Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities

In this paper we are concerned with a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using a completely variational method, we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. In order to prove this fact, we show that for every positive integer k, when a suitable parameter is large enough, there exists a solution which presents k peaks. Under the assumptions we consider in this paper, new (unexpected) phenomena are observed in the study of this problem and new methods are required to construct the k-peaks solutions and describe their asymptotic behavior (weak limits of the rescaled solutions, localization of the concentration points of the peaks, asymptotic profile of the rescaled peaks, etc.).     

A global compactness result for an elliptic equation with double singular terms

In this paper, we establish a global compactness result for (P.S.) sequences of the variational functional of the elliptic problem $\left\{\begin{array}{cc}-\Delta u-\frac{\mu }{|x{|}^2}u=\frac{1}{|x{|}^s}|u{|}^{2_s^1-2}u+\lambda u,\hfill & x\in \Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$where $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, is a bounded smooth domain with $0\in \Omega$, $\mu \in [0,{(n-2)}^2∕4)$, $s\in [0,2)$ and $\lambda \in \mathbb{R}$ are constants. This extends the global compactness result of Cao and Peng (2003) to the case of elliptic problems with double singular critical terms. Our arguments adapt some refined Sobolev inequalities systematically developed quite recently by Palatucci and Pisante (2014) and blow-up analysis. In this way, our arguments turn out to be quite transparent and easy to be applied to many other problems.     

Multiple solutions of semilinear elliptic equations with one-sided growth conditions

Multiplicity results for semilinear elliptic equations are obtained under one-sided growth conditions on the nonlinearity. Techniques of nonsmooth critical point theory are employed.     

Two solutions for a singular elliptic equation with critical growth at infinity

We look for positive solutions for the singular equation $-\Delta u-\frac{1}{2}\left(x\cdot \nabla u\right)=\mu h\left(x\right)u^{q-1}+\lambda u+u^{(N+2)/(N-2)},\text{in }{\mathbb{R}}^N,$ where $N\ge 3$, $\lambda >0$, $\mu >0$ is a parameter, $0<q<1$ and $h$ has some summability properties. By using a perturbation method and critical point theory, we obtain two solutions when $max\{1,N/4\}<\lambda <N/2$ and the parameter $\mu >0$ is small.     

Multiple solutions for semilinear elliptic equations with Neumann boundary condition and jumping nonlinearities

We obtain nonconstant solutions of semilinear elliptic Neumann boundary value problems with jumping nonlinearities when the asymptotic limits of the nonlinearity fall in the type (Il), l>2 and (IIl), l?1 regions formed by the curves of the Fucik spectrum. Furthermore, we have at least two nonconstant solutions in every order interval under resonance case. In this paper, we apply the sub-sup solution method, Fucik spectrum, mountain pass theorem in order intervals, degree theory and Morse theory to get the conclusions.     

On a variational problem for radial solutions to extremal elliptic equations

On the ball |x| ≤ 1 of R ^m , m ≥ 2, a radial variational problem, related to a priori estimates for solutions to extremal elliptic equations with fixed ellipticity constant α is investigated. Such a problem has been studied and solved [see Manselli Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] in L ^p spaces, with p ≤ m. In this paper, we assume p > m and we prove the existence of a positive number α ₀ = α ₀(p,m) such that if there exists a smooth function maximizing the problem, whose representation is explicitly determined as in Manselli [Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] This fact is no longer true if 0 < α < α ₀.      

Multiplicity and asymptotic behavior of solutions for quasilinear elliptic equations with small perturbations

This paper considers the following general form of quasilinear elliptic equation with a small perturbation:$\{\begin{array}{c}?{\sum }_{i,j=1}^N{D}_j(a_{ij}(x,u)D_{i}u)+\frac{1}{2}{\sum }_{i,j=1}^N{D}_t{a}_{ij}(x,u)D_{i}u{D}_{j}u\hfill \\ =f(x,u)+\epsilon g(x,u),x\in \mathrm{\Omega },u\in H_0^1(\mathrm{\Omega }),\hfill \end{array}$ where $\mathrm{\Omega }?{\mathbb{R}}^N(N\ge 3)$ is a bounded domain with smooth boundary and $|\epsilon |$ small enough. We assume the main term in the equation to have a mountain pass structure but do not suppose any conditions for the perturbation term $\epsilon g(x,u)$. Then we prove the equation possesses a positive solution, a negative solution and a sign-changing solution. Moreover, we are able to obtain the asymptotic behavior of these solutions as $\epsilon \to 0$.     

Problems for elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order nonlinear elliptic partial differential equations with singular data. In particular, we study the asymptotic behaviour of the solution near the boundary up to the second order under various assumptions on the growth of the coefficients of the equation.     

A variational approach to singular quasilinear elliptic problems with sign changing nonlinearities

We obtain positive solutions of singular p-Laplacian problems with sign changing nonlinearities using variational methods.     

Nonlinear singular elliptic problem with gradient term and general datum

In this paper we deal with an elliptic problem, with a first order term in natural growth, a singular nonlinearity and a L¹$L^1$ datum. We obtain an existence result without any smallness condition on the datum. We prove also a multiplicity result of positive solutions under additional conditions on the data.     

Gradient estimates for viscosity solutions of singular fully nonlinear elliptic equations

We establish Lipschitz regularity for solutions to a family of non-isotropic fully nonlinear partial differential equations of elliptic type. In general such a regularity is optimal. No sign constraint is imposed on the solution, thus limiting free boundaries may have two-phases. Our estimates are then employed in combination with fine regularizing techniques to prove existence of viscosity solutions to singular nonlinear PDEs.     

Problems for elliptic singular equations with a quadratic gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order elliptic partial differential equations with a quadratic gradient term and singular data. In particular, we study the asymptotic behaviour of the solution near the boundary under suitable assumptions on the growth of the coefficients of the equation.     

Existence and multiplicity results for quasilinear elliptic differential systems

We use a nonsmooth critical point theory to prove existence results for a variational system of quasilinear elliptic equations in both the sublinear and superlinear cases. We extend a technique of Bartsch to obtain multiplicity results when the system is invariant under the action of a compact Lie group. The problem is rather different from its scalar version, because a suitable condition on the coefficients of the system seems to be necessary in order to prove the convergence of the Palais-Smale sequences. Such condition is in some sense a restriction to the "distance" between the quasilinear operator and a semilinear one.     

Minimax systems

The variational approach to solving nonlinear problems eventually leads to the search for critical points of related functionals. In case of semibounded functionals, one can look for extrema. Otherwise, one is forced to use minimax methods. There are several approaches to such methods. In this paper we unify these approaches providing one theory that works for all of them. The usual approach has used Palais-Smale sequences. We show that all of them lead to Cerami sequences as well. Applications are given.     

Boundary behavior of solutions to some singular elliptic boundary value problems

Let Ω be a bounded domain with smooth boundary in . For the more general weight b, some nonlinearities f and singularities g, by two kinds of nonlinear transformations, a new perturbation method, which was introduced by García Melián in [J. García Melián, Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal. 67 (2007) 818–826], and comparison principles, we show that the boundary behavior of solutions to a boundary blow-up elliptic problem Δw=b(x)f(w),w>0,xΩ,w|_∂Ω=∞ and a singular Dirichlet problem −Δu=b(x)g(u),u>0,xΩ,u|_∂Ω=0 has the same form under the nonlinear transformations, which can be determined in terms of the inverses of the transformations.     

On a perturbed non-local boundary value problem: a multiplicity result

In the present paper, we deal with a non-local boundary value problem with an exponential non-linearity. The existence of one, two or three solutions is investigated under the presence of a suitable perturbation. Our approach is variational and combines results from critical point theory.