In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain Ω ⊂ R N , N > 2s:Here 0 < s < 1, δ > 0, λ > 0 and f : R + → R + is a positive C 2 function. K : Ω → R + is a Hölder continuous function in Ω which behave as dist(x, ∂Ω) −β near the boundary with 0 ≤ β < 2s. First, for any δ > 0 and for λ > small enough, we prove the existence of solutions to (P λ ). Next, for a suitable range of values of δ, we show the existence of an unbounded connected branch of solutions to (P λ ) emanating from the trivial solution at λ = 0. For a certain class of nonlinearities f , we derive a global multiplicity result that extends results proved in [5]. To establish the results, we prove new properties which are of independent interest and deal with the behavior and Hölder regularity of solutions to (P λ ).
In this article, we study the following fractional elliptic equation with critical growth and singular nonlinearity:(-\Delta)^{s}u=u^{-q}+\lambda u^{{2^{*}_{s}}-1},\qquad u>0\quad\text{in }%
\Omega,\qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega,where Ω is a bounded domain in {\mathbb{R}^{n}} with smooth boundary {\partial\Omega}, {n>2s}, {s\in(0,1)}, {\lambda>0}, {q>0} and {2^{*}_{s}=\frac{2n}{n-2s}}.
We use variational methods to show the existence and multiplicity of positive solutions with respect to the parameter λ.
This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see (KC) below). We use the variational method in the light of Moser-Trudinger inequality to show the existence of weak solutions to (KC). Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex-concave problem (P λ,M ) below.
We study the existence of positive solutions for fractional elliptic equations of the type
(-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1)
where h is a real valued function that behaves like eu2 as
u → ∞ . Here (-Δ)1/2 is the
fractional Laplacian operator. We show the existence of
mountain-pass solution when the nonlinearity is superlinear near
t = 0. In case h is concave near t = 0, we show the existence of
multiple solutions for suitable range of λ by analyzing the
fibering maps and the corresponding Nehari manifold.
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