

The Dirichlet problem for singular elliptic equations with general nonlinearities

et al. 2019

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In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the formis a continuous function which may become singular at s = 0 + , and f is a nonnegative datum in L N,∞ (Ω) with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and f > 0. As a by-product of our method a general theory for the same problem involving the p-laplacian as principal part, which is missed in the literature, is established. The main assumptions we use are also further discussed in order to show their optimality.

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“…Once more using the Hölder inequality gives f n w dx ≤ C 2 ( |w| 2 dx) 1 2 for some C 2 > 0. Thus combining this inequality with ( 17), (18), and (19), we obtain…”

Section: Proposition 31 If G(x)mentioning

confidence: 95%

“…where is a bounded domain of R N , N ≥ 2, γ > 0, and μ is a general Radon measure in . See [17][18][19][20][21][22][23][24][25][26] for more related problems. These types of problems have been extensively studied for their relations with some physical phenomena in the theory of pseudoplastic fluids [27].…”

Section: Introductionmentioning

confidence: 99%

Nonlocal Lazer–McKenna-type problem perturbed by the Hardy’s potential and its parabolic equivalence

et al. 2021

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In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: $$\begin{aligned} \textstyle\begin{cases} (-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{\mu }{u^{\gamma }}+f & \text{in } \Omega, \\ u>0 & \text{in } \Omega, \\ u=0 & \text{in } (\mathbb{R}^{N} \setminus \Omega ). \end{cases}\displaystyle \end{aligned}$$ { ( − Δ ) s u = λ u | x | 2 s + μ u γ + f in Ω , u > 0 in Ω , u = 0 in ( R N ∖ Ω ) . Here $0 < s<1$ 0 < s < 1 , $\lambda >0$ λ > 0 , $\gamma >0$ γ > 0 , and $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N ($N > 2s$ N > 2 s ) is a bounded smooth domain such that $0 \in \Omega $ 0 ∈ Ω . Moreover, $0 \leq \mu,f \in L^{1}(\Omega )$ 0 ≤ μ , f ∈ L 1 ( Ω ) . For $0< \lambda \leq \Lambda _{N,s}$ 0 < λ ≤ Λ N , s , $\Lambda _{N,s}$ Λ N , s being the best constant in the fractional Hardy inequality, we find a necessary and sufficient condition for the existence of a positive weak solution to the problem with respect to the data μ and f. Also, for a regular datum of f, under suitable assumptions, we obtain some existence and uniqueness results and calculate the rate of growth of solutions. Moreover, we mention a nonexistence and a complete blowup result for the case $\lambda > \Lambda _{N,s}$ λ > Λ N , s . Besides, we consider the parabolic equivalence of the above problem in the case $\mu \equiv 1$ μ ≡ 1 and some suitable $f(x,t)$ f ( x , t ) , that is, $$\begin{aligned} \textstyle\begin{cases} u_{t}+(-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{1}{u^{\gamma }}+f(x,t) & \text{in } \Omega \times (0,T), \\ u>0 & \text{in } \Omega \times (0,T), \\ u =0 & \text{in } (\mathbb{R}^{N} \setminus \Omega ) \times (0,T), \\ u(x,0)=u_{0} & \text{in } \mathbb{R}^{N}, \end{cases}\displaystyle \end{aligned}$$ { u t + ( − Δ ) s u = λ u | x | 2 s + 1 u γ + f ( x , t ) in Ω × ( 0 , T ) , u > 0 in Ω × ( 0 , T ) , u = 0 in ( R N ∖ Ω ) × ( 0 , T ) , u ( x , 0 ) = u 0 in R N , where $u_{0} \in X_{0}^{s}(\Omega )$ u 0 ∈ X 0 s ( Ω ) satisfies an appropriate cone condition. In the case $0<\gamma \leq 1$ 0 < γ ≤ 1 or $\gamma >1$ γ > 1 with $2s(\gamma -1)<(\gamma +1)$ 2 s ( γ − 1 ) < ( γ + 1 ) , we show the existence of a unique solution for any $0< \lambda < \Lambda _{N,s}$ 0 < λ < Λ N , s and prove a stabilization result for certain range of λ.

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Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

Cited by 32 publications

References 40 publications

On some singular nonlinear problems for monotone elliptic operators

On some singular nonlinear problems for monotone elliptic operators

The Dirichlet problem for singular elliptic equations with general nonlinearities

Nonlocal Lazer–McKenna-type problem perturbed by the Hardy’s potential and its parabolic equivalence

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