A bifurcation-type theorem for the positive solutions of a nonlinear Neumann problem with concave and convex terms

We consider a nonlinear elliptic Neumann problem driven by the p-Laplacian with a reaction that involves the combined effects of a “concave” and of a “convex” terms. The convex term (p-superlinear term) need not satisfy the Ambrosetti-Rabinowitz condition. Employing variational methods based on the critical point theory together with truncation techniques, we prove a bifurcation type theorem for the equation

Nonlinear elliptic equations with nonsmooth potential : variational and topological methods

Doutoramento em MatemáticaNesta tese de doutoramento, estudamos a existência e a multiplicidade de soluções para algumas classes de equações elípticas não lineares com potencial não suave. Os resultados originais foram obtidos, utilizando métodos variacionais e da teoria de grau. A nossa abordagem variacional é baseada em descobertas recentes na teoria não suave (nonsmooth) dos pontos críticos. A teoria de grau é aplicada a determinadas perturbações multívocas de operadores de tipo monótono (operadores do tipo (S)+ ). O primeiro problema que consideramos é um problema de valor próprio semi-linear com potencial não suave (ver Capítulo 3). O resultado de existência obtido estende para uma versão não suave, e sob hipóteses de crescimento mais fracas, um resultado obtido por Rabinowitz para potenciais suaves. Mais, sob condições no potencial que permitem ressonância, quer em zero, quer no infinito, provamos um resultado de multiplicidade. Para um problema elíptico não linear derivado do p-Laplaciano e com um potencial não suave (ver Capítulo 4), estabelecemos a existência de, pelo menos, três soluções suaves, não triviais e distintas, sendo duas delas de sinal constante (uma positiva e uma negativa). Problemas semi-lineares de Neumann, que são duplamente ressonantes na origem, relativamente a qualquer intervalo espectral [λk,λk+1], são estudados no Capítulo 5. O resultado de multiplicidade obtido para um potencial não suave estende resultados existentes para o caso do potencial suave, nos quais a ressonância é completa relativamente a λk, mas incompleta relativamente a λk+1. Respondemos afirmativamente à questão aberta em relação à validade do resultado de multiplicidade, quando ocorre, também, ressonância completa relativamente a λk+1 (situação de dupla ressonância). A última parte da tese (Capítulo 6) é dedicada ao estudo de uma classe de problemas de Neumann, em que o operador diferencial não é homogéneo, nem variacional. Portanto, os métodos mini-max da teoria dos pontos críticos (suave e não-suave) não podem ser utilizados. Usando o espectro do operador diferencial assimptótico, juntamente com métodos da teoria de grau, estabelecemos a existência de soluções suaves não triviais.In this Ph.D. thesis, we study the existence and the multiplicity of solutions to some classes of nonlinear elliptic equations with a nonsmooth potential. Our new results were obtained by using variational and degree theoretic methods. The variational approach we used is based on recent developments in nonsmooth critical point theory. The degree theory we used concerns certain multivalued perturbations of a class of monotone type operators (the (S)+ type operators). The first problem we consider is a semilinear eigenvalue problem with a nonsmooth potential (see Chapter 3). The existence result we obtained extends to nonsmooth setting and under weaker growth assumptions, a result obtained by Rabinowitz for smooth potentials. Moreover, under conditions on the potential which allow resonance both at zero and at infinity, we prove a multiplicity result. For a nonlinear elliptic problem driven by the p-Laplacian and with a nonsmooth potential (see Chapter 4), we establish the existence of at least three distinct nontrivial smooth solutions, two of them with constant sign (one positive and one negative). Semilinear Neumann problems which are doubly resonant at the origin with respect to any spectral interval [λk,λk+1] were studied in Chapter 5. The multiplicity result we obtained for nonsmooth potential, extend results known for the case of smooth potential, where the resonance is complete with respect to λk, but incomplete (nonuniform nonresonance) with respect to λk+1. We give a positive answer to an open question asking whether the multiplicity result also holds when complete resonance occurs also with respect to λk+1 (double resonance situation). The last part of the thesis (Chapter 6) is devoted to the study of a class of Neumann problems where the differential operator driving the problem is neither homogeneous, nor variational. So the minimax methods of critical point theory (smooth and nonsmooth alike) fail. Using the spectrum of the asymptotic differential operator together with degree theoretic methods, we establish the existence of nontrivial smooth solutions

On a Dirichlet problem with (p,q)(p,q)-Laplacian and parametric concave-convex nonlinearity

A homogeneous Dirichlet problem with $(p,q)$-Laplace differential operator and reaction given by a parametric $p$-convex term plus a $q$-concave one is investigated. A bifurcation-type result, describing changes in the set of positive solutions as the parameter $\lambda>0$ varies, is proven. Since for every admissible $\lambda$ the problem has a smallest positive solution $\bar u_{\lambda}$, both monotonicity and continuity of the map $\lambda \mapsto \bar u_{\lambda}$ are studied.Comment: 12 pages, comments are welcom

Nonlinear Dirichlet problems with unilateral growth on the reaction

We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, $p=2$), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multiplicity results existing for a class of parametric, coercive Dirichlet problems

Asymmetric (p, 2)-equations with double resonance

We consider a nonlinear Dirichlet elliptic problem driven by the sum of a p-Laplacian and a Laplacian [a (p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti-Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance situation. Using variational methods based on the critical point theory and Morse theory (critical groups), we establish the existence of at least three nontrivial smooth solutions

Minimax methods for finding multiple saddle critical points in Banach spaces and their applications

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study