We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is 
$$\begin{cases} -\Delta u = \frac{f(x)}{u^{\gamma}} +\mu & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\ \Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$. Here $\gamma > 0$, $f$ is a nonnegative function on $\Omega$, and $\mu$ is a nonnegative bounded Radon measure on $\Omega$.

On singular elliptic equations with measure sources

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity

In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal $p$-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular $p$-Laplace and mixed local and nonlocal $p$-Laplace equation is established. Finally, we investigate the existence, uniqueness, regularity and symmetry properties of weak solutions for such problems.

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

Existence, uniqueness and multiplicity of positive solutions for Schrödinger–Poisson system with singularity

In this paper we consider a semilinear elliptic equation with a strong singularity at u = 0, namely {u ≥ 0 in Ω, −div A(x)Du = F (x, u) in Ω, u = 0 on ∂Ω, with F (x, s) a Caratheodory function such that 0 ≤ F (x, s) ≤ h(x)/Γ(s) a.e. x ∈ Ω, ∀s > 0, with h in some L^r(Ω) and Γ a C^1 ([0, +∞[) function such that Γ(0) = 0 and Γ'(s) > 0 for every s > 0. We introduce a notion of solution to this problem in the spirit of the solutions defined by transposition. This definition allows us to prove the existence and the stability of this solution, as well as its uniqueness when F (x, s) is nonincreasing in s.

https://hal.archives-ouvertes.fr/hal-01348682/document

Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0

We study existence and regularity of positive solutions of problems like 

$$\left\{\begin{array}{cl} - {\rm div}([a(x) + u^q] \nabla u) + b(x)\frac{1}{u^\theta}|\nabla u|^2 = f & {\rm in} \, \Omega,\\ u > 0 &{\rm in} \, \Omega, \\ u = 0 & {\rm on} \, \partial\Omega ,\end{array}\right$$

depending on the values of q > 0, 0 < θ < 1, and on the summability of the datum f ≥ 0 in Lebesgue spaces

A Class of Quasilinear Dirichlet Problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms

For an open, bounded set Ω ⊂ R N , measurable bounded functions a ( x ) , b ( x ) which are strictly positive and p , q > 0 , we prove the existence of a weak solution of the quasilinear b.v.p { − div [ ( a ( x ) + | u | q ) ∇ u ( x ) ] + b ( x ) u | u | p − 1 | ∇ u | 2 = f ( x ) , x ∈ Ω ; u ( x ) = 0 , x ∈ ∂ Ω . The datum f is assumed to be in L 1 ( Ω ) and does not satisfy any sign assumption.

A quasilinear Dirichlet problem with quadratic growth respect to the gradient and L1 data

For an open bounded set $\Omega \subset \mathbb {R}^N$, $N\ge 3$, $0\lvertneqq f\in L^m(\Omega )$ with $m> 1$, $0\le \mu (x)\in L^\infty (\Omega )$ and assuming in addition $\Vert \mu \Vert _\infty < \frac{N(m-1)}{N-2m}$ if $m<\frac{N}{2}$, we prove the existence of a positive solution for the singular b.v.p 

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda u + \mu (x)\,\frac{|\nabla u |^2}{u}\, + f(x), &{} \text { in } \Omega , \\ u=0, &{} \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

provided that $\lambda <\lambda _1/(1+\Vert \mu \Vert _{ L^\infty (\Omega ) })$ (extending the previous results in [3] for $\lambda =0$). The model case $\mu (x)\equiv B<1$ is studied in more detail obtaining in addition the uniqueness (resp. nonexistence) of positive solution if the parameter $\lambda <\frac{\lambda _1}{B+1}$ (resp. $\lambda \ge \frac{\lambda _1}{B+1}$). Even more, the solutions constitute a continuum of solutions bifurcating from infinity at $\lambda =\frac{\lambda _1}{B+1}$. This is in contrast with [5], where the multiplicity of solutions of the nonsingular problem ($\frac{1}{u}$ do not appear in the equation) is deduced due to the bifurcation from infinity at $\lambda =0$.

The effect of a singular term in a quadratic quasi-linear problem

Abstract In this paper, we study the regularizing effect of lower order terms in elliptic problems involving a Hardy potential. Concretely, our model problem is the differential equation - Δ ⁢ u + h ⁢ ( x ) ⁢ | u | p - 1 ⁢ u = λ ⁢ u | x | 2 + f ⁢ ( x )   in ⁢ Ω , -\\Delta u+h(x)|u|^{p-1}u=\\lambda\\frac{u}{|x|^{2}}+f(x)\\quad\\text{in }\\Omega, with Dirichlet boundary condition on ∂ ⁡ Ω {\\partial\\Omega} , where p > 1 {p>1} and f ∈ L h m ⁢ ( Ω ) {f\\in L^{m}_{h}(\\Omega)} (i.e. | f | m ⁢ h ∈ L 1 ⁢ ( Ω ) {|f|^{m}h\\in L^{1}(\\Omega)} ) with m ≥ p + 1 p {m\\geq\\frac{p+1}{p}} . We prove that there is a solution of the above problem even for λ greater than the Hardy constant; i.e., λ ≥ ℋ = ( N - 2 ) 2 4 {\\lambda\\geq\\mathcal{H}=\\frac{(N-2)^{2}}{4}} and nonnegative functions h ∈ L 1 ⁢ ( Ω ) {h\\in L^{1}(\\Omega)} which could vanish in a subset of Ω. Moreover, we show that all the solutions are in L h p ⁢ m ⁢ ( Ω ) {L^{pm}_{h}(\\Omega)} . These results improve and generalize the case h ⁢ ( x ) ≡ h 0 {h(x)\\equiv h_{0}} treated in [2, 10].

Existence and Regularizing Effect of Degenerate Lower Order Terms in Elliptic Equations Beyond the Hardy Constant

In this paper we introduce some of the main tools to study non-linear boundary value problems whose simplest model is $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle - \Delta _p(u)= -\mathrm{div}(|\nabla u|^{p-2}\nabla u)=f(x), &{} \text{ in } \; \Omega ; \\ u=0, &{} \text{ on } \; \partial \Omega \end{array} \right. \end{aligned}$$
 and $$f(x)$$
 belongs to L
 $$^m(\Omega ), m \ge 1$$
 .

Existence and regularity results for p-Laplacian boundary value problems

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