SUPERLINEAR ELLIPTIC PROBLEMS WITH SIGN CHANGING COEFFICIENTS

Via variational methods, we study multiplicity of solutions for the problem {-Delta u = lambda b(x)vertical bar u vertical bar(q-2)u + au + g(x, u) in Omega, u - 0 on partial derivative Omega, where a simple example for g( x, u) is |u|(p-2)u; here a, lambda are real parameters, 1 < q < 2 < p <= 2* and b(x) is a function in a suitable space L-sigma. We obtain a class of sign changing coefficients b(x) for which two non-negative solutions exist for any lambda > 0, and a total of five nontrivial solutions are obtained when lambda is small and a >= lambda(1). Note that this type of results are valid even in the critical case.

Existence and concentration of solutions for a class of biharmonic equations

Some superlinear fourth order elliptic equations are considered. A family of solutions is proved to exist and to concentrate at a point in the limit. The proof relies on variational methods and makes use of a weak version of the Ambrosetti-Rabinowitz condition. The existence and concentration of solutions are related to a suitable truncated equation. (C) 2012 Elsevier Inc. All rights reserved.

MULTIPLICITY OF NONRADIAL SOLUTIONS FOR A CLASS OF QUASILINEAR EQUATIONS ON ANNULUS WITH EXPONENTIAL CRITICAL GROWTH

In this paper, we establish the existence of many rotationally non-equivalent and nonradial solutions for the following class of quasilinear problems (p) {-Delta(N)u = lambda f(vertical bar x vertical bar, u) x is an element of Omega(r), u > 0 x is an element of Omega(r), u = 0 x is an element of Omega(r), where Omega(r) = {x is an element of R-N : r < vertical bar x vertical bar < r + 1}, N >= 2, N not equal 3, r >0, lambda > 0, Delta(N)u = div(vertical bar del u vertical bar(N-2)del u) is the N-Laplacian operator and f is a continuous function with exponential critical growth.

A nonlinear elliptic problem with terms concentrating in the boundary

In this paper, we investigate the behavior of a family of steady-state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a e-neighborhood of a portion G of the boundary. We assume that this e-neighborhood shrinks to G as the small parameter e goes to zero. Also, we suppose the upper boundary of this e-strip presents a highly oscillatory behavior. Our main goal here was to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on G, which depends on the oscillating neighborhood. Copyright (C) 2012 John Wiley & Sons, Ltd.

MULTI-BUMP SOLUTIONS FOR A CLASS OF QUASILINEAR EQUATIONS ON R

This paper is concerned with the existence of multi-bump solutions to a class of quasilinear Schrodinger equations in R. The proof relies on variational methods and combines some arguments given by del Pino and Felmer, Ding and Tanaka, and Sere.