Existence Of Nontrivial Nonnegative Solutions Research Articles

Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N

Abstract In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.

Solutions for nonhomogeneous fractional (p,q)-Laplacian systems with critical nonlinearities

AbstractIn this article, we aimed to study a class of nonhomogeneous fractional (p,q)-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities inRN{{\mathbb{R}}}^{N}. By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities in the second part of this article. More precisely, with the help of Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions for the related systems based on the same approach. Finally, we obtain the corresponding existence results for the fractional (p,q)-Laplacian systems in the case ofN=sp=lqN=sp=lq. It is worth pointing out that using fixed point argument to seek solutions for a class of nonhomogeneous fractional (p,q)-Laplacian systems is the main novelty of this article.

Existence of fixed points in conical shells of a Banach space for sum of two operators and application in ODEs

In this work a new functional expansion-compression fixed point theorem of Leggett--Williams type is developed for a class of mappings of the form $T+F,$ where $(I-T)$ is Lipschitz invertible map and $F$ is a $k$-set contraction. The arguments are based upon recent fixed point index theory in cones of Banach spaces for this class of mappings. As application, our approach is applied to prove the existence of nontrivial nonnegative solutions for three-point boundary value problem.

Positive solutions for second-order differential equations o Kirchhoff type on the half-line

The aim of the present paper is to study the existence of nontrivial nonnegative solutions for a second-order boundary value problem of Kirchhoff type on the half-line. Our approach is based on variational methods, a monotonicity trick related to the mountain pass lemma, cut-off functional technique, and a Pohozaev type identity.

Classical solutions for a class of nonlinear wave equations

We study a class of initial value problems subject to nonlinear partial differential equations of hyperbolic type. A new topological approach is applied to prove the existence of nontrivial nonnegative solutions. More precisely, we propose a new integral representation of the solutions for the considered initial value problems and using this integral representation we establish existence of classical solutions for the considered classes of nonlinear wave equations.

Existence of solution for a quasilinear elliptic Neumann problem involving multiple critical exponents

In this paper, we study the Neumann boundary value problem to a quasilinear elliptic equation with the critical Sobolev exponent and critical Hardy–Sobolev exponent, and prove the existence of nontrivial nonnegative solution by means of variational method.

Schrödinger–Hardy systems involving two Laplacian operators in the Heisenberg group

In this paper we first prove existence of nontrivial nonnegative solutions of a Schrödinger–Hardy system in the Heisenberg group, driven by two possibly different Laplacian operators. The main originality of the paper is to work in the Heisenberg group. In fact several new theorems have to be proved in order to overcome the difficulties arising in the new framework, also due to the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Finally, we discuss and prove existence even for systems in the Heisenberg group, including critical nonlinear terms.

Existence of entire solutions for Schrödinger–Hardy systems involving two fractional operators

This paper deals with the existence of nontrivial nonnegative solutions of Schrödinger–Hardy systems driven by two possibly different fractional ℘-Laplacian operators, via various variational methods. The main features of the paper are the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Moreover, we consider systems including critical nonlinear terms, as treated very recently in literature, and present radial versions of the main theorems. Finally, we briefly show how to extend the previous results when the fractional Laplacian operators are replaced by more general elliptic nonlocal integro–differential operators.

Semiclassical states for weakly coupled fractional Schrödinger systems

In this paper, we consider the nonlinear system of two weakly coupled fractional Schrödinger equations with nonconstant potentials{ε2s(−Δ)su+P1(x)u=(|u|2p+b|u|p−1|v|p+1)u,x∈RN,ε2s(−Δ)sv+P2(x)v=(|v|2p+b|v|p−1|u|p+1)v,x∈RN. We investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials.

Existence of multiple solutions for modified Schrödinger–Kirchhoff–Poisson type systems via perturbation method with sign-changing potential

In this paper, we prove the existence of positive solutions and negative solutions for the following modified Schrödinger–Kirchhoff–Poisson type systems {−(a+b∫R3∣∇u∣2)Δu+V(x)u+ϕu−12uΔ(u2)=f(x,u),inR3,−Δϕ=u2,inR3, where a>0, b≥0, V and f are continuous functions and V(x) is allowed to be sign-changing. Under some certain assumptions on V and f, we prove the existence of nontrivial nonnegative solutions, nontrivial nonpositive solutions and sequence of high energy solutions via the perturbation method.

Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$

In this paper, we apply the sharp Adams-type inequalities for the Sobolev space $W^{m,\frac{n}{m}}\left( \mathbb{R} ^{n}\right) $ for any positive real number $m$ less than $n$, established by Ruf and Sani [46] and Lam and Lu [30,31], to study polyharmonic equations in $\mathbb{R}^{2m}$. We will consider the polyharmonic equations in $\mathbb{R}^{2m}$ of the form \[ \left( I-\Delta\right) ^{m}u=f(x,u)\text{ in }% \mathbb{R} ^{2m}. \] We study the existence of the nontrivial solutions when the nonlinear terms have the critical exponential growth in the sense of Adams' inequalities on the entire Euclidean space. Our approach is variational methods such as the Mountain Pass Theorem ([5]) without Palais-Smale condition combining with a version of a result due to Lions ([39,40]) for the critical growth case. Moreover, using the regularity lifting by contracting operators and regularity lifting by combinations of contracting and shrinking operators developed in [14] and [11], we will prove that our solutions are uniformly bounded and Lipschitz continuous. Finally, using the moving plane method of Gidas, Ni and Nirenberg [22,23] in integral form developed by Chen, Li and Ou [12] together with the Hardy-Littlewood-Sobolev type inequality instead of the maximum principle, we prove our positive solutions are radially symmetric and monotone decreasing about some point. This appears to be the first work concerning existence of nontrivial nonnegative solutions of the Bessel type polyharmonic equation with exponential growth of the nonlinearity in the whole Euclidean space.

Singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities

We consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities. The dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect. We investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function. Necessary conditions to locate the possible concentration points are also given.

Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

In this paper, we prove the existence of nontrivial nonnegative solutions to a class of elliptic equations and systems which do not satisfy the Ambrosetti–Rabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth at ∞. The known results without the AR condition in the literature only involve nonlinear terms of polynomial growth. We will use suitable versions of the Mountain Pass Theorem and Linking Theorem introduced by Cerami (Istit. Lombardo Accad. Sci. Lett. Rend. A, 112(2):332–336, 1978 Ann. Mat. Pura Appl., 124:161–179, 1980). The Moser–Trudinger inequality plays an important role in establishing our results. Our theorems extend the results of de Figueiredo, Miyagaki, and Ruf (Calc. Var. Partial Differ. Equ., 3(2):139–153, 1995) and of de Figueiredo, do O, and Ruf (Indiana Univ. Math. J., 53(4):1037–1054, 2004) to the case where the nonlinear term does not satisfy the AR condition. Examples of such nonlinear terms are given in Appendix A. Thus, we have established the existence of nontrivial nonnegative solutions for a wider class of nonlinear terms.

On Variational Inequalities Driven by Elliptic Operators Not in Divergence Form

Abstract In this paper we study semilinear variational inequalities driven by an elliptic operator not in divergence form modeled by where Ω is a bounded domain of RN, N ≥ 3, with smooth boundary, A is the elliptic operator, not in divergence form, given by Here aij, ai , i,j = 1,...,N , and a0 satisfy suitable regularity conditions, while 1 < s < 4/(N − 2) and the obstacle ψ is a function sufficiently smooth. Even if this problem is not variational in nature, we will prove the existence of non-trivial non-negative solutions for it, performing a variational approach combined with a penalization technique. This kind of approach seems to be new for problems of this type. We also prove a C1,α-regularity result for the solutions of our problem.

On Radial Solutions of the Schrödinger Type Equation

Abstract We establish compact embeddings of the radial Sobolev space Hrad 1,p(ℝN) into weighted Lebesgue spaces Lq w (ℝN) under various assumptions on the weight function w. We use these, along with a variety of variational techniques, to prove the existence of nontrivial nonnegative solutions for a class of nonlinear Schrödinger type equations.

Time periodic solutions of porous medium equation

In this article, we study the time periodic solutions to the following porous medium equation under the homogeneous Dirichlet boundary condition: The existence of nontrivial nonnegative solution is established provided that 0≤α<m. The existence is also proved in the case α=m but with an additional assumption \documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document} $\mathop{\min}\nolimits_{\overline{\Omega}\times[0,T]}a(x,t){>}{\lambda}_1$\end{document}, where λ1 is the first eigenvalue of the operator −Δ under the homogeneous Dirichlet boundary condition. We also show that the support of these solutions is independent of time by providing a priori estimates for their upper bounds using Moser iteration. Further, we establish the attractivity of maximal periodic solution using the monotonicity method. Copyright © 2010 John Wiley & Sons, Ltd.

Semiclassical states for weakly coupled nonlinear Schrödinger systems

We consider systems of weakly coupled Schrodinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Time periodic solutions of a class of degenerate parabolic equations

This paper is devoted to the time periodic solutions to the degenerate parabolic equations of the form $$\frac{{\partial u}}{{\partial t}} = \vartriangle u^m + u^p (a(x,t) - b(x,t)u) in \Omega \times R$$ under the Dirichlet boundary value condition, wherem>1,p≥0, Ω∈R N is a bounded domain with smooth boundary σΩ anda,b are positive, smooth functions which are periodic int with period ω>0. The existence of nontrivial nonnegative solutions is established provided that 0≤p<m. The existence is also proved in the casep=m but with an additional assumption $$\begin{array}{*{20}c} {\min } \\ {\bar Q} \\ \end{array} a(x,t) > \lambda _1 $$ , where $$\lambda _1 $$ is the first eigenvalue of the operator −Δ under the homogeneous Dirichlet boundary condition.