Singular equations with variable exponents and concave-convex nonlinearities

Multiple Positive Solutions for Singular Elliptic Equations with Concave-Convex Nonlinearities and Sign-Changing Weights

Recommended by Pavel Drabek We study existence and multiplicity of positive solutions for the following Dirichlet equations: −Δu − μ/|x| 2 u λfx|u| q−2 u gx|u| 2 * −2 u in Ω, u 0 on ∂Ω, where 0 ∈ Ω ⊂ R N N ≥ 3 is a bounded domain with smooth boundary ∂Ω, λ > 0, 0 ≤ μ < μ N − 2 2 /4, 2 * 2N/N − 2, 1 ≤ q < 2, and f, g are continuous functions on Ω which are somewhere positive but which may change...

Multiplicity of Positive Solutions for Weighted Quasilinear Elliptic Equations Involving Critical Hardy-Sobolev Exponents and Concave-Convex Nonlinearities

and Applied Analysis 3 When a 0, we set s dp∗ 0, d and t bp∗ 0, b , then 1.1 is equivalent to the following quasilinear elliptic equations: −div ( |∇u|p−2∇u ) − μ |u| p−2u |x| |u|p t −2u |x| λ |u|q−2u |x| in Ω, u 0 on ∂Ω, 1.7 where λ > 0, 1 < p < N, 0 ≤ μ < μ N − p /p , 0 ≤ s, t < p, 1 ≤ q < p and p∗ t p N − t / N − p . Such kind of problem relative with 1.7 has been extensively studied by many...

Degenerate elliptic equations with singular nonlinearities

The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of RN , and compactness holds below a critical dimension N #. The nonlinearity f (u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of p-Laplace operator, for p = 2 it is crucial to...

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR DEGENERATE p(x)-LAPLACE EQUATIONS INVOLVING CONCAVE-CONVEX TYPE NONLINEARITIES WITH TWO PARAMETERS

We show the existence of two nontrivial nonnegative solutions and infinitely many solutions for degenerate p(x)-Laplace equations involving concaveconvex type nonlinearities with two parameters. By investigating the order of concave and convex terms and using a variational method, we determine the existence according to the range of each parameter. Some Caffarelli-Kohn-Nirenberg type problems w...

Subelliptic equations with singular nonlinearities on the Heisenberg group