Solutions For Singular Elliptic Problems Research Articles

Existence of solutions for singular elliptic problems with singular nonlinearities and critical Caffarelli-Kohn-Nirenberg exponent

In this article, we consider a singular elliptic problem with singular nonlinearities and critical Caffarelli-Kohn-Nirenberg exponent. By using variational methods and Palais-Smale condition, we show the existence of at least two nontrivial solutions. The result depends crucially on the parameters \(a,b,N,\beta,\gamma,\lambda,\mu\). For more information see https://ejde.math.txstate.edu/Volumes/2023/54/abstr.html

Multiplicity of positive solutions for singular elliptic problems

AbstractWe consider the following nonlinear singular elliptic equation where , . Our main purpose is to prove the existence of a large number of positive solutions with the asymptotic decay as . We also investigate the rate of decay of . These results are based on the sub and supersolution method and cover both sublinear and superlinear cases of f.

Multiple positive solutions for singular elliptic problems involving concave-convex nonlinearities and sign-changing potential

In this paper, we are interested in considering the following singular elliptic problem with concaveconvex nonlinearities $$\left\{ {\begin{array}{*{20}{l}} { - \Delta u - \frac{\mu }{{|x{|^2}}}u = f(x)|u{|^{p - 2}}u + g(x)|u{|^{q - 2}}u,}&{in\;\;\Omega \backslash \{ 0\} ,} \\ {u = 0,}&{on\;\;2\Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN(N ≥ 3) is a smooth bounded domain with 0 ∈ Ω, $$0 < \mu < \bar\mu = \frac{{{{(N - 2)}^2}}}{4}$$, 1 < q < 2 < p < 2* and $$2* = \frac{{2N}}{{N - 2}}$$ is the Sobolev critical exponent, the coefficient functions f, g may change sign on Ω. By the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.

About positive $W_{loc}^{1,\Phi}(\Omega)$-solutions to quasilinear elliptic problems with singular semilinear term

This paper deals with the existence, uniqueness and regularity of positive$W_{\rom loc}^{1,\Phi}(\Omega)$-solutions of singular elliptic problems on a~smooth bounded domain with Dirichlet boundary conditions involving the $\Phi$-Laplacian operator.The proof of the existence is based on a variant of the generalized Galerkin method that we developed inspired by ideas of Browder \cite{Browder} and a comparison principle. By the use of a kind of Moser's iteration scheme we show the$L^{\infty}(\Omega)$-regularity for positive solutions.

Finite and infinite energy solutions of singular elliptic problems: Existence and uniqueness

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by−Δu=h(u)finΩ, where f is an irregular datum, possibly a measure, and h is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.

Positive evanescent solutions of singular elliptic problems in exterior domains

Positive evanescent solutions of singular elliptic problems in exterior domains

Existence of multiple solutions for singular elliptic problems with nonlinear boundary conditions

In this paper, the existence and multiplicity results of solutions for the following quasilinear elliptic problem(0.1){−div(|x|−ap|∇u|p−2∇u)+h(x)|u|p−2u=g(x)|u|r−2u,x∈Ω,|x|−ap|∇u|p−2∂u∂ν=λf(x)|u|q−2u,x∈∂Ω are established, where Ω is an exterior domain in RN with the compact and smooth boundary ∂Ω. By the variational methods, we prove that the problem (0.1) has at least two positive solutions. At the last part of the paper, we also consider the critical case and prove the existence of solution by concentration-compactness principle.

Infinitely many arbitrarily small solutions for singular elliptic problems with critical Sobolev–Hardy exponents

AbstractLet Ω ⊂ ℝN be a bounded domain such that 0 ∈ Ω, N ≥ 3, 2*(s) = 2(N − s)/(N − 2), 0 ≤ s &lt; 2, $0\leq\mu\lt\bar{\mu}=\frac{14}(N-2)^{2}$. We obtain the existence of infinitely many solutions for the singular critical problem $\smash{-\Delta u-\mu(u/|x|^2)=(|u|^{2^*(s)-2/|x|^s)u+\lambda f(x,u)$ with Dirichlet boundary condition for suitable positive number λ.

The effect of domain topology on the number of positive solutions for singular elliptic problems involving the Caffarelli–Kohn–Nirenberg inequalities

In this paper we consider the critical singular equation involving the Caffarelli–Kohn–Nirenberg inequalities of the type ( P 1 ) { − div ( | x | − 2 a ∇ u ) − μ u | x | 2 ( 1 + a ) = λ | u | q − 2 u | x | d D + | u | p − 2 u | x | b p , x ∈ Ω , u > 0 , x ∈ Ω , u = 0 , x ∈ ∂ Ω . Here Ω is a bounded domain with smooth boundary in R N and contains 0 in its interior, 0 ⩽ μ < ( μ ¯ − a ) 2 , μ ¯ = ( N − 2 2 ) 2 , N ⩾ 3 , a ⩽ b < a + 1 , a ⩽ d < a + 1 , p = p ( a , b ) ≜ 2 N N − 2 ( 1 + a − b ) , D = D ( a , d ) ≜ 2 N N − 2 ( 1 + a − d ) , λ is a positive parameter and 2 ⩽ q < D . By Lusternik–Schnirelmann category theory, we prove that the problem ( P 1 ) has at least cat ( Ω ) positive solutions.

Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities

We study the existence of multiple positive solutions of $ -\Delta u = \lambda u^{-q} +u^p $ in $\Omega$ with homogeneous Dirichlet boundary condition, where $\Omega$ is a bounded domain in $\mathbb R^N$, $\lambda >0$, and $0 < q \leq 1 < p \leq (N+2)/(N-2)$. We show by a variational method that if $\lambda$ is less than some positive constant then the problem has at least two positive, weak solutions including the cases of $q=1$ or $p=(N+2)/(N-2)$. We also study the regularity of positive weak solutions.