Multiplicity results for nonlinear mixed boundary value problem

Boundary Value Problems20122012:134

DOI: 10.1186/1687-2770-2012-134

Received: 20 July 2012

Accepted: 26 October 2012

Published: 14 November 2012

Abstract

The aim of this paper is to establish multiplicity results of nontrivial and nonnegative solutions for mixed boundary value problems with the Sturm-Liouville equation. The approach is based on variational methods.

MSC: 34B15.

Keywords

boundary value problem mixed conditions

1 Introduction

The aim of this paper is to establish existence results of two and three nontrivial solutions for Sturm-Liouville problems with mixed conditions involving the ordinary p-Laplacian. We consider the following problem:
{ ( q | u | p 2 u ) + r | u | p 2 u = λ f ( x , u ) on  ] a , b [ , u ( a ) = u ( b ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ1_HTML.gif
(P)

with p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq1_HTML.gif, q , r L ( [ a , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq2_HTML.gif, with q 0 = ess inf [ a , b ] q > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq3_HTML.gif and r 0 = ess inf [ a , b ] r 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq4_HTML.gif. Here the nonlinearity f : [ a , b ] × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq5_HTML.gif is an L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq6_HTML.gif-Carathéodory function and λ is a real positive parameter.

The existence of at least one solution for problem (P) has been obtained in [1], where only a unique algebraic condition on the nonlinear term is assumed (see [[1], Theorem 1.1]). In the present paper, first we obtain the existence of two solutions by combining an algebraic condition on f of type contained in [1] with the classical Ambrosetti-Rabinowitz condition.

(AR): There exist ν > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq7_HTML.gif and R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq8_HTML.gif such that
0 < ν F ( x , t ) t f ( x , t ) , for all  | t | R  and for all  x [ a , b ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equa_HTML.gif

The role of (AR) is to ensure the boundness of the Palais-Smale sequences for the Euler-Lagrange functional associated to the problem. This is very crucial in the applications of critical point theory. Subsequently, an existence result of three positive solutions is obtained combining two algebraic conditions which guarantee the existence of two local minima for the Euler-Lagrange functional and applying the mountain pass theorem as given by Pucci and Serrin (see [2]) to ensure the existence of the third critical point.

Many mathematical models give rise to problems for which only nonnegative solutions make sense; therefore, many research articles on the theory of positive solutions have appeared. For a complete overview on this subject, we refer to the monograph [3].

In this paper, we also present, as a consequence of our main theorems, some results on the existence of nonnegative solutions for a particular problem of type
{ u = λ α ( x ) g ( u ) on  ] a , b [ , u ( a ) = u ( b ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ2_HTML.gif
(P1)

where α L 1 ( [ a , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq9_HTML.gif is such that α ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq10_HTML.gif a.e. x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq11_HTML.gif, α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq12_HTML.gif, and g : R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq13_HTML.gif is a nonnegative continuous function. In particular, we obtain for such a problem the existence of at least three nonnegative solutions by requiring that the function g has a superlinear behavior at zero, a sublinear behavior at infinity, and a particular growth in a suitable interval [ c , d ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq14_HTML.gif. By a similar approach, in [4], the authors obtain the existence of multiple solutions for a Neumann elliptic problem.

Multiplicity results for a mixed boundary value problem have been studied by several authors (see, for instance, [58] and references therein). In [5], the authors establish multiplicity results for problem (P), when p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq15_HTML.gif, and, in particular, they obtain the existence of three solutions, one of which can be trivial. On the contrary, our results (Theorems 3.5 and 3.6) guarantee the existence of three nonnegative and nontrivial solutions.

In [7], by using a fixed point theorem, the existence of at least three solutions for a mixed boundary problem with the equation ( | u | p 2 u ) = q ( x ) f ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq16_HTML.gif is obtained, by requiring, among other things, the boundness of f in a right neighborhood of zero (hypothesis (H6), Theorem 3.1), instead in our results (Theorems 3.5 and 3.6) the nonlinearity can blow up at zero.

Here, as an example, we present the following result which is a particular case of Theorem 3.6.

Theorem 1.1 Let g : R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq17_HTML.gif be a nonnegative continuous function such that
lim ξ 0 + g ( ξ ) ξ = + , lim ξ + g ( ξ ) ξ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ3_HTML.gif
(1.1)
and
0 1 g ( ξ ) d ξ < 1 16 0 2 g ( ξ ) d ξ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ4_HTML.gif
(1.2)
Then, for each λ ] 8 0 2 g ( ξ ) d ξ , 1 2 0 1 g ( ξ ) d ξ [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq18_HTML.gif, the problem
{ u = λ g ( u ) in  ] 0 , 1 [ , u ( 0 ) = u ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equb_HTML.gif

admits at least three classical nonnegative and nontrivial solutions.

2 Preliminaries and basic notations

Our main tools are Theorems 2.1 and 2.2, consequences of the existence result of a local minimum [[9], Theorem 3.1] which is inspired by the Ricceri variational principle (see [10]). For more information on this topic see, for instance, [11] and [12].

Given a set X and two functionals Φ , Ψ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq19_HTML.gif, put
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ5_HTML.gif
(2.1)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ6_HTML.gif
(2.2)
for all r 1 , r 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq20_HTML.gif, with r 1 < r 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq21_HTML.gif, and
ρ ( r ) = sup v Φ 1 ( ] r , + [ ) Ψ ( v ) sup u Φ 1 ( ] , r ] ) Ψ ( u ) Φ ( v ) r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ7_HTML.gif
(2.3)

for all r R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq22_HTML.gif.

Theorem 2.1 [[9], Theorem 5.1]

Let X be a reflexive real Banach space; Φ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq23_HTML.gif be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq24_HTML.gif; Ψ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq25_HTML.gif be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact. Put I λ = Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq26_HTML.gif and assume that there are r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq27_HTML.gif, r 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq28_HTML.gif, with r 1 < r 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq21_HTML.gif, such that
β ( r 1 , r 2 ) < ρ 2 ( r 1 , r 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ8_HTML.gif
(2.4)

where β and ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq29_HTML.gif are given by (2.1) and (2.2).

Then, for each λ ] 1 ρ 2 ( r 1 , r 2 ) , 1 β ( r 1 , r 2 ) [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq30_HTML.gif, there is u 0 , λ Φ 1 ( ] r 1 , r 2 [ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq31_HTML.gif such that I λ ( u 0 , λ ) I λ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq32_HTML.gif for all u Φ 1 ( ] r 1 , r 2 [ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq33_HTML.gif and I λ ( u 0 , λ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq34_HTML.gif.

Theorem 2.2 [[9], Theorem 5.3]

Let X be a real Banach space; Φ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq35_HTML.gif be a continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq24_HTML.gif; Ψ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq25_HTML.gif be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact. Fix inf X Φ < r < sup X Φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq36_HTML.gif and assume that
ρ ( r ) > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ9_HTML.gif
(2.5)

where ρ is given by (2.3), and for each λ > 1 ρ ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq37_HTML.gif, the function I λ = Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq38_HTML.gif is coercive.

Then, for each λ > 1 ρ ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq39_HTML.gif, there is u 0 , λ Φ 1 ( ] r , + [ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq40_HTML.gif such that I λ ( u 0 , λ ) I λ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq41_HTML.gif for all u Φ 1 ( ] r , + [ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq42_HTML.gif and I λ ( u 0 , λ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq43_HTML.gif.

Now, consider problem (P) and assume that q , r L ( [ a , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq44_HTML.gif with
ess inf [ a , b ] q > 0 and ess inf [ a , b ] r 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equc_HTML.gif
Denote by X = { u W 1 , p ( [ a , b ] ) : u ( a ) = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq45_HTML.gif endowed with the norm
u = ( a b q ( x ) | u ( x ) | p d x + a b r ( x ) | u ( x ) | p d x ) 1 p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equd_HTML.gif

Throughout the sequel, f : [ a , b ] × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq46_HTML.gif is an L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq6_HTML.gif-Carathéodory function. We recall that a function f : [ a , b ] × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq47_HTML.gif is said to be an L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq6_HTML.gif-Carathéodory function if x f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq48_HTML.gif is measurable for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq49_HTML.gif, t f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq50_HTML.gif is continuous for almost every x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq51_HTML.gif, and for all M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq52_HTML.gif, one has sup | t | M | f ( x , t ) | L 1 ( [ a , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq53_HTML.gif. Clearly, if f is continuous in [ a , b ] × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq54_HTML.gif, then it is L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq6_HTML.gif-Carathéodory.

Put
F ( x , ξ ) : = 0 ξ f ( x , t ) d t for all  ( x , ξ ) [ a , b ] × R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Eque_HTML.gif
Moreover, it is well known that ( X , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq55_HTML.gif is compactly embedded in ( C 0 ( [ a , b ] ) , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq56_HTML.gif and one has
u < ( b a q 0 ) p 1 p u for all  u X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ10_HTML.gif
(2.6)
We use the following notations:
q : = ess sup x [ a , b ] q ( x ) , r : = ess sup x [ a , b ] r ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equf_HTML.gif
In order to study problem (P), we introduce the functionals Φ , Ψ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq57_HTML.gif defined as follows:
Φ ( u ) = 1 p u p and Ψ ( u ) = a b F ( x , u ( x ) ) d x , u X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ11_HTML.gif
(2.7)
Clearly, the critical points of the functional Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq58_HTML.gif on X are weak solutions of problem (P). We recall that u : [ a , b ] R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq59_HTML.gif is a weak solution of problem (P) if u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq60_HTML.gif satisfies the following condition:
a b q ( x ) | u ( x ) | p 2 u ( x ) v ( x ) d x + a b r ( x ) | u ( x ) | p 2 u ( x ) v ( x ) d x = λ a b f ( x , u ( x ) ) v ( x ) d x , for all  v X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equg_HTML.gif

Clearly, if f is continuous, q C 1 ( [ a , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq61_HTML.gif, and r C 0 ( [ a , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq62_HTML.gif, the weak solutions for (P) are classical solutions.

3 Main results

In this section we present our main results.

Given two nonnegative constants c, d such that q ¯ c p r ¯ d p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq63_HTML.gif, where
q ¯ = 1 p ( q 0 b a ) p 1 and r ¯ = 1 p ( 2 b a ) p 1 [ q + p + 2 p + 1 ( b a 2 ) p r ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equh_HTML.gif
put
a d ( c ) : = a b max | ξ | c F ( x , ξ ) d x a + b 2 b F ( x , d ) d x q ¯ c p r ¯ d p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equi_HTML.gif
Theorem 3.1 Under the following conditions:
  1. (i)
    there exist three constants c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq64_HTML.gif, c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq65_HTML.gif, d, with
    ( q 0 p 2 2 p 1 ) 1 p c 1 < d < ( q 0 p 1 2 p 1 ) 1 p 1 [ q + p + 2 p + 1 ( b a 2 ) p r ] 1 p c 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ12_HTML.gif
    (3.1)
     
such that
a d ( c 2 ) < a d ( c 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equj_HTML.gif
and a a + b 2 F ( x , t ) d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq66_HTML.gif t [ 0 , d ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq67_HTML.gif;
  1. (ii)
    there exist ν > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq7_HTML.gif and R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq8_HTML.gif such that
    http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equk_HTML.gif
     

For each λ ] 1 a d ( c 1 ) , 1 a d ( c 2 ) [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq68_HTML.gif, problem (P) admits at least two nontrivial weak solutions u ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq69_HTML.gif, u ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq70_HTML.gif, with u ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq69_HTML.gif such that ( q 0 b a ) p 1 p c 1 < u ¯ 1 < ( q 0 b a ) p 1 p c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq71_HTML.gif.

Proof The proof of this theorem is divided into two steps. In the first part, by applying Theorem 2.1, we prove the existence of a local minimum for the functional Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq58_HTML.gif, where Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq72_HTML.gif and Ψ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq73_HTML.gif are functionals given in (2.7) for all u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq60_HTML.gif. Obviously, Φ and Ψ satisfy all regularity assumptions requested in Theorem 2.1, and the critical points in X of the functional Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq74_HTML.gif are exactly the weak solutions of problem (P). To this end, we verify condition (2.4) of Theorem 2.1.

Define the following function u 0 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq75_HTML.gif, by setting
u 0 ( x ) = { 2 d b a ( x a ) if  x [ a , a + b 2 [ , d if  x [ a + b 2 , b ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equl_HTML.gif
and estimate Ψ ( u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq76_HTML.gif and Φ ( u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq77_HTML.gif as follows:
Ψ ( u 0 ) a + b 2 b F ( x , d ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ13_HTML.gif
(3.2)
and
Φ ( u 0 ) 1 p ( 2 b a ) p 1 [ q + p + 2 p + 1 ( b a 2 ) p r ] d p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ14_HTML.gif
(3.3)

Fix c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq64_HTML.gif, d, c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq65_HTML.gif satisfying (3.1) and put r 1 = 1 p ( q 0 b a ) p 1 c 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq78_HTML.gif and r 2 = 1 p ( q 0 b a ) p 1 c 2 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq79_HTML.gif.

From (3.1), one has r 1 < Φ ( u 0 ) < r 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq80_HTML.gif.

Moreover, for all u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq60_HTML.gif such that u Φ 1 ( ] , r 2 [ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq81_HTML.gif, one has
Ψ ( u ) = a b F ( x , u ( x ) ) d x a b max | ξ | c 2 F ( x , ξ ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ15_HTML.gif
(3.4)
Hence,
sup u Φ 1 ( ] , r 2 [ ) Ψ ( u ) a b max | ξ | c 2 F ( x , ξ ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ16_HTML.gif
(3.5)
Now, arguing as before, we obtain
sup u Φ 1 ( ] , r 1 ] ) Ψ ( u ) a b max | ξ | c 1 F ( x , ξ ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ17_HTML.gif
(3.6)
From hypothesis (i) and bearing in mind (3.3), (3.2), (3.5), and (3.6), we obtain
β ( r 1 , r 2 ) < ρ 2 ( r 1 , r 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equm_HTML.gif
From Theorem 2.1, for each λ ] 1 a d ( c 1 ) , 1 a d ( c 2 ) [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq82_HTML.gif, Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq83_HTML.gif admits at least one critical point u ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq69_HTML.gif which is a local minimum such that
( q 0 b a ) p 1 p c 1 < u ¯ 1 < ( q 0 b a ) p 1 p c 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equn_HTML.gif

Now, we prove the existence of the second local minimum distinct from the first one. To this end, we must show that the functional Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq84_HTML.gif satisfies the hypotheses of the mountain pass theorem.

Clearly, the functional Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq58_HTML.gif is of class C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq85_HTML.gif and ( Φ λ Ψ ) ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq86_HTML.gif.

From the first part of the proof, we can assume that u ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq69_HTML.gif is a strict local minimum for Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq74_HTML.gif in X. Therefore, there is ρ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq87_HTML.gif such that inf u u 1 = ρ ( Φ λ Ψ ) ( u ) > ( Φ λ Ψ ) ( u ¯ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq88_HTML.gif, so condition [[13], ( I 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq89_HTML.gif), Theorem 2.2] is verified.

Now, choosing any u X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq90_HTML.gif, from (ii) one has
( Φ λ Ψ ) ( t u ) = 1 p t u p λ a b F ( x , t u ( x ) ) d x t p p u p λ t μ a 3 a b | u | μ + a 4 ( b a ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equo_HTML.gif

as t + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq91_HTML.gif, so condition [[13], ( I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq92_HTML.gif), Theorem 2.2] is verified. Moreover, by standard computations, Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq74_HTML.gif satisfies the Palais-Smale condition. Hence, the classical theorem of Ambrosetti and Rabinowitz ensures a critical point u ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq70_HTML.gif of Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq93_HTML.gif such that ( Φ λ Ψ ) ( u ¯ 2 ) > ( Φ λ Ψ ) ( u ¯ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq94_HTML.gif. So, u ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq69_HTML.gif and u ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq70_HTML.gif are two distinct weak solutions of (P) and the proof is complete. □

Remark 3.1 We observe that in literature the existence of at least one nontrivial solution for differential problems is obtained associating to the classical Ambrosetti-Rabinowitz condition a hypothesis on the nonlinear term of type f ( x , t ) = o ( | t | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq95_HTML.gif as t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq96_HTML.gif. This implies that the problem possesses also the trivial solution u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq97_HTML.gif. In Theorem 3.1, we find a nontrivial solution of the problem that actually is a proper local minimum of the Euler-Lagrange functional associated to the problem different from zero.

Now, we present an application of Theorem 2.2 which we will use to obtain multiple solutions.

Theorem 3.2 Assume that there exist two constants c ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq98_HTML.gif, d ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq99_HTML.gif, with ( q 0 p 2 2 p 1 ) 1 p c ¯ < d ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq100_HTML.gif, such that
a b max | ξ | c ¯ F ( x , ξ ) d x < a + b 2 b F ( x , d ¯ ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ18_HTML.gif
(3.7)
and
lim sup | ξ | + F ( x , ξ ) | ξ | p 0 uniformly in  X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ19_HTML.gif
(3.8)
Then, for each λ > λ ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq101_HTML.gif, where
λ ˜ = r ¯ d ¯ p q ¯ c ¯ p a + b 2 b F ( x , d ¯ ) d x a b max | ξ | c ¯ F ( x , ξ ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equp_HTML.gif

problem (P) admits at least one nontrivial weak solution u ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq102_HTML.gif such that u ˜ > c ¯ ( q 0 b a ) p 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq103_HTML.gif.

Proof The functionals Φ and Ψ satisfy all regularity assumptions requested in Theorem 2.2. Moreover, by standard computations, condition (3.8) implies that Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq74_HTML.gif, λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq104_HTML.gif, is coercive. So, our aim is to verify condition (2.5) of Theorem 2.2. To this end, put
r = c ¯ p p ( q 0 b a ) p 1 , and u 0 ( x ) = { 2 d ¯ b a ( x a ) if  x [ a , a + b 2 [ , d ¯ if  x [ a + b 2 , b ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equq_HTML.gif
Arguing as in the proof of Theorem 3.1, we obtain that
ρ ( r ) a + b 2 b F ( x , d ¯ ) d x a b max | ξ | c ¯ F ( x , ξ ) d x 1 p ( 2 b a ) p 1 d ¯ p [ q + p + 2 p + 1 ( b a 2 ) p r ] 1 p ( q 0 b a ) p 1 c ¯ p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equr_HTML.gif

So, from our assumption, it follows that ρ ( r ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq105_HTML.gif.

Hence, from Theorem 2.2 for each λ > λ ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq106_HTML.gif, the functional Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq74_HTML.gif admits at least one local minimum u ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq102_HTML.gif such that u ˜ > c ¯ ( q 0 b a ) p 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq103_HTML.gif and our conclusion is achieved. □

Remark 3.2 We point out that the same statement of above given result can be obtained by using a classical direct methods theorem (see [14]), but in addition we get the location of the solution, hence in particular the solution is nontrivial.

Now, we point out some results when the nonlinear term is with separable variables. To be precise, let

  • α L 1 ( [ a , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq9_HTML.gif such that α ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq10_HTML.gif a.e. x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq11_HTML.gif, α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq12_HTML.gif, and

  • g : R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq13_HTML.gif be a nonnegative continuous function,

consider the following boundary value problem:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equs_HTML.gif

We observe that the following results give the existence of multiple nonnegative solutions since the nonlinear term is supposed to be nonnegative. In order to justify what has been said above, we point out the following weak maximum principle.

Lemma 3.1 Suppose that u ¯ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq107_HTML.gif is a weak solution of problem (P1), then u ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq108_HTML.gif is nonnegative.

Proof We claim that a weak solution u ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq108_HTML.gif is nonnegative. In fact, arguing by a contradiction and setting A = { x [ a , b ] : u ¯ ( x ) < 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq109_HTML.gif, one has A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq110_HTML.gif. Put u ¯ = min { u ¯ , 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq111_HTML.gif, one has u ¯ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq112_HTML.gif (see, for instance, [[15], Lemma 7.6]). So, taking into account that u ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq113_HTML.gif is a weak solution and by choosing v = u ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq114_HTML.gif, one has
A | u ¯ ( x ) | u ¯ ( x ) d x = λ A α ( x ) g ( u ¯ ( x ) ) u ¯ ( x ) d x 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equt_HTML.gif

that is, u ¯ W 1 , 2 ( A ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq115_HTML.gif which is absurd. Hence, our claim is proved. □

Corollary 3.1 Assume that

(i′) there exist three nonnegative constants c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq64_HTML.gif, c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq65_HTML.gif, d, with c 1 < 2 d < c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq116_HTML.gif, such that
α L 1 ( [ a , b ] ) G ( c 2 ) α L 1 ( [ a + b 2 , b ] ) G ( d ) c 2 2 2 d 2 < α L 1 ( [ a + b 2 , b ] ) G ( d ) α L 1 ( [ a , b ] ) G ( c 1 ) 2 d 2 c 1 2 ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ20_HTML.gif
(3.9)
(ii′) there exist ν > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq117_HTML.gif and R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq118_HTML.gif such that
0 < ν G ( t ) t g ( t ) , for all  | t | R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equu_HTML.gif
Then, for each λ Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq119_HTML.gif, where
Λ = ] 2 d 2 c 1 2 2 ( b a ) [ α L 1 ( [ a + b 2 , b ] ) G ( d ) α L 1 ( [ a , b ] ) G ( c 1 ) ] , c 2 2 2 d 2 2 ( b a ) [ α L 1 ( [ a , b ] ) G ( c 2 ) α L 1 ( [ a + b 2 , b ] ) G ( d ) ] [ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equv_HTML.gif

problem (P1) admits at least two nonnegative weak solutions u ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq69_HTML.gif and u ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq70_HTML.gif such that 1 b a c 1 < u ¯ 1 < 1 b a c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq120_HTML.gif.

Theorem 3.3 Assume that there exist two positive constants c, d, with 2 d < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq121_HTML.gif, such that
G ( c ) c 2 < ( α L 1 ( [ a + b 2 , b ] ) 2 α L 1 ( [ a , b ] ) ) G ( d ) d 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ21_HTML.gif
(3.10)
Further, suppose that there exist ν > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq117_HTML.gif and R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq118_HTML.gif such that
0 < ν G ( t ) t g ( t ) , for all  | t | R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equw_HTML.gif

Then, for each λ ] 2 d 2 2 ( b a ) α L 1 ( [ a + b 2 , b ] ) G ( d ) , c 2 2 ( b a ) α L 1 ( [ a , b ] ) G ( c ) [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq122_HTML.gif, problem (P1) admits at least two nonnegative weak solutions.

Proof Our aim is to apply Corollary 3.1. To this end, we pick c 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq123_HTML.gif and c 2 = c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq124_HTML.gif. From (3.10), one has
α L 1 ( [ a , b ] ) G ( c 2 ) α L 1 ( [ a + b 2 , b ] ) G ( d ) c 2 2 2 d 2 < 1 c 2 [ c 2 α L 1 ( [ a , b ] ) G ( c ) 2 d 2 α L 1 ( [ a , b ] ) G ( c ) ] c 2 2 d 2 = α L 1 ( [ a , b ] ) G ( c ) c 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equx_HTML.gif
On the other hand, one has
α L 1 ( [ a + b 2 , b ] ) G ( d ) α L 1 ( [ a , b ] ) G ( c 1 ) 2 d 2 c 1 2 = α L 1 ( [ a + b 2 , b ] ) G ( d ) 2 d 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equy_HTML.gif

Hence, from Corollary 3.1 and taking (2.6) into account, the conclusion follows. □

A further consequence of Theorem 3.1 is the following result.

Theorem 3.4 Assume that
lim t 0 + g ( t ) t = + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ22_HTML.gif
(3.11)
and there are constants μ > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq125_HTML.gif and R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq118_HTML.gif such that, for all | ξ | R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq126_HTML.gif, one has
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equz_HTML.gif

Then, for each λ ] 0 , λ [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq127_HTML.gif, where λ = 1 2 ( b a ) α L 1 ( [ a , b ] ) sup c > 0 c 2 G ( c ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq128_HTML.gif, problem (P1) admits at least two nonnegative weak solutions.

Proof Fix λ ] 0 , λ [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq129_HTML.gif. Then there is c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq130_HTML.gif such that λ < 1 2 ( b a ) α L 1 ( [ a , b ] ) c 2 G ( c ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq131_HTML.gif. From (3.11), there is d < c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq132_HTML.gif such that 2 ( b a ) α L 1 ( [ a + b 2 , b ] ) G ( d ) 2 d 2 > 1 λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq133_HTML.gif. Hence, Theorem 3.3 ensures the conclusion. □

Next, as a consequence of Theorems 3.3 and 3.2, the following theorem of the existence of three solutions is obtained.

Theorem 3.5 Assume that
lim sup | ξ | + G ( ξ ) ξ 2 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ23_HTML.gif
(3.12)
Moreover, assume that there exist four positive constants c, d, c ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq98_HTML.gif, d ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq99_HTML.gif, with 2 d < c c ¯ < 2 d ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq134_HTML.gif, such that (3.10),
G ( c ¯ ) < ( α L 1 ( [ a + b 2 , b ] ) α L 1 ( [ a , b ] ) ) 1 2 G ( d ¯ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equaa_HTML.gif
and
G ( c ) c 2 < α L 1 ( [ a + b 2 , b ] ) G ( d ¯ ) α L 1 ( [ a , b ] ) G ( c ¯ ) 1 b a d ¯ 2 1 2 ( b a ) c ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ24_HTML.gif
(3.13)

are satisfied.

Then, for each λ Λ = ] max { λ ˜ , 2 d 2 2 ( b a ) α L 1 ( [ a + b 2 , b ] ) G ( d ) } , c 2 2 ( b a ) α L 1 ( [ a , b ] ) G ( c ) [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq135_HTML.gif, problem (P1) admits at least three weak nonnegative solutions.

Proof First, we observe that Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq136_HTML.gif owing to (3.13). Next, fix λ Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq119_HTML.gif. Theorem 3.3 ensures a nontrivial weak solution u ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq108_HTML.gif such that u ¯ < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq137_HTML.gif which is a local minimum for the associated functional Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq74_HTML.gif, as well as Theorem 3.2 guarantees a nontrivial weak solution u ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq102_HTML.gif such that u ˜ > c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq138_HTML.gif which is a local minimum for Φ λ Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq74_HTML.gif. Hence, the mountain pass theorem as given by Pucci and Serrin (see [2]) ensures the conclusion. □

Theorem 3.6 Assume that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ25_HTML.gif
(3.14)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ26_HTML.gif
(3.15)
Further, assume that there exist two positive constants c ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq139_HTML.gif, d ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq99_HTML.gif, with c ¯ < 2 d ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq140_HTML.gif, such that
G ( c ¯ ) c ¯ 2 < ( α L 1 ( [ a + b 2 , b ] ) 2 α L 1 ( [ a , b ] ) ) G ( d ¯ ) d ¯ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equ27_HTML.gif
(3.16)

Then, for each λ ] 2 d ¯ 2 2 ( b a ) α L 1 ( [ a + b 2 , b ] ) G ( d ¯ ) , c ¯ 2 2 ( b a ) α L 1 ( [ a , b ] ) G ( c ¯ ) [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq141_HTML.gif, problem (P1) admits at least three weak nonnegative solutions.

Proof Clearly, (3.15) implies (3.8). Moreover, by choosing d small enough and c = c ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq142_HTML.gif, simple computations show that (3.14) implies (3.10). Finally, from (3.16) we get (3.7) and also (3.13). Hence, Theorem 3.5 ensures the conclusion. □

Finally, we present two examples of problems that admit multiple solutions owing to Theorems 3.4 and 3.6.

Example 3.1 Owing to Theorem 3.4, for each λ ] 0 , 1 2 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq143_HTML.gif, the problem
{ u = λ ( u 4 + 1 ) in  ] 0 , 1 [ , u ( 0 ) = u ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equab_HTML.gif

admits at least two nonnegative solutions. In fact, one has lim u 0 + g ( u ) u = lim u 0 + u 4 + 1 u = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq144_HTML.gif and (AR) is satisfied as a simple computation shows. Moreover, one has λ = 1 2 ( b a ) α 1 sup c > 0 c 2 G ( c ) = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq145_HTML.gif.

Example 3.2 Consider the following problem:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equac_HTML.gif
It has three nonnegative solutions. In fact, let g : R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq17_HTML.gif be a function defined as
g ( u ) = { ( u 7 e u + 1 ) if  u 0 , 1 if  u < 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equad_HTML.gif
Owing to Theorem 3.6, the following problem
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equae_HTML.gif
admits three nonnegative classical solutions. In fact, one has
lim u 0 + g ( u ) u = lim u 0 + ( u 7 e u + 1 ) u = + and lim u + g ( u ) u = lim u + ( u 7 e u + 1 ) u = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_Equaf_HTML.gif

Moreover, taking into account that G ( u ) = u i = 0 7 7 ! i ! u i e u + 7 ! http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq146_HTML.gif, by choosing c ¯ = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq147_HTML.gif and d ¯ = 9 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq148_HTML.gif, one has G ( 2 ) 2 2 < 1 4 G ( 9 ) 9 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq149_HTML.gif and 2 9 2 G ( 9 ) < 1 10 < 2 2 2 G ( 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq150_HTML.gif.

So, it is clear that any nonnegative solution u of problem ( P g http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq151_HTML.gif) is also a solution of problem ( P E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-134/MediaObjects/13661_2012_Article_235_IEq152_HTML.gif).

Declarations

Authors’ Affiliations

(1)
Department of Civil, Computer, Construction, Environmental Engineering and Applied Mathematics, University of Messina

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