In this survey paper, by using variational methods, we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type (ProQuest: Formulae and/or non-USASCII text ...omitted) where Omega is a bounded or an exterior domain of dbl-struck R super(N) and q is a continuous positive function. The results presented in this paper extend several contributions concerning the Lane-Emden equation and we focus on new phenomena which are due to the presence of variable exponents.
This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic ...treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.
In this paper, we study the existence of multiple ground state solutions for a class of parametric fractional Schrödinger equations whose simplest prototype is
(
-
Δ
)
s
u
+
V
(
x
)
u
=
λ
f
(
x
,
u
)
...in
R
n
,
where
n
>
2
,
(
-
Δ
)
s
stands for the fractional Laplace operator of order
s
∈
(
0
,
1
)
, and
λ
is a positive real parameter. The nonlinear term
f
is assumed to have a superlinear behaviour at the origin and a sublinear decay at infinity. By using variational methods, we establish the existence of a suitable range of positive eigenvalues for which the problem admits at least two nontrivial solutions in a suitable weighted fractional Sobolev space.
The purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p-Laplacian operator as ...follows:(−Δ)psu+V(x)|u|p−2u=λa(x)|u|r−2u−b(x)|u|q−2uin RN, where λ is a real parameter, (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<p<∞, p<r<min{q,ps⁎} and V,a,b:RN→(0,∞) are three positive weights. Using variational methods, we obtain nonexistence and multiplicity results for the above-mentioned equations depending on λ and according to the integrability properties of the ratio aq−p/br−p. Our results extend the previous work of Autuori and Pucci (2013) 5 to the fractional p-Laplacian setting. Furthermore, we weaken one of the conditions used in their paper. Hence the results of this paper are new even in the fractional Laplacian case.
We study the existence and concentration of positive solutions for the following class of fractional p-Kirchhoff type problems:
$$ \left\{\begin{array}{@{}ll} ...\left(\varepsilon^{sp}a+\varepsilon^{2sp-3}b \,u_{s, p}^{p}\right)(-\Delta)_{p}^{s}u+V(x)u^{p-1}=f(u) & \text{in}\ \mathbb{R}^{3},\\ \noalign{ u\in W^{s, p}(\mathbb{R}^{3}), \quad u>0 & \text{in}\ \mathbb{R}^{3}, \end{array}\right.$$
where ɛ is a small positive parameter, a and b are positive constants, s ∈ (0, 1) and p ∈ (1, ∞) are such that $sp \in (\frac {3}{2}, 3)$, $(-\Delta )^{s}_{p}$ is the fractional p-Laplacian operator, f: ℝ → ℝ is a superlinear continuous function with subcritical growth and V: ℝ3 → ℝ is a continuous potential having a local minimum. We also prove a multiplicity result and relate the number of positive solutions with the topology of the set where the potential V attains its minimum values. Finally, we obtain an existence result when f(u) = uq−1 + γur−1, where γ > 0 is sufficiently small, and the powers q and r satisfy 2p < q < p*s ⩽ r. The main results are obtained by using some appropriate variational arguments.
We consider the following class of fractional problems with unbalanced growth:{(−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=f(u)in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0in RN, where ε>0 is a small parameter, s∈(0,1), ...2≤p<q<Ns, (−Δ)ts (with t∈{p,q}) is the fractional t-Laplacian operator, V:RN→R is a continuous potential satisfying local conditions, and f:R→R is a continuous nonlinearity with subcritical growth. Applying suitable variational and topological arguments, we obtain multiple positive solutions for ε>0 sufficiently small as well as related concentration properties, in relationship with the set where the potential V attains its minimum.
Dans cet article on considère la classe suivante de problèmes fractionnaires à double phase :{(−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=f(u)dans RN,u∈Ws,p(RN)∩Ws,q(RN),u>0dans RN, où ε>0 est un petit paramètre, s∈(0,1), 2≤p<q<Ns, (−Δ)ts (avec t∈{p,q}) est l'opérateur t-fractionnaire de Laplace, V:RN→R est un potentiel continu satisfaisant des conditions locales et f:R→R est une non linéarité continue à croissance sous-critique. En appliquant des arguments variationnels et topologiques appropriés, nous obtenons l'existence de plusieurs solutions positives pour ε>0 suffisamment petit ainsi que des propriétés de concentration connexes, en relation avec l'ensemble où le potentiel V atteint son minimum.
We consider a nonlinear Robin problem driven by a nonlinear, nonhomogeneous differential operator, and with a Carathéodory reaction term which is
-superlinear near
without satisfying the ...Ambrosetti–Rabinowitz condition and which does not have a standard subcritical polynomial growth. Using a combination of variational methods and Morse theoretic techniques, we prove a multiplicity theorem producing three nontrivial solutions (two of which have constant sign). In the process we establish some useful facts about the boundedness of the weak solutions of critical equations and the relation of Sobolev and Hölder local minimizers for functionals with a critical perturbation term.
We are concerned with the qualitative analysis of positive solutions to the fractional Choquard equation{(−Δ)su+a(x)u=(Iα⁎|u|2α,s⁎)|u|2α,s⁎−2u,x∈RN,u∈Ds,2(RN),u(x)>0,x∈RN, where Iα(x) is the Riesz ...potential, s∈(0,1), N>2s, 0<α<min{N,4s}, and 2α,s⁎=2N−αN−2s is the fractional critical Hardy-Littlewood-Sobolev exponent. We first establish a nonlocal global compactness property in the framework of fractional Choquard equations. In the second part of this paper, we prove that the equation has at least one positive solution in the case of small perturbations of the potential that describes the linear term.
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