A class of elliptic systemsinvolving N-functions

AbstractIn this work, we state a result of compactness due to Lions in Orlicz spaces. Wegive an application proving an existence result for a gradient type elliptic systems in ℝN involving N-functions

A population biological model with a singular nonlinearity

summary:We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$\begin {cases} -{\rm div}(|x|^{-\alpha p}|\nabla u|^{p-2}\nabla u)=|x|^{-(\alpha +1)p+\beta } \Big (a u^{p-1}-f(u)-\dfrac {c}{u^{\gamma }}\Big ), \quad x\in \Omega ,\\ u=0, \quad x\in \partial \Omega , \end {cases}$$ where $\Omega$ is a bounded smooth domain of ${\mathbb R}^N$ with $0\in \Omega$, $1<p<N$, $0\leq \alpha < {(N-p)}/{p}$, $\gamma \in (0,1)$, and $a$, $\beta$, $c$ and $\lambda$ are positive parameters. Here $f\colon [0,\infty )\to {\mathbb R}$ is a continuous function. This model arises in the studies of population biology of one species with $u$ representing the concentration of the species. We discuss the existence of a positive solution when $f$ satisfies certain additional conditions. We use the method of sub-supersolutions to establish our results

Comparison and Positive Solutions for Problems with the (p, q)-Laplacian and a Convection Term

International audienceThe aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the (p,q)-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bounds and regularity up to the boundary. The positivity of the solution is shown by applying a new comparison principle, which is established here

Semilinear Elliptic Equations With The Primitive Of The Nonlinearity Away From The Spectrum

[No abstract available]171212011219Ahmad, Multiple nontrivial solutions of resonant and nonresonant asymptotically linear problems (1986) Proceedings of the American Mathematical Society, 96, pp. 405-409Li, Liu, Nontrivial critical points for asymptotically quadratic function (1986) Preprint ICTP IC 86/390Amann, A note on degree theory for gradient mappings (1982) Proceedings of the American Mathematical Society, 85, pp. 591-595Amann, Zehnder, Nontrivial solutions for a class of nonresonant problems and applications to nonlinear differential equations (1980) Ann. Sci. Normale Sup. Pisa, 7, pp. 539-603Berestycki, Figueiredo, Double resonance in similinear elliptic problems (1981) Communications in Partial Differential Equations, 6 (1), pp. 91-120Cambini, Sul lemma di Morse (1973) Bol. U.M.I., 7, pp. 87-93Costa, Oliveira, Existence of solution for a class of semilinear elliptic problems at double resonance (1988) Bol. Soc. Bras. Mat., 19 (1), pp. 21-37Dolph, Nonlinear integral equations of the Hammerstein type (1949) Transactions of the American Mathematical Society, 60, pp. 289-307Figueiredo, Positive solutions of semilinear elliptic problems (1982) Lecture Notes in Mathematics, 957. , Springer, New York, (Differential Equations Proc. São Paulo)Figueiredo, Ekeland variational principle with applications and detours (1989) Tata Institute Lectures on Mathematics, , Springer, New YorkFigueiredo, Gossez, Conditions de non-résonance pour certains problèmes elliptiques semilinéaires (1986) C.r. Acad. Sci. Paris, 302, pp. 543-545Figueiredo, Lions, Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations (1982) J. Math. pures Appl., 61, pp. 41-63Figueiredo D. G. de & Massabó I., Semilinear elliptic equations with the primitive of the nonlinearity interacting with the first eigenvalue. J. Math. Analysis Applic. (to appear)Fonda, Gossez, Semicoercive variational problems at resonance: an abstract approach (1988) Louvain-la-Neuve Report No. 143Hirano, Existence of nontrivial solutions of semilinear elliptic equations (1989) Nonlinear Analysis, 13, pp. 695-705Hofer, A note on the topological degree at a critical point of mountain pass type (1984) Proceedings of the American Mathematical Society, 90, pp. 309-315Lazer, Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min–max type (1988) Nonlinear Analysis, 12, pp. 761-775Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations (1978) Nonlinear Analysis, pp. 161-177. , L. Cesari, R. Kannan, H. Weinberger, Academic Press, New Yor

Existence results for nonlinear elliptic equations with Leray–Lions operator and dependence on the gradient

International audienceThe existence of positive solutions for nonlinear elliptic problems under Dirichlet boundary condition is studied as well as the compactness and directness of the solution set. The main novelties consist in the presence of a Leray–Lions operator in the differential part and in the dependence of the reaction term on the gradient of the solution. Our approach relies on the method of sub-supersolution, nonlinear regularity theory and strong maximum principle

Elliptic Equations In R2 With Nonlinearities In The Critical Growth Range

[No abstract available]3213915