1,721,007 research outputs found
Recent developments in problems with nonstandard growth and nonuniform ellipticity
No full textWe provide an overview of recent results concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators. Regularity theory is at the center of this paper
Multiplicity of concentrating solutions for (p, q)-Schrödinger equations with lack of compactness
Full text linkWe study the multiplicity of concentrating solutions for the following class of (p, q)-Laplacian problems (Formula presented.) where ε > 0 is a small parameter, γ∈{0,1},1 infΛV for some bounded open set Λ ⊂ RN, and f: R → R is a continuous nonlinearity with subcritical growth. The main results are obtained by combining minimax theorems, penalization technique and Ljusternik–Schnirelmann category theory. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. As far as we know, all these results are new
A characterization for elliptic problems on fractal sets
No full textIn this paper we prove a characterization theorem on the existence of one non-zero strong solution for elliptic equations defined on the Sierpinski gasket. More generally, the validity of our result can be checked studying elliptic equations defined on self-similar fractal domains whose spectral dimension nu is an element of (0, 2). Our theorem can be viewed as an elliptic version on fractal domains of a recent contribution obtained in a recent work of Ricceri for a two-point boundary value problem
Fractional double-phase patterns: concentration and multiplicity of solutions
Full text linkWe 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), [Formula presented], (−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
Applications of Local Linking to Nonlocal Neumann Problems
No full textWe study a nonlocal Neumann problem driven by a nonhomogeneous elliptic differential operator. The reaction term is a nonlinearity function that exhibits p-superlinear growth but need not satisfy the Ambrosetti-Rabinowitz condition. By using an abstract linking theorem for smooth functionals, we prove a multiplicity result on the existence of weak solutions for such problems. An explicit example illustrates the main abstract result of this paper
Qualitative Analysis of Gradient-Type Systems with Oscillatory Nonlinearities on the Sierpinski Gasket
No full textUnder an appropriate oscillating behavior either at zero or at infinity of the nonlinear data, the existence of a sequence of weak solutions for parametric quasilinear systems of the gradient-type on the Sierpinski gasket is proved. Moreover, by adopting the same hypotheses on the potential and in presence of suitable small perturbations, the same conclusion is achieved. The approach is based on variational methods and on certain analytic and geometrical properties of the SierpiA"ski fractal as, for instance, a compact embedding result due to Fukushima and Shima
Positive homoclinic solutions for the discrete p-Laplacian with a coercive weight function
No full textWe study a p-Laplacian difference equation on the set of integers, involving a coercive weight function and a reaction term satisfying the Ambrosetti–Rabinowitz condition. By means of critical-point theory and a discrete maximum principle, we prove the existence of a positive homoclinic solution
Infinitely many solutions for a class of nonlinear eigenvalue problems in Orlicz-Sobolev spaces
No full textWe study a Neumann problem in a bounded Euclidean domain. The main result in this Note establishes that this problem has infinitely many solutions that converge to zero in the Orlicz-Sobolev space under suitable conditions on the nonlinear term
Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces
No full textIn this paper, we are interested in the existence of infinitely many weak solutions for a non-homogeneous eigenvalue Dirichlet problem. By using variational methods, in an appropriate Orlicz-Sobolev setting, we determine intervals of parameters such that our problem admits either a sequence of non-negative weak solutions strongly converging to zero provided that the non-linearity has a suitable behaviour at zero or an unbounded sequence of non-negative weak solutions if a similar behaviour occurs at infinity
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